Complex Functions and Some of Their Functional Transforms

**Chapter 2**

s-Plane

**Abstract**

s-plane.

**11**

**1. Introduction**

Interference Pattern

*Martín Javier Martínez Silva*

Representation on the Complex

*Mario Alberto García Ramírez, Gustavo Adolfo Vega Gómez,*

In this work, the normalized interference pattern produced by a coherence interferometer system was represented as a complex function. The Laplace transform was applied for the transformation. Poles and zeros were determined from this complex function, and then, its pole-zero map and its Bode diagram were proposed. Both graphical representations were implemented numerically. From our numerical results, pole location and zero location depend on the optical path difference (OPD), while the Bode diagram gives us information about the OPD parameter. Based on the results obtained from the graphical representations, the coherence interferometer systems, the low-coherence interferometer systems, the interferometric sensing systems, and the fiber optic sensors can be analyze on the complex

**Keywords:** coherence interferometer system, Laplace transform, complex

Many coherence interferometers systems find practical applications for the physical parameter measurement, such as are temperature, strain, humidity, pressure, level, current, voltage, and vibration [1–10]. Physical implementation and signal demodulation are very important for the good measurement. Many implementations are based on the Bragg gratings, fiber optics, vacuum, mirrors, crystals, polarizer, and their combinations [11–15]; whereas in the signal demodulation, has been applied commonly the Fourier transform [16–20]. This transform permits us to know all frequency components of any interference pattern, doing

The Laplace transform has many practical applications in topics such as control

systems, electronic circuit analysis, mechanic systems, electric circuit system, pure mathematics, and communications. The linear transformation permits us to

function, pole-zero map, Bode diagram, graphical representations

possible the signal demodulation for the interferometer systems.

*José Trinidad Guillen Bonilla, Alex Guillen Bonilla,*

*Héctor Guillen Bonilla, María Susana Ruiz Palacio,*

*and Verónica María Bettancourt Rodriguez*

#### **Chapter 2**

## Interference Pattern Representation on the Complex s-Plane

*José Trinidad Guillen Bonilla, Alex Guillen Bonilla, Mario Alberto García Ramírez, Gustavo Adolfo Vega Gómez, Héctor Guillen Bonilla, María Susana Ruiz Palacio, Martín Javier Martínez Silva and Verónica María Bettancourt Rodriguez*

#### **Abstract**

In this work, the normalized interference pattern produced by a coherence interferometer system was represented as a complex function. The Laplace transform was applied for the transformation. Poles and zeros were determined from this complex function, and then, its pole-zero map and its Bode diagram were proposed. Both graphical representations were implemented numerically. From our numerical results, pole location and zero location depend on the optical path difference (OPD), while the Bode diagram gives us information about the OPD parameter. Based on the results obtained from the graphical representations, the coherence interferometer systems, the low-coherence interferometer systems, the interferometric sensing systems, and the fiber optic sensors can be analyze on the complex s-plane.

**Keywords:** coherence interferometer system, Laplace transform, complex function, pole-zero map, Bode diagram, graphical representations

#### **1. Introduction**

Many coherence interferometers systems find practical applications for the physical parameter measurement, such as are temperature, strain, humidity, pressure, level, current, voltage, and vibration [1–10]. Physical implementation and signal demodulation are very important for the good measurement. Many implementations are based on the Bragg gratings, fiber optics, vacuum, mirrors, crystals, polarizer, and their combinations [11–15]; whereas in the signal demodulation, has been applied commonly the Fourier transform [16–20]. This transform permits us to know all frequency components of any interference pattern, doing possible the signal demodulation for the interferometer systems.

The Laplace transform has many practical applications in topics such as control systems, electronic circuit analysis, mechanic systems, electric circuit system, pure mathematics, and communications. The linear transformation permits us to

transform any time function into a complex function whose variable is *s* ¼ *iω* þ *σ*, where *i* is the complex operator, *ω* is the angular frequency, and *σ* is a real value. The complex function can represent in the complex s-plane, where their axes represent the real and imaginary parts of the complex variable *s*. This complex plane does feasible the study of dynamic systems, and some applications are the tuning closed-loop, stability, mathematical methods, fault detection, optimization, and filter design [21–23]. In addition, the s-plane permits graphical methods such as pole-zero map, Bode diagrams, root locus, polar plots, gain margin and phase margin, Nichols charts, and N circles [24].

two ways and its difference produces the optical path difference. The first one will

*<sup>i</sup> <sup>ω</sup>t*þ*φ<sup>R</sup>* ð Þ*:* (1)

*<sup>i</sup>*ð Þ *<sup>ω</sup>t*þ*φ<sup>D</sup> :* (2)

*φ<sup>R</sup>* ¼ 2*kxR*, (3)

*φ<sup>D</sup>* ¼ 2*kxD:* (4)

�*<sup>i</sup> <sup>ω</sup>t*þ*φ<sup>R</sup>* ð Þ <sup>þ</sup> *ADe*

*<sup>i</sup>*ð Þ *<sup>ω</sup>t*þ*φ<sup>D</sup> e* �*<sup>i</sup> <sup>ω</sup>t*þ*φ<sup>R</sup>* ð Þ h i (7)

> �*i*ð Þ *φR*�*φ<sup>D</sup>* h i*:* (8)

*<sup>D</sup>* þ 2*ERED* cosð Þ *φ<sup>R</sup>* � *φ<sup>D</sup> :* (9)

<sup>p</sup> cosð Þ *<sup>φ</sup><sup>R</sup>* � *<sup>φ</sup><sup>D</sup> :* (10)

*IT* ¼ 2*Io*½ � 1 þ cosð Þ *φ<sup>R</sup>* � *φ<sup>D</sup> :* (11)

�*i*ð Þ *<sup>ω</sup>t*þ*φ<sup>D</sup>* <sup>þ</sup> *<sup>e</sup>*

*<sup>i</sup>*ð Þ *<sup>φ</sup>R*�*φ<sup>D</sup>* <sup>þ</sup> *<sup>e</sup>*

<sup>2</sup> , Eq. (8) takes the form

In terms of intensity, the interferometer system produces the next interference

*IRID*

If both beams have the same intensity *IR* ¼ *ID* ¼ *Io*, the total intensity will take

As seen in Eq. (11), the phase difference is due to the optical path difference between the two beams. Substituting Eqs. (3) and (4) into Eq. (11), the interference

*<sup>λ</sup>* is the

*<sup>i</sup>*ð Þ *<sup>ω</sup>t*þ*φ<sup>D</sup> :* (5)

�*i*ð Þ *ωt*þ*φ<sup>D</sup>* h i*:* (6)

*ER*ðÞ¼ *t ARe*

*ED*ðÞ¼ *t ADe*

*AR* and *AD* are amplitudes, *ω* is the angular frequency, *t* is the time, and *φ<sup>R</sup>* and

The second one is the signal measurement and its electrical field is

*xR* and *xD* are the distances traveled by both beams and *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*π<sup>n</sup>*

*<sup>T</sup>* will be

*i*ð Þ *ωt*þ*φ<sup>D</sup>*

*<sup>i</sup> <sup>ω</sup>t*þ*φ<sup>R</sup>* ð Þ*e*

*<sup>D</sup>* þ *ERED e*

From **Figure 1**, when the photodetector detects the total field, its signal is

*<sup>i</sup> <sup>ω</sup>t*þ*φ<sup>R</sup>* ð Þ <sup>þ</sup> *ADe*

wavenumber: *λ* is the wavelength and *n* is the refraction index.

*ET*ðÞ¼ *t ARe*

h i *ARe*

*<sup>i</sup> <sup>ω</sup>t*þ*φ<sup>R</sup>* ð Þ <sup>þ</sup> *ADe*

*<sup>D</sup>* þ *ERED e*

*<sup>R</sup>* <sup>þ</sup> *<sup>E</sup>*<sup>2</sup>

*<sup>R</sup>* <sup>þ</sup> *<sup>E</sup>*<sup>2</sup>

*IT* <sup>¼</sup> *IR* <sup>þ</sup> *ID* <sup>þ</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffi

be the reference. Its electrical field is

*DOI: http://dx.doi.org/10.5772/intechopen.89491*

*Interference Pattern Representation on the Complex s-Plane*

Following, the irradiance *E*<sup>2</sup>

*<sup>T</sup>* ¼ *ARe*

Developing, we will obtain

*<sup>R</sup>* <sup>þ</sup> *<sup>E</sup>*<sup>2</sup>

*E*2 *<sup>T</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup>

Using the identity *cos* ð Þ¼ *<sup>θ</sup> <sup>e</sup>iθ*þ*e*�*i<sup>θ</sup>*

*E*2 *<sup>T</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup>

pattern in terms of intensity can be written as

*E*2

*E*2 *<sup>T</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup>

or

pattern

the form

**13**

*φ<sup>D</sup>* are given by

and

In dynamic system analysis, pole-zero map and Bode diagrams are two graphical methods which have many practical applications. Both methods require a complex function, where the frequency response plays a very important role. In the polezero map, poles and zeros have been calculated from the complex function, and then, their locations are represented on the complex s-plane. It is usual to mark a zero location by a circle ð Þ ⋄ and a pole location a cross ð Þ � [24]. In the Bode diagram, the magnitude and phase are calculated from the complex function, and then, both parameters are graphed. The graphic is logarithmic, and it shows the frequency response of our system under study.

Under our knowledge, the coherence interferometer system was not studied on the s-plane, and as a consequence, its interference pattern was not represented over the pole-zero map or Bode diagrams. In this work, the complex s-plane was used to represent the output signal of an interferometer system. Applying two graphical methods, such as pole-zero plot and Bode plot, the optical signal was represented. Numerically was verified that the pole location and the zero location depend directly on the optical path difference, while a Bode diagram shows the stability/ instability of the interferometer.

#### **2. Interference pattern**

**Figure 1** shows a schematic example of a Michelson interferometer [25]. The interferometer consists of a coherent source, an oscilloscope, a generator function, a beam splitter 50/50 and a PZT optical element. This interferometric system has

**Figure 1.** *Michelson interometer system.*

two ways and its difference produces the optical path difference. The first one will be the reference. Its electrical field is

$$E\_R(t) = A\_R e^{i(at + \varphi\_R)}.\tag{1}$$

The second one is the signal measurement and its electrical field is

$$E\_D(t) = A\_D e^{i(at + q\_D)}.\tag{2}$$

*AR* and *AD* are amplitudes, *ω* is the angular frequency, *t* is the time, and *φ<sup>R</sup>* and *φ<sup>D</sup>* are given by

$$
\rho\_R = \mathfrak{L}\mathfrak{x}\_R,\tag{3}
$$

and

transform any time function into a complex function whose variable is *s* ¼ *iω* þ *σ*, where *i* is the complex operator, *ω* is the angular frequency, and *σ* is a real value. The complex function can represent in the complex s-plane, where their axes represent the real and imaginary parts of the complex variable *s*. This complex plane does feasible the study of dynamic systems, and some applications are the tuning closed-loop, stability, mathematical methods, fault detection, optimization, and filter design [21–23]. In addition, the s-plane permits graphical methods such as pole-zero map, Bode diagrams, root locus, polar plots, gain margin and phase

In dynamic system analysis, pole-zero map and Bode diagrams are two graphical methods which have many practical applications. Both methods require a complex function, where the frequency response plays a very important role. In the polezero map, poles and zeros have been calculated from the complex function, and then, their locations are represented on the complex s-plane. It is usual to mark a zero location by a circle ð Þ ⋄ and a pole location a cross ð Þ � [24]. In the Bode diagram, the magnitude and phase are calculated from the complex function, and then, both parameters are graphed. The graphic is logarithmic, and it shows the

Under our knowledge, the coherence interferometer system was not studied on the s-plane, and as a consequence, its interference pattern was not represented over the pole-zero map or Bode diagrams. In this work, the complex s-plane was used to represent the output signal of an interferometer system. Applying two graphical methods, such as pole-zero plot and Bode plot, the optical signal was represented. Numerically was verified that the pole location and the zero location depend directly on the optical path difference, while a Bode diagram shows the stability/

**Figure 1** shows a schematic example of a Michelson interferometer [25]. The interferometer consists of a coherent source, an oscilloscope, a generator function, a beam splitter 50/50 and a PZT optical element. This interferometric system has

margin, Nichols charts, and N circles [24].

*Advances in Complex Analysis and Applications*

frequency response of our system under study.

instability of the interferometer.

**2. Interference pattern**

**Figure 1.**

**12**

*Michelson interometer system.*

$$
\rho\_D = 2k\chi\_D.\tag{4}
$$

*xR* and *xD* are the distances traveled by both beams and *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*π<sup>n</sup> <sup>λ</sup>* is the wavenumber: *λ* is the wavelength and *n* is the refraction index.

From **Figure 1**, when the photodetector detects the total field, its signal is

$$E\_T(t) = A\_R e^{i(at + q\_R)} + A\_D e^{i(at + q\_D)}.\tag{5}$$

Following, the irradiance *E*<sup>2</sup> *<sup>T</sup>* will be

$$E\_T^2 = \left[A\_R e^{i(at+\varphi\_R)} + A\_D e^{i(at+\varphi\_D)}\right] \left[A\_R e^{-i(at+\varphi\_R)} + A\_D e^{-i(at+\varphi\_D)}\right].\tag{6}$$

Developing, we will obtain

$$E\_T^2 = E\_R^2 + E\_D^2 + E\_R E\_D \left[ e^{i(at + \varphi\_R)} e^{-i(at + \varphi\_D)} + e^{i(at + \varphi\_D)} e^{-i(at + \varphi\_R)} \right] \tag{7}$$

or

$$E\_T^2 = E\_R^2 + E\_D^2 + E\_R E\_D \left[ e^{i(\varphi\_R - \varphi\_D)} + e^{-i(\varphi\_R - \varphi\_D)} \right].\tag{8}$$

Using the identity *cos* ð Þ¼ *<sup>θ</sup> <sup>e</sup>iθ*þ*e*�*i<sup>θ</sup>* <sup>2</sup> , Eq. (8) takes the form

$$E\_T^2 = E\_R^2 + E\_D^2 + 2E\_R E\_D \cos\left(\varphi\_R - \varphi\_D\right). \tag{9}$$

In terms of intensity, the interferometer system produces the next interference pattern

$$I\_T = I\_R + I\_D + 2\sqrt{I\_R I\_D} \cos\left(\varphi\_R - \varphi\_D\right). \tag{10}$$

If both beams have the same intensity *IR* ¼ *ID* ¼ *Io*, the total intensity will take the form

$$I\_T = 2I\_o[1 + \cos\left(\varphi\_R - \varphi\_D\right)].\tag{11}$$

As seen in Eq. (11), the phase difference is due to the optical path difference between the two beams. Substituting Eqs. (3) and (4) into Eq. (11), the interference pattern in terms of intensity can be written as

$$I\_T(t) = 2I\_o \left[ 1 + \cos\left(\frac{4\pi n}{\lambda} \Delta x(t)\right) \right],\tag{12}$$

*In*ðÞ¼ *s*

Using the algebraic procedure, we obtain

*Interference Pattern Representation on the Complex s-Plane*

*DOI: http://dx.doi.org/10.5772/intechopen.89491*

2 *s* þ

*In*ðÞ¼ *s*

**4. Graphical representation**

complex function in terms of those factors

**4.1 Pole-zero plot**

we require

**15**

function as

2 *s* þ

*In*ðÞ¼ *s*

It is possible since both Eqs. (13) and (20) contain the same information.

basis for determining important system response characteristics.

*In*ðÞ¼ *s*

*s* <sup>2</sup> <sup>¼</sup> �*ω*<sup>2</sup> *m* <sup>2</sup> ! *<sup>s</sup>* <sup>¼</sup>

*PN*ð Þ*s PD*ð Þ*<sup>s</sup>* <sup>¼</sup> <sup>4</sup>*<sup>s</sup>*

*PN*ðÞ¼ *s* 0 ¼ 4*s*

Solving last polynomial function, the roots (zeros) are localized at

where the numerator and denominator polynomials, *PN*ð Þ*s* and *PD*ð Þ*s* , have real coefficient defined by the system's characteristic. To calculate the zeros,

1 *s* � *iω<sup>m</sup>*

*s* þ *iω<sup>m</sup>* þ *s* � *iω<sup>m</sup> s*<sup>2</sup> � *i* 2 *ω*<sup>2</sup> *m*

As seen in Eq. (19), the first term was produced by the direct component and the second term was produced by cosine function. Now, let us represent the complex

> *s s*<sup>2</sup> þ *ω*<sup>2</sup> *m*

Because the Laplace transform was used for the transformation, the normalized interference pattern can be studied in the time domain and on a complex s-plane.

In mathematics and engineering, the s-plane is the complex plane which Laplace transform is graphed. It is a mathematical domain where, instead of view processes in the time domain modeled with time-based functions, they are viewed as equations in the frequency domain. Then, the function *In*ð Þ*s* can be graphed using the pole-zero map and the Bode diagrams. These graphical representations provide a

In general, the poles and zeros of a complex function may be complex, and the system dynamics may be represented graphically by plotting their locations on the complex *s*-plane, whose axes represent the real and imaginary parts of the complex variable *s*. Such graphics are known as pole-zero plots. It is usual to mark a zero location by a circle ð Þ ⋄ and a pole location a cross ð Þ � *:* In this study, it is convenient to factor the polynomials in the numerator and denominator and to write the

> <sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup> *m*

� � , (21)

*<sup>m</sup>:* (22)

*:* (23)

*s s*<sup>2</sup> þ *ω*<sup>2</sup> *m*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup>

r

ffiffiffiffiffiffiffiffiffiffiffi � *ω*2 *m* 2

4*s* <sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup> *m*

þ

1 *s* þ *iω<sup>m</sup>*

¼ 2 *s* þ

2*s s*<sup>2</sup> þ *ω*<sup>2</sup> *m*

� � *:* (20)

*:* (18)

*:* (19)

where Δ*x* ¼ *xR* � *xD* is the length difference between the distances *xR* and *xD*. Basically, the irradiance is an interference pattern which is formed by two functions: enveloped and modulate. The enveloped function is *fenv* <sup>¼</sup> <sup>2</sup>*Io <sup>W</sup> m*<sup>2</sup> � � and this function contains information from the optical source. The modulate function is given by *<sup>f</sup> mod* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> cos <sup>4</sup>*π<sup>n</sup> <sup>λ</sup>* <sup>Δ</sup>*x t*ð Þ � � and it contains information about the interference pattern. The modulate function consist of a constant (direct component) and a trigonometric function (cosine function) whose frequency depends on the optical path difference.

#### **3. Complex function**

Observing **Figure 1** and Eq. (11), the phase difference *φ<sup>R</sup>* � *φ<sup>D</sup>* is a time-varying function, and as a consequence, the phase <sup>4</sup>*π<sup>n</sup> <sup>λ</sup>* Δ*x t*ð Þ is also a time-varying function. In this case, the instantaneous output voltage (or current) of our photodector is proportional to the normalized optical intensity *IT*ð Þ*<sup>t</sup> Io* , where *Io* is the LASER intensity *W m*<sup>2</sup> � � [25]. Mathematically, the normalized interference pattern can be written as

$$I\_n(t) = \frac{I\_T(t)}{I\_o} = 2[1 + \cos\left(\alpha\_m t\right)].\tag{13}$$

Here, the angular frequency *ω<sup>m</sup>* was proposed from the phase <sup>4</sup>*π<sup>n</sup> <sup>λ</sup>* Δ*x t*ð Þ and the interferometer system has not external perturbations.

To determinate the complex function *In*ð Þ*s* , we calculate the unilateral Laplace transform for our last expression

$$I\_n(\epsilon) = \int\_0^\infty I\_n(t)e^{-\epsilon t}dt = 2\int\_0^\infty e^{-\epsilon t}dt + 2\int\_0^\infty \cos\left(\alpha\_n t\right)e^{-\epsilon t}dt.\tag{14}$$

Substituting the trigonometric identity cosð Þ¼ *<sup>ω</sup>mt ei<sup>ω</sup>mt* <sup>þ</sup>*e*�*iωmt* <sup>2</sup> into Eq. (14), the complex function can be estimated through

$$I\_n(s) = 2\int\_0^\infty e^{-st}dt + 2\int\_0^\infty \left(\frac{e^{io\_mt} + e^{-io\_mt}}{2}\right)e^{-st}dt,\tag{15}$$

or

$$I\_n(s) = 2\int\_0^\infty e^{-st}dt + \int\_0^\infty e^{io\_nt}e^{-st}dt + \int\_0^\infty e^{-io\_nt}e^{-st}dt.\tag{16}$$

Solving the integrals, the complex function will be

$$I\_n(\boldsymbol{s}) = -\frac{2}{s}e^{-\boldsymbol{s}t}\Big|\_{0}^{\infty} + \frac{e^{-(-i\boldsymbol{o}\_m\boldsymbol{t}+\boldsymbol{s})\mathbf{t}}}{-(-i\boldsymbol{o}\_m\boldsymbol{t}+\boldsymbol{s})}\Big|\_{0}^{\infty} + \frac{e^{-(i\boldsymbol{o}\_m+\boldsymbol{s})\mathbf{t}}}{-(i\boldsymbol{o}\_m+\boldsymbol{s})}\Big|\_{0}^{\infty}.\tag{17}$$

Evaluating the limits,

*Interference Pattern Representation on the Complex s-Plane DOI: http://dx.doi.org/10.5772/intechopen.89491*

$$I\_n(s) = \frac{2}{s} + \frac{1}{s - i o o\_m} + \frac{1}{s + i o o\_m}.\tag{18}$$

Using the algebraic procedure, we obtain

*IT*ðÞ¼ *t* 2*Io* 1 þ cos

tions: enveloped and modulate. The enveloped function is *fenv* <sup>¼</sup> <sup>2</sup>*Io <sup>W</sup>*

given by *<sup>f</sup> mod* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> cos <sup>4</sup>*π<sup>n</sup>*

**3. Complex function**

function, and as a consequence, the phase <sup>4</sup>*π<sup>n</sup>*

*Advances in Complex Analysis and Applications*

proportional to the normalized optical intensity *IT*ð Þ*<sup>t</sup>*

*In*ðÞ¼ *t*

interferometer system has not external perturbations.

*In*ð Þ*t e*

Substituting the trigonometric identity cosð Þ¼ *<sup>ω</sup>mt ei<sup>ω</sup>mt*

∞ð

0 *e* �*stdt* <sup>þ</sup> <sup>2</sup>

∞ð

0 *e* �*stdt* <sup>þ</sup>

Solving the integrals, the complex function will be

∞

0 þ

2 *s e* �*st* � � � �

∞ð

0

*In*ðÞ¼ *s* 2

*In*ðÞ¼ *s* 2

*In*ðÞ¼� *s*

Evaluating the limits,

complex function can be estimated through

transform for our last expression

*In*ðÞ¼ *s*

path difference.

*W m*<sup>2</sup>

or

**14**

4*πn λ*

� � � �

where Δ*x* ¼ *xR* � *xD* is the length difference between the distances *xR* and *xD*. Basically, the irradiance is an interference pattern which is formed by two func-

function contains information from the optical source. The modulate function is

ence pattern. The modulate function consist of a constant (direct component) and a trigonometric function (cosine function) whose frequency depends on the optical

Observing **Figure 1** and Eq. (11), the phase difference *φ<sup>R</sup>* � *φ<sup>D</sup>* is a time-varying

In this case, the instantaneous output voltage (or current) of our photodector is

� � [25]. Mathematically, the normalized interference pattern can be written as

To determinate the complex function *In*ð Þ*s* , we calculate the unilateral Laplace

∞ð

0

*<sup>e</sup><sup>i</sup>ωmt* <sup>þ</sup> *<sup>e</sup>*�*iωmt* 2 � �

∞ð

0 *e* �*iωmt e*

� � � �

∞

0 þ

∞ð

0 *e* �*stdt* <sup>þ</sup> <sup>2</sup>

∞ð

0

∞ð

0 *e iωmt e* �*stdt* <sup>þ</sup>

*e*� �ð Þ *<sup>i</sup>ωmt*þ*<sup>s</sup> <sup>t</sup>* � �ð Þ *iωmt* þ *s*

*IT*ð Þ*t Io*

Here, the angular frequency *ω<sup>m</sup>* was proposed from the phase <sup>4</sup>*π<sup>n</sup>*

�*stdt* <sup>¼</sup> <sup>2</sup>

Δ*x t*ð Þ

*<sup>λ</sup>* <sup>Δ</sup>*x t*ð Þ � � and it contains information about the interfer-

, (12)

*m*<sup>2</sup> � � and this

*<sup>λ</sup>* Δ*x t*ð Þ is also a time-varying function.

¼ 2 1½ � þ cosð Þ *ωmt :* (13)

cosð Þ *ωmt e*

*e*

*e*�ð Þ *<sup>i</sup>ωm*þ*<sup>s</sup> <sup>t</sup>* �ð Þ *iω<sup>m</sup>* þ *s*

� � � �

∞

0

<sup>þ</sup>*e*�*iωmt*

*Io* , where *Io* is the LASER intensity

*<sup>λ</sup>* Δ*x t*ð Þ and the

�*stdt:* (14)

<sup>2</sup> into Eq. (14), the

�*stdt*, (15)

�*stdt:* (16)

*:* (17)

$$I\_n(\mathfrak{s}) = \frac{2}{\mathfrak{s}} + \frac{\mathfrak{s} + io\mathfrak{o}\_m + \mathfrak{s} - io\mathfrak{o}\_m}{\mathfrak{s}^2 - \mathfrak{i}^2 o\mathfrak{o}\_m^2} = \frac{2}{\mathfrak{s}} + \frac{2\mathfrak{s}}{\mathfrak{s}^2 + o\mathfrak{o}\_m^2}. \tag{19}$$

As seen in Eq. (19), the first term was produced by the direct component and the second term was produced by cosine function. Now, let us represent the complex function as

$$I\_n(s) = \frac{4s^2 + 2\alpha\_m^2}{s\left(s^2 + \alpha\_m^2\right)}.\tag{20}$$

Because the Laplace transform was used for the transformation, the normalized interference pattern can be studied in the time domain and on a complex s-plane. It is possible since both Eqs. (13) and (20) contain the same information.

#### **4. Graphical representation**

In mathematics and engineering, the s-plane is the complex plane which Laplace transform is graphed. It is a mathematical domain where, instead of view processes in the time domain modeled with time-based functions, they are viewed as equations in the frequency domain. Then, the function *In*ð Þ*s* can be graphed using the pole-zero map and the Bode diagrams. These graphical representations provide a basis for determining important system response characteristics.

#### **4.1 Pole-zero plot**

In general, the poles and zeros of a complex function may be complex, and the system dynamics may be represented graphically by plotting their locations on the complex *s*-plane, whose axes represent the real and imaginary parts of the complex variable *s*. Such graphics are known as pole-zero plots. It is usual to mark a zero location by a circle ð Þ ⋄ and a pole location a cross ð Þ � *:* In this study, it is convenient to factor the polynomials in the numerator and denominator and to write the complex function in terms of those factors

$$I\_n(s) = \frac{P\_N(s)}{P\_D(s)} = \frac{4s^2 + 2\alpha\_m^2}{s\left(s^2 + \alpha\_m^2\right)},\tag{21}$$

where the numerator and denominator polynomials, *PN*ð Þ*s* and *PD*ð Þ*s* , have real coefficient defined by the system's characteristic. To calculate the zeros, we require

$$P\_N(\mathfrak{s}) = \mathbf{0} = 4\mathfrak{s}^2 + 2a\_m^2. \tag{22}$$

Solving last polynomial function, the roots (zeros) are localized at

$$s^2 = \frac{-\alpha\_m^2}{2} \to s = \sqrt{-\frac{\alpha\_m^2}{2}}.\tag{23}$$

From our last results, the zeros are imaginary values

$$\begin{aligned} s\_1 &= i \frac{\alpha\_m}{\sqrt{2}}\\ s\_2 &= -i \frac{\alpha\_m}{\sqrt{2}} \end{aligned} \tag{24}$$

As was mentioned, the complex interference pattern can be represented through the Bode diagram. To represent it, the term *s* is substituted by the term *iω*: *i* is the complex number and *ω* is the angular frequency. Such that, Eq. (20) takes the form

The magnitude (in decibels) of the transference function above is given by

� � � � � �

�

*m*

*In*ð Þ¼ *<sup>i</sup><sup>ω</sup>* �4*ω*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup>

*In*ð Þ¼� *<sup>i</sup><sup>ω</sup> <sup>i</sup>* �4*ω*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup>

As the Laplace transform is a linear transformation, during the transformation: *In*ðÞ!*t In*ð Þ*s* and *In*ðÞ!*s In*ð Þ*t* , the information is not lost and then the interference pattern can be studied in the time domain and on the complex s-plane. In Section 2, it was explained the transformation *In*ðÞ!*t In*ð Þ*s* , and their poles and zeros were graphed over the pole-zero map. In addition, we developed interference pattern on the frequency plane, being possible to implement the Bode plot. Following, we recover the time function from the complex function *In*ðÞ!*s In*ð Þ*t :* This section is didactic since the objective is to verify that the complex function's information can

To recover the time function, we calculate the inverse Laplace transform

*i* �*ω*<sup>3</sup> þ *ωω*<sup>2</sup>

�*ω*<sup>3</sup> þ *ωω*<sup>2</sup>

� �

<sup>2</sup> ¼ �1 was used. Last expression can be written as

2 2ð Þ *<sup>i</sup><sup>ω</sup>* <sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

*<sup>i</sup><sup>ω</sup>* ð Þ *<sup>i</sup><sup>ω</sup>* <sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

2 2ð Þ *<sup>i</sup><sup>ω</sup>* <sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

*<sup>i</sup><sup>ω</sup>* ð Þ *<sup>i</sup><sup>ω</sup>* <sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

h i

h i

<sup>2</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> m

*m*

*m*

*m*

*m*

� �

h i

*m*

h i *:* (27)

*:* (29)

*m*

*AvdB* ¼ 20 *log I*j j *<sup>n</sup>*ð Þ *iω :* (28)

� � � � � �

*m*

*m*

� � � � <sup>20</sup> *log* ð Þ j j <sup>i</sup><sup>ω</sup>

� � �*:* (30)

� � *:* (31)

� � *:* (32)

*In*ð Þ¼ *iω In*ð Þ¼ *ω*

*Interference Pattern Representation on the Complex s-Plane*

*DOI: http://dx.doi.org/10.5772/intechopen.89491*

Substituting Eq. (27) into Eq. (28), the magnitude will be

*AvdB* ¼ 20 *log*

Applying the logarithm rules, Eq. (29) can express as

AvdB ¼ 20 *log* ð Þþ 2 20 *log* 2 ið Þ ω

�<sup>20</sup> *log i*ð Þ *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

�

To determine the phase, Eq. (27) will express as

From Eq. (32), the phase can also be determined.

also be represented in the time domain.

decibels gain expression:

Here, *i*

**5.** *In*ð Þ*t* **retrieval**

through

**17**

By the similar way,

$$P\_D(\mathfrak{s}) = \mathbf{0} = \mathfrak{s} \left(\mathfrak{s}^2 + a\_m^2\right). \tag{25}$$

To calculate the roots,

$$\begin{aligned} s\_1 &= 0\\ s\_2 &= i o \rho\_m \quad . \\ s\_3 &= -i o \rho\_m \end{aligned} \tag{26}$$

Using our previous results presented at Eq. (24) and Eq. (26), we represent a pole-zero plot for the interference pattern, see **Figure 2**.

From **Figure 2**, the interference pattern produces two zeros and three poles. Both zeros ð Þ *s*<sup>1</sup> and *s*<sup>2</sup> and two poles ð Þ *s*<sup>2</sup> and *s*<sup>3</sup> are over the imaginary axes and their locations depend on the angular frequency. The pole ð Þ *s*<sup>1</sup> was obtained by the direct component; our normalized interference pattern and the location are over the origin.

#### **4.2 Bode diagram**

Based on the system theory and system graphic representation, the complex interference pattern can be represented through the Bode diagram. The graphical representation permits us to graph the frequency response of our interferometer system. It combines a Bode magnitude plot, expressing the magnitude (decibels) of the frequency response, and a Bode phase plot, expressing the phase shift.

**Figure 2.** *Polo-zero map obtained from the interference pattern.*

From our last results, the zeros are imaginary values

*Advances in Complex Analysis and Applications*

pole-zero plot for the interference pattern, see **Figure 2**.

By the similar way,

To calculate the roots,

origin.

**Figure 2.**

**16**

*Polo-zero map obtained from the interference pattern.*

**4.2 Bode diagram**

*s*<sup>1</sup> ¼ *i*

*s*<sup>2</sup> ¼ �*i*

*PD*ðÞ¼ *<sup>s</sup>* <sup>0</sup> <sup>¼</sup> *s s*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

*s*<sup>1</sup> ¼ 0 *s*<sup>2</sup> ¼ *iω<sup>m</sup> s*<sup>3</sup> ¼ �*iω<sup>m</sup>*

Using our previous results presented at Eq. (24) and Eq. (26), we represent a

From **Figure 2**, the interference pattern produces two zeros and three poles. Both zeros ð Þ *s*<sup>1</sup> and *s*<sup>2</sup> and two poles ð Þ *s*<sup>2</sup> and *s*<sup>3</sup> are over the imaginary axes and their locations depend on the angular frequency. The pole ð Þ *s*<sup>1</sup> was obtained by the direct component; our normalized interference pattern and the location are over the

Based on the system theory and system graphic representation, the complex interference pattern can be represented through the Bode diagram. The graphical representation permits us to graph the frequency response of our interferometer system. It combines a Bode magnitude plot, expressing the magnitude (decibels) of

the frequency response, and a Bode phase plot, expressing the phase shift.

*ω<sup>m</sup>* ffiffi 2 p

*ω<sup>m</sup>* ffiffi 2 p

*m*

*:* (24)

� �*:* (25)

*:* (26)

As was mentioned, the complex interference pattern can be represented through the Bode diagram. To represent it, the term *s* is substituted by the term *iω*: *i* is the complex number and *ω* is the angular frequency. Such that, Eq. (20) takes the form

$$I\_n(ioo) = I\_n(oo) = \frac{2\left[2(ioo)^2 + o\_m^2\right]}{ioo\left[(ioo)^2 + o\_m^2\right]}.\tag{27}$$

The magnitude (in decibels) of the transference function above is given by decibels gain expression:

$$A\_{vdB} = 20 \log |I\_n(io)|. \tag{28}$$

Substituting Eq. (27) into Eq. (28), the magnitude will be

$$A\_{vdB} = 20\log\left|\frac{2\left[2\left(i\nu\right)^2 + \alpha\_m^2\right]}{i\alpha\left[\left(i\nu\right)^2 + \alpha\_m^2\right]}\right|\,. \tag{29}$$

Applying the logarithm rules, Eq. (29) can express as

$$\mathbf{A\_{vdB}} = 20\log\left(2\right) + 20\log\left(\left|2\left(\text{ioo}\right)^2 + \alpha\_m^2\right|\right) - 20\log\left(\left|\text{ioo}\right|\right)$$

$$- 20\log\left(\left|\left(\text{ioo}\right)^2 + \alpha\_m^2\right|\right). \tag{30}$$

To determine the phase, Eq. (27) will express as

$$I\_n(i\alpha) = \frac{-4\alpha^2 + 2\alpha\_m^2}{i\left[-\alpha^3 + \alpha\alpha\_m^2\right]}.\tag{31}$$

Here, *i* <sup>2</sup> ¼ �1 was used. Last expression can be written as

$$I\_n(io) = -i \frac{-4o\rho^2 + 2o\rho\_m^2}{\left[-o\rho^3 + aoo\rho\_m^2\right]}.\tag{32}$$

From Eq. (32), the phase can also be determined.

### **5.** *In*ð Þ*t* **retrieval**

As the Laplace transform is a linear transformation, during the transformation: *In*ðÞ!*t In*ð Þ*s* and *In*ðÞ!*s In*ð Þ*t* , the information is not lost and then the interference pattern can be studied in the time domain and on the complex s-plane. In Section 2, it was explained the transformation *In*ðÞ!*t In*ð Þ*s* , and their poles and zeros were graphed over the pole-zero map. In addition, we developed interference pattern on the frequency plane, being possible to implement the Bode plot. Following, we recover the time function from the complex function *In*ðÞ!*s In*ð Þ*t :* This section is didactic since the objective is to verify that the complex function's information can also be represented in the time domain.

To recover the time function, we calculate the inverse Laplace transform through

*Advances in Complex Analysis and Applications*

$$I\_n(t) = \frac{1}{2\pi i} \lim\_{T \to \infty} \int\_{r-iT}^{r+iT} I\_n(s)e^s ds = \mathcal{L}^{-1} \{ I\_n(s) \}. \tag{33}$$

Applying **Table 1**, the solved inverse Laplace transform is

*Interference Pattern Representation on the Complex s-Plane*

*DOI: http://dx.doi.org/10.5772/intechopen.89491*

To verify our proposal, we consider the next interference pattern

*t* 2

From Eq. (40), the enveloped *<sup>f</sup> env* is a Gaussian function *<sup>f</sup> env* <sup>¼</sup> <sup>2</sup>*Io* <sup>¼</sup> <sup>2</sup>*e<sup>t</sup>*

*IT*ðÞ¼ *t* 2*e*

and the modulate function is *f mod* ¼ 1 þ cos 10 ð Þ*t* , where the angular

*IT*ð Þ*t Io*

**Figure 3** shows the interference pattern and the modulate function.

*PN*ð Þ*s PD*ð Þ*<sup>s</sup>* <sup>¼</sup> <sup>4</sup>*<sup>s</sup>*

Using Expressions (24) and (42), the zeros are localized at the points

*s*<sup>1</sup> ¼ *i*

*s*<sup>2</sup> ¼ �*i*

*s*<sup>1</sup> ¼ 0 *s*<sup>2</sup> ¼ *i*10 *s*<sup>3</sup> ¼ �*i*10*:*

As seen in **Figure 4**, the zeros ð Þ *s*<sup>1</sup> and *s*<sup>2</sup> and the poles ð Þ *s*<sup>2</sup> and *s*<sup>3</sup> are over the imaginary axis. Their positions depend on the angular frequency, and therefore, their positions change due to the variations of the optical path difference. The pole *s*<sup>1</sup> is over the origin (of the complex s-plane), and it was generated by the direct

10ffiffi 2 p

> 10ffiffi 2 p

*In*ðÞ¼ *t*

*In*ðÞ¼ *s*

Now, using Eqs. (26) and (42), the poles are

Finally, its pole-zero map can observe in **Figure 4**.

component of our interference pattern.

**19**

and Bode diagrams.

frequency is *<sup>ω</sup><sup>m</sup>* <sup>¼</sup> <sup>10</sup> *radians*

**6.1 Results**

we obtain

**6. Numerical results and discussion**

Calculating the Laplace transform,

Observing both Eq. (13) and Eq. (39), we recover the time function from the complex function. Thus, we confirm that the complex s-plane permits us to study the interferometer system through the complex s-plane, using the pole-zero map

*In*ðÞ¼ *t* 2 1½ � þ cosð Þ *ωmt :* (39)

½ � 1 þ cos 10 ð Þ*t :* (40)

¼ 2 1½ � þ cos 10 ð Þ*t :* (41)

*s s*ð Þ <sup>2</sup> <sup>þ</sup> <sup>100</sup> *:* (42)

*:* (43)

(44)

*se* � �. If the interference pattern is normalized as Eq. (12),

<sup>2</sup> <sup>þ</sup> <sup>200</sup>

<sup>2</sup> � �

The integral complex is in the s-plane; their limits are *γ* � *iT* and *γ* þ *iT*; the symbol L�<sup>1</sup> f g� <sup>¼</sup> <sup>1</sup> <sup>2</sup>*π<sup>i</sup>* lim *T*!∞ Ð *γ*þ*iT γ*�*iT* f g� *<sup>e</sup>stds* indicates the inverse Laplace transform. Substituting Eq. (21) into Eq. (33), the normalized interference pattern can be obtained by

$$I\_n(t) = \frac{1}{2\pi i} \lim\_{T \to \infty} \int\_{-iT}^{+iT} \frac{4s^2 + 2\alpha\_m^2}{s(s^2 + \alpha\_m^2)} e^{st} ds = \mathcal{L}^{-1} \left\{ \frac{4s^2 + 2\alpha\_m^2}{s(s^2 + \alpha\_m^2)} \right\}.\tag{34}$$

Applying the partial fraction, Eq. (34) can be expressed as

$$I\_n(t) = \frac{1}{2\pi i} \lim\_{T \to \infty} \int\_{-iT}^{+iT} \left(\frac{A}{s} + \frac{Bs + C}{s^2 + o\_m^2}\right) \epsilon^t ds = \mathcal{L}^{-1} \left\{\frac{A}{s} + \frac{Bs + C}{s^2 + o\_m^2}\right\}.\tag{35}$$

Here, *A*, *B*, and *C* are constants. To calculate the constant, we use next equality

$$\frac{4s^2 + 2\alpha\_m^2}{s\left(s^2 + \alpha\_m^2\right)} = \frac{A}{s} + \frac{Bs + C}{s^2 + \alpha\_m^2} \to 4s^2 + 2\alpha\_m^2 = As^2 + As\alpha\_m^2 + Bs^2 + Cs \tag{36}$$

Using Eq. (36), we obtain the next equation system and their solutions as

$$\begin{aligned} (A+B)s^2 &= 4s^2 & A &= 2\\ \text{Cs} &= \text{0} & \rightarrow & B &= 2 \ . \\ Ao\_m^2 &= 2o\_m^2 & \text{C} &= \text{0} \end{aligned} \tag{37}$$

Substituting all constants into Eq. (37), the time function will be

$$I\_n(t) = \mathcal{L}^{-1}\left\{\frac{2}{s}\right\} + \mathcal{L}^{-1}\left\{\frac{2s}{s^2 + \alpha\_m^2}\right\}.\tag{38}$$


#### **Table 1.** *Fundamental Laplace transform [24].*

Applying **Table 1**, the solved inverse Laplace transform is

$$I\_n(t) = 2[1 + \cos\left(\alpha\_m t\right)].\tag{39}$$

Observing both Eq. (13) and Eq. (39), we recover the time function from the complex function. Thus, we confirm that the complex s-plane permits us to study the interferometer system through the complex s-plane, using the pole-zero map and Bode diagrams.

#### **6. Numerical results and discussion**

#### **6.1 Results**

*In*ðÞ¼ *t*

Ð *γ*þ*iT γ*�*iT*

symbol L�<sup>1</sup>

obtained by

f g� <sup>¼</sup> <sup>1</sup>

*In*ðÞ¼ *t*

1 2*πi* lim *T*!∞

*In*ðÞ¼ *t*

4*s* <sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup> *m*

*s s*<sup>2</sup> þ *ω*<sup>2</sup> *m* � � <sup>¼</sup> *<sup>A</sup>*

<sup>2</sup>*π<sup>i</sup>* lim *T*!∞

*Advances in Complex Analysis and Applications*

1 2*πi* lim *T*!∞

ð*γ*þ*iT*

*γ*�*iT*

*s* þ

**Laplace transform Inverse Laplace transform**

*f t*ðÞ¼ *ku t*ð Þ *F s*ðÞ¼ *<sup>k</sup>*

*f t*ðÞ¼ *tu t*ð Þ *F s*ðÞ¼ <sup>1</sup>

*f t*ðÞ¼ *cos* ð Þ *<sup>ω</sup><sup>t</sup> u t*ð Þ *F s*ðÞ¼ *<sup>s</sup>*

*f t*ðÞ¼ *sen* ð Þ *<sup>ω</sup><sup>t</sup> u t*ð Þ *F s*ðÞ¼ *<sup>ω</sup>*

*f t*ðÞ¼ *cosh* ð Þ *<sup>ω</sup><sup>t</sup> u t*ð Þ *F s*ðÞ¼ *<sup>s</sup>*

*f t*ðÞ¼ *senh* ð Þ *<sup>ω</sup><sup>t</sup> u t*ð Þ *F s*ðÞ¼ *<sup>ω</sup>*

*Note: u t*ð Þ *is the Heaviside function.*

*Fundamental Laplace transform [24].*

*nu t*ð Þ *F s*ðÞ¼ *<sup>n</sup>*!

*f t*ðÞ¼ *t*

**Table 1.**

**18**

1 2*πi* lim *T*!∞

ð*γ*þ*iT*

4*s* <sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup> *m*

*s s*<sup>2</sup> þ *ω*<sup>2</sup> *m* � � *<sup>e</sup>*

*Bs* þ *C s*<sup>2</sup> þ *ω*<sup>2</sup> *m*

*e*

Here, *A*, *B*, and *C* are constants. To calculate the constant, we use next equality

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup>

Using Eq. (36), we obtain the next equation system and their solutions as

<sup>2</sup> <sup>¼</sup> <sup>4</sup>*<sup>s</sup>* 2

!

þ L�<sup>1</sup> <sup>2</sup>*<sup>s</sup>*

� �

*γ*�*iT*

*A s* þ

*Bs* þ *C s*<sup>2</sup> þ *ω*<sup>2</sup> *m* ! 4*s*

ð Þ *A* þ *B s*

*Aω*<sup>2</sup>

*In*ðÞ¼L *<sup>t</sup>* �<sup>1</sup> <sup>2</sup>

*Cs* ¼ 0

Substituting all constants into Eq. (37), the time function will be

*<sup>m</sup>* <sup>¼</sup> <sup>2</sup>*ω*<sup>2</sup> *m*

> *s* � �

**Time function Complex function Complex function Time function**

*<sup>s</sup> F s*ðÞ¼ *<sup>k</sup>*

*<sup>s</sup>*<sup>2</sup> *F s*ðÞ¼ *<sup>k</sup>*

*sn* <sup>þ</sup> <sup>1</sup> *F s*ðÞ¼ *<sup>n</sup>*!

*<sup>s</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *F s*ðÞ¼ *<sup>s</sup>*

*<sup>s</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *F s*ðÞ¼ *<sup>ω</sup>*

*<sup>s</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> *F s*ðÞ¼ *<sup>s</sup>*

*<sup>s</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> *F s*ðÞ¼ *<sup>ω</sup>*

Applying the partial fraction, Eq. (34) can be expressed as

ð*γ*þ*iT*

*In*ð Þ*s e*

The integral complex is in the s-plane; their limits are *γ* � *iT* and *γ* þ *iT*; the

Substituting Eq. (21) into Eq. (33), the normalized interference pattern can be

*stds* ¼ L�<sup>1</sup>

f g� *<sup>e</sup>stds* indicates the inverse Laplace transform.

*stds* ¼ L�<sup>1</sup> <sup>4</sup>*<sup>s</sup>*

*stds* ¼ L�<sup>1</sup> *<sup>A</sup>*

*<sup>m</sup>* <sup>¼</sup> *As*<sup>2</sup> <sup>þ</sup> *<sup>A</sup>ω*<sup>2</sup>

*A* ¼ 2 *B* ¼ 2 *C* ¼ 0

*s*<sup>2</sup> þ *ω*<sup>2</sup> *m*

� �

*s* þ <sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup> *m*

*Bs* þ *C s*<sup>2</sup> þ *ω*<sup>2</sup> *m*

� �

*s s*<sup>2</sup> þ *ω*<sup>2</sup> *m* � � ( )

f g *In*ð Þ*s :* (33)

*:* (34)

*:* (35)

*<sup>m</sup>* <sup>þ</sup> *Bs*<sup>2</sup> <sup>þ</sup> *Cs* (36)

*:* (37)

*:* (38)

*nu t*ð Þ

*<sup>s</sup> f t*ðÞ¼ *ku t*ð Þ

*<sup>s</sup> f t*ðÞ¼ *tu t*ð Þ

*<sup>s</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *f t*ðÞ¼ *cos* ð Þ *<sup>ω</sup> u t*ð Þ

*<sup>s</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *f t*ðÞ¼ *sen* ð Þ *<sup>ω</sup><sup>t</sup> u t*ð Þ

*<sup>s</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> *f t*ðÞ¼ *cosh* ð Þ *<sup>ω</sup> u t*ð Þ

*<sup>s</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> *f t*ðÞ¼ *senh* ð Þ *<sup>ω</sup><sup>t</sup> u t*ð Þ

*sn* <sup>þ</sup> <sup>1</sup> *f t*ðÞ¼ *t*

*γ*�*iT*

To verify our proposal, we consider the next interference pattern

$$I\_T(t) = 2\mathbf{\dot{z}}^{t^2} [\mathbf{1} + \cos(10t)].\tag{40}$$

From Eq. (40), the enveloped *<sup>f</sup> env* is a Gaussian function *<sup>f</sup> env* <sup>¼</sup> <sup>2</sup>*Io* <sup>¼</sup> <sup>2</sup>*e<sup>t</sup>* <sup>2</sup> � � and the modulate function is *f mod* ¼ 1 þ cos 10 ð Þ*t* , where the angular frequency is *<sup>ω</sup><sup>m</sup>* <sup>¼</sup> <sup>10</sup> *radians se* � �. If the interference pattern is normalized as Eq. (12), we obtain

$$I\_n(t) = \frac{I\_T(t)}{I\_o} = 2[\mathbf{1} + \cos\left(\mathbf{10t}\right)].\tag{41}$$

**Figure 3** shows the interference pattern and the modulate function. Calculating the Laplace transform,

$$I\_n(s) = \frac{P\_N(s)}{P\_D(s)} = \frac{4s^2 + 200}{s(s^2 + 100)}.\tag{42}$$

Using Expressions (24) and (42), the zeros are localized at the points

$$\begin{aligned} s\_1 &= i \frac{10}{\sqrt{2}}\\ s\_2 &= -i \frac{10}{\sqrt{2}} \end{aligned} \tag{43}$$

Now, using Eqs. (26) and (42), the poles are

$$\begin{aligned} s\_1 &= \mathbf{0} \\ s\_2 &= i\mathbf{1} \mathbf{0} \\ s\_3 &= -i\mathbf{1} \mathbf{0} .\end{aligned} \tag{44}$$

Finally, its pole-zero map can observe in **Figure 4**.

As seen in **Figure 4**, the zeros ð Þ *s*<sup>1</sup> and *s*<sup>2</sup> and the poles ð Þ *s*<sup>2</sup> and *s*<sup>3</sup> are over the imaginary axis. Their positions depend on the angular frequency, and therefore, their positions change due to the variations of the optical path difference. The pole *s*<sup>1</sup> is over the origin (of the complex s-plane), and it was generated by the direct component of our interference pattern.

**Figure 3.** *(a) The simulated interference pattern. (b) The modulate function.*

To generate the Bode diagram, we consider the next complex function

$$I\_n(s) = \frac{4(io)^2 + 200}{io \left[ \left( io \right)^2 + 100 \right]}. \tag{45}$$

AvdB ¼ 20 *log* 4 ið Þ ω

**Figure 4.**

**Figure 5.**

**21**

*Bode diagram obtained from the interference pattern.*

<sup>2</sup> <sup>þ</sup> <sup>200</sup> 

*Pole-zero plot determined from the complex modulate function (42).*

*Interference Pattern Representation on the Complex s-Plane*

*DOI: http://dx.doi.org/10.5772/intechopen.89491*

of 10<sup>0</sup> to 100.7 and from 10<sup>1</sup> to 10<sup>2</sup> while the phase is �90<sup>0</sup>

the point 100.7 and the phase has transition from �90<sup>o</sup> to 90<sup>0</sup>

� <sup>20</sup> *log* ð Þ� j j <sup>i</sup><sup>ω</sup> <sup>20</sup> *log i*ð Þ *<sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>100</sup>

Using the Scientific MatLab software, we represent its Bode plot, see **Figure 5**. Observing **Figure 5**, the magnitude has a small variation between the intervals

the integrative action indicated by Eq. (47). The interferometric system produces two asymptotics for the magnitude. First asymptotic is negative, its location is at

 *:* (47)

. These results confirm

. Second asymptotic

Combining Eqs. (28) and (45), the magnitude (in decibels) is

$$A\_{vdB} = 20\log\left|\frac{4(io)^2 + 200}{io\left[(io)^2 + 100\right]}\right|,\tag{46}$$

where *s* ¼ *iω* was used. Applying the logarithm rules, the magnitude can calculate as

*Interference Pattern Representation on the Complex s-Plane DOI: http://dx.doi.org/10.5772/intechopen.89491*

**Figure 4.** *Pole-zero plot determined from the complex modulate function (42).*

$$\mathbf{A\_{vdB}} = 20\log\left(\left|4(\text{i}\alpha)^2 + 200\right|\right) - 20\log\left(|\text{i}\alpha|\right) - 20\log\left(\left|(\text{i}\alpha)^2 + 100\right|\right). \tag{47}$$

Using the Scientific MatLab software, we represent its Bode plot, see **Figure 5**. Observing **Figure 5**, the magnitude has a small variation between the intervals of 10<sup>0</sup> to 100.7 and from 10<sup>1</sup> to 10<sup>2</sup> while the phase is �90<sup>0</sup> . These results confirm the integrative action indicated by Eq. (47). The interferometric system produces two asymptotics for the magnitude. First asymptotic is negative, its location is at the point 100.7 and the phase has transition from �90<sup>o</sup> to 90<sup>0</sup> . Second asymptotic

**Figure 5.** *Bode diagram obtained from the interference pattern.*

To generate the Bode diagram, we consider the next complex function

*AvdB* <sup>¼</sup> <sup>20</sup> *log* <sup>4</sup>ð Þ *<sup>i</sup><sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>200</sup>

where *s* ¼ *iω* was used. Applying the logarithm rules, the magnitude can

� � � � � �

<sup>4</sup>ð Þ *<sup>i</sup><sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>200</sup> *<sup>i</sup><sup>ω</sup>* ð Þ *<sup>i</sup><sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>100</sup>

> *<sup>i</sup><sup>ω</sup>* ð Þ *<sup>i</sup><sup>ω</sup>* <sup>2</sup> <sup>þ</sup> <sup>100</sup> h i

h i *:* (45)

, (46)

� � � � � �

*In*ðÞ¼ *s*

*(a) The simulated interference pattern. (b) The modulate function.*

*Advances in Complex Analysis and Applications*

Combining Eqs. (28) and (45), the magnitude (in decibels) is

calculate as

**20**

**Figure 3.**

is positive, its location is 10<sup>1</sup> and the phase has transition from 90<sup>0</sup> to �90<sup>o</sup> . Last interval is between 100.7 and 10<sup>1</sup> . In this case, the magnitude has small variation again but the phase is constant to 90<sup>o</sup> . These results confirm that the interferometer system will have integrative action and derivative action.

representations give us information about the interferometer. The optical Path Difference (OPD) information can measure through the pole-zero map and the behavior of interferometer can understand through the Bode diagram. Therefore, the interferometers can be studied on the complex s-plane, being possible measures physical parameters when those interferometers were disturbed. Also, the signal demodulation can be implemented for the quasi-distributed fiber sensor when the local sensors are interferometers. Some measurable parameters are temperature,

If a low-coherence interferometer is studied on the s-plane, then the fringe

Authors thank the Mexico's National Council for Science and Technology

visibility and the magnitude of coherence grade can be measured.

(CONACyT) and university of Guadalajara for the support.

The authors declare no conflict of interest.

string, displacement, voltage and pressure.

*Interference Pattern Representation on the Complex s-Plane*

*DOI: http://dx.doi.org/10.5772/intechopen.89491*

**Acknowledgements**

**Conflict of interest**

**23**

#### **6.2 Discussion**

Here, a complex function was obtained from the interference pattern produced by a coherence interferometer. Considering the pole-zero map, poles and zeros depend directly on the optical path difference of an interferometer. The interference pattern generates three poles and two zeros. A pole is over the origin and two poles are over the points �*iωm*. The zeros are over the points �*<sup>i</sup> <sup>ω</sup><sup>m</sup>*ffiffi 2 <sup>p</sup> . Now, considering the Bode plot, the interferometer can act as an integrator and as a derivator since the phase can take the value of �90<sup>o</sup> or 90<sup>o</sup> , see **Figure 5**. Both graphical representations permit to know the optical path difference through the angular frequency and its dynamic response.

From our analysis and results, it is possible to infer a few key point of our novel method.


Based on our results, the interference pattern can be studied by both graphical methods. Those graphical representations can be applied to low-coherence interferometric systems, optical fiber sensors, communication systems, and optical source characterization.

#### **7. Conclusion**

In this work, applying the Laplace transform and inverse Laplace transform, we confirm that the interference pattern produced by an Interferometer, can study in the time domain and on the complex s-plane. The pole-zero plot and the Bode diagram were obtained from the complex interference pattern. Both graphical

*Interference Pattern Representation on the Complex s-Plane DOI: http://dx.doi.org/10.5772/intechopen.89491*

representations give us information about the interferometer. The optical Path Difference (OPD) information can measure through the pole-zero map and the behavior of interferometer can understand through the Bode diagram. Therefore, the interferometers can be studied on the complex s-plane, being possible measures physical parameters when those interferometers were disturbed. Also, the signal demodulation can be implemented for the quasi-distributed fiber sensor when the local sensors are interferometers. Some measurable parameters are temperature, string, displacement, voltage and pressure.

If a low-coherence interferometer is studied on the s-plane, then the fringe visibility and the magnitude of coherence grade can be measured.

#### **Acknowledgements**

is positive, its location is 10<sup>1</sup> and the phase has transition from 90<sup>0</sup> to �90<sup>o</sup>

Here, a complex function was obtained from the interference pattern produced by a coherence interferometer. Considering the pole-zero map, poles and zeros depend directly on the optical path difference of an interferometer. The interference pattern generates three poles and two zeros. A pole is over the origin and two

ing the Bode plot, the interferometer can act as an integrator and as a derivator since

tations permit to know the optical path difference through the angular frequency

• The complex s-plane permits us to study the interferometric systems.

• Pole-zero map gives information about the optical path difference.

From our analysis and results, it is possible to infer a few key point of our novel

• The interference pattern can represent as a complex function whose poles are

• The pole *s*<sup>1</sup> is over the origin and it was generated by the direct component of

• *s*<sup>2</sup> and *s*<sup>3</sup> poles are over the imaginary axis and their position are �*iωm*, where

• *s*<sup>1</sup> and *s*<sup>2</sup> zeros are over the imaginary axes and their locations depend on the

• Based on the Bode diagram (Phase information), the interferometer can act as

Based on our results, the interference pattern can be studied by both graphical methods. Those graphical representations can be applied to low-coherence interferometric systems, optical fiber sensors, communication systems, and optical

In this work, applying the Laplace transform and inverse Laplace transform, we confirm that the interference pattern produced by an Interferometer, can study in the time domain and on the complex s-plane. The pole-zero plot and the Bode diagram were obtained from the complex interference pattern. Both graphical

• Bode diagram gives us information about the dynamic response of any

ometer system will have integrative action and derivative action.

poles are over the points �*iωm*. The zeros are over the points �*<sup>i</sup> <sup>ω</sup><sup>m</sup>*ffiffi

Last interval is between 100.7 and 10<sup>1</sup>

**6.2 Discussion**

tion again but the phase is constant to 90<sup>o</sup>

*Advances in Complex Analysis and Applications*

the phase can take the value of �90<sup>o</sup> or 90<sup>o</sup>

and its dynamic response.

three and zeros are two.

the interference pattern.

interference pattern.

source characterization.

**7. Conclusion**

**22**

*ω<sup>m</sup>* is the angular frequency.

angular frequency, see **Figure 4**.

an integrative action and as a derivative action.

method.

.

<sup>p</sup> . Now, consider-

. In this case, the magnitude has small varia-

. These results confirm that the interfer-

2

, see **Figure 5**. Both graphical represen-

Authors thank the Mexico's National Council for Science and Technology (CONACyT) and university of Guadalajara for the support.

#### **Conflict of interest**

The authors declare no conflict of interest.

### **Author details**

José Trinidad Guillen Bonilla1,2\*, Alex Guillen Bonilla3 , Mario Alberto García Ramírez<sup>1</sup> , Gustavo Adolfo Vega Gómez<sup>1</sup> , Héctor Guillen Bonilla<sup>4</sup> , María Susana Ruiz Palacio<sup>1</sup> , Martín Javier Martínez Silva<sup>1</sup> and Verónica María Bettancourt Rodriguez<sup>5</sup>

**References**

[1] Zhao N, Lin Q, Jiang Z, Yao K, Tian B, Fang X, et al. High temperature high sensitivity multipoint sensing system based on three cascade Mach-Zehnder interferometers. Sensors. 2018; **18**(8):2688. DOI: 10.3390/s18082688

*DOI: http://dx.doi.org/10.5772/intechopen.89491*

*Interference Pattern Representation on the Complex s-Plane*

[9] Dong C, Li K, Jiang Y, Arola D, Zhang D. Evaluation of thermal expansion coefficient carbon fiber reinforced composites using electronic speckle interferometry. Optics Express.

2018;**26**(1):531. DOI: 10.1364/

[10] Wang S, Gao Z, Li G, Feng Z, Feng Q. Continual mechanical vibration trajectory tracking based on electrooptical heterodyne interferometric. Optics Express. 2014;**22**(7):7799. DOI:

[11] Wan X, Ge J, Chen Z. Development

of stable monolithic wide-field Michelson interferometers. Applied Optics. 2011;**50**(21):4105-4114. DOI:

[12] Hassan MA, Martin H, Jiang X. Development of a spatially dispersed short-coherence interferometry sensor using diffraction grating orders: Publisher's note. Applied Optics. 2018; **57**(1):5. DOI: 10.1364/AO.57.000005

[13] Guillen Bonilla JT, Guillen Bonilla A, Rodríguez Betancourtt VM, Guillen Bonilla H, Casillas Zamora A. A theoretical study and numerical

simulation of a quasi-distributed sensor based on the low-finesse Fabry-Perot interferometer: Frequency-division multiplexing. Sensors. 2017;**17**(4):859.

DOI: 10.3390/s17040859

s.18052502

[14] Liang Y, Zhao M, Wu Z,

Morthier G. Investigation of gratingassisted trimodal interferometer

biosensors based on a polymer platform. Sensors. 2018;**18**(5):1502. DOI: 10.3390/

[15] Guillen Bonilla JT, Guillen Bonilla H, Casillas Zamora A, Vega Gómez GA, Franco Rodríguez NE, Guillen Bonilla A, et al. Twin-grating fiber optic sensors applied on wavelength-division multiplexing and its numerical

OE.26.000531

10.1364/OE.22.007799

10-1364/AO.50.004105

[2] Jia X, Liu Z, Deng Z, Deng W, Wzng Z, Zhen Z. Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model. Optics Communication. 2019;**430**:163-169

[3] Peng J, Lyu D, Huang Q, Qu Y, Wang W, Sun T, et al. Dielectric film based optical fiber sensor using Fabry-Perot resonator structure. Optics Communication. 2019;**430**:63-69

[4] Vigneswaran D, Ayyanar VN, Sharma M, Sumahí M, Mani Rajan MS, Porsezian K. Salinity sensor using photonic crystral fiber. Sensors and Actuors A: Physical. 2018;**269**(1):22-28.

DOI: 10.1016/j.sna2017.10.052

[5] Kamenev O, Kulchin YN, Petrov YS, Khiznyak RV, Romashko RV. Fiberoptic seismometer on the basis of Mach-Zehnder interfometer. Sensors and Actuors A: Physical. 2016;**244**:133-137. DOI: 10.1016/j.sna.2016.04.006

[6] Li L, Xia L, Xie Z, Liu D. All-fiber Mach-Zehnder interferometers for sensing applications. Optic Express. 2012;**20**(10):11109-11120. DOI: 10.1364/

[7] Yu Q, Zhou X. Pressure sensor based on the fiber-optic extrinsic Fabry-Perot interferometer. Photonic Sensors. 2011; **1**(1):72-83. DOI: 10.1007/s13320-010-

[8] Yeo TL, Sun T, Grattan KTV. Fibreoptic sensor technologies for humidity and moisture measurement. Sensors and Actuators: Physical. 2008;**144**(2): 280-295. DOI: 10.1016/j.sna.2008.01.017

OE.20.011109

0017-9

**25**

1 Electronic Department, CUCEI, University of Guadalajara, Guadalajara, Jaliscos, Mexico

2 Mathematic Department, CUCEI, University of Guadalajara, Guadalajara, Jaliscos, Mexico

3 Department of Computer Science and Engineering, CUVAlles, University of Guadalajara, Ameca, Jalisco, Mexico

4 Department of Engineering Projects, CUCEI, University of Guadalajara, Guadalajara, Jaliscos, Mexico

5 Chemical Department, CUCEI, University of Guadalajara, Guadalajara, Jaliscos, Mexico

\*Address all correspondence to: trinidad.guillen@academicos.udg.mx

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Interference Pattern Representation on the Complex s-Plane DOI: http://dx.doi.org/10.5772/intechopen.89491*

#### **References**

[1] Zhao N, Lin Q, Jiang Z, Yao K, Tian B, Fang X, et al. High temperature high sensitivity multipoint sensing system based on three cascade Mach-Zehnder interferometers. Sensors. 2018; **18**(8):2688. DOI: 10.3390/s18082688

[2] Jia X, Liu Z, Deng Z, Deng W, Wzng Z, Zhen Z. Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model. Optics Communication. 2019;**430**:163-169

[3] Peng J, Lyu D, Huang Q, Qu Y, Wang W, Sun T, et al. Dielectric film based optical fiber sensor using Fabry-Perot resonator structure. Optics Communication. 2019;**430**:63-69

[4] Vigneswaran D, Ayyanar VN, Sharma M, Sumahí M, Mani Rajan MS, Porsezian K. Salinity sensor using photonic crystral fiber. Sensors and Actuors A: Physical. 2018;**269**(1):22-28. DOI: 10.1016/j.sna2017.10.052

[5] Kamenev O, Kulchin YN, Petrov YS, Khiznyak RV, Romashko RV. Fiberoptic seismometer on the basis of Mach-Zehnder interfometer. Sensors and Actuors A: Physical. 2016;**244**:133-137. DOI: 10.1016/j.sna.2016.04.006

[6] Li L, Xia L, Xie Z, Liu D. All-fiber Mach-Zehnder interferometers for sensing applications. Optic Express. 2012;**20**(10):11109-11120. DOI: 10.1364/ OE.20.011109

[7] Yu Q, Zhou X. Pressure sensor based on the fiber-optic extrinsic Fabry-Perot interferometer. Photonic Sensors. 2011; **1**(1):72-83. DOI: 10.1007/s13320-010- 0017-9

[8] Yeo TL, Sun T, Grattan KTV. Fibreoptic sensor technologies for humidity and moisture measurement. Sensors and Actuators: Physical. 2008;**144**(2): 280-295. DOI: 10.1016/j.sna.2008.01.017 [9] Dong C, Li K, Jiang Y, Arola D, Zhang D. Evaluation of thermal expansion coefficient carbon fiber reinforced composites using electronic speckle interferometry. Optics Express. 2018;**26**(1):531. DOI: 10.1364/ OE.26.000531

[10] Wang S, Gao Z, Li G, Feng Z, Feng Q. Continual mechanical vibration trajectory tracking based on electrooptical heterodyne interferometric. Optics Express. 2014;**22**(7):7799. DOI: 10.1364/OE.22.007799

[11] Wan X, Ge J, Chen Z. Development of stable monolithic wide-field Michelson interferometers. Applied Optics. 2011;**50**(21):4105-4114. DOI: 10-1364/AO.50.004105

[12] Hassan MA, Martin H, Jiang X. Development of a spatially dispersed short-coherence interferometry sensor using diffraction grating orders: Publisher's note. Applied Optics. 2018; **57**(1):5. DOI: 10.1364/AO.57.000005

[13] Guillen Bonilla JT, Guillen Bonilla A, Rodríguez Betancourtt VM, Guillen Bonilla H, Casillas Zamora A. A theoretical study and numerical simulation of a quasi-distributed sensor based on the low-finesse Fabry-Perot interferometer: Frequency-division multiplexing. Sensors. 2017;**17**(4):859. DOI: 10.3390/s17040859

[14] Liang Y, Zhao M, Wu Z, Morthier G. Investigation of gratingassisted trimodal interferometer biosensors based on a polymer platform. Sensors. 2018;**18**(5):1502. DOI: 10.3390/ s.18052502

[15] Guillen Bonilla JT, Guillen Bonilla H, Casillas Zamora A, Vega Gómez GA, Franco Rodríguez NE, Guillen Bonilla A, et al. Twin-grating fiber optic sensors applied on wavelength-division multiplexing and its numerical

**Author details**

Mexico

Mexico

Mexico

**24**

Mario Alberto García Ramírez<sup>1</sup>

Guadalajara, Ameca, Jalisco, Mexico

provided the original work is properly cited.

Guadalajara, Jaliscos, Mexico

Héctor Guillen Bonilla<sup>4</sup>

José Trinidad Guillen Bonilla1,2\*, Alex Guillen Bonilla3

and Verónica María Bettancourt Rodriguez<sup>5</sup>

*Advances in Complex Analysis and Applications*

,

,

, Martín Javier Martínez Silva<sup>1</sup>

, Gustavo Adolfo Vega Gómez<sup>1</sup>

, María Susana Ruiz Palacio<sup>1</sup>

1 Electronic Department, CUCEI, University of Guadalajara, Guadalajara, Jaliscos,

2 Mathematic Department, CUCEI, University of Guadalajara, Guadalajara, Jaliscos,

3 Department of Computer Science and Engineering, CUVAlles, University of

5 Chemical Department, CUCEI, University of Guadalajara, Guadalajara, Jaliscos,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

4 Department of Engineering Projects, CUCEI, University of Guadalajara,

\*Address all correspondence to: trinidad.guillen@academicos.udg.mx

simulation. In: Rao SP, editor. Book of Numerical Simulation in Engineering and Science. 1st ed. IntechOpen. DOI: 10.5772/intechopen.75586

[16] de SAuza JC, Oliveira ME, Dos Santos PAM. Brach-cut algorithm for optical phase unwrapping. Optics Letters. 2015;**40**(15):3456-3459. DOI: 10.1364OL.40.003456

[17] Mizuno T, Kitoh T, Oguma M, Inoue Y, Shibata T, Hiroshi T. Mach-Zehnder interferometer with uniform wavelength period. Optics Letters. 2004;**29**(5):454-456. DOI: 10.1364/ OL.29.000454

[18] Miridonov SV, Shlyagin M, Tentori D. Twin-grating fiber optic sensor demodulation. Optics Communication. 2001;**191**(3–6): 253-362. DOI: 10.1016/S0030-4018(01) 01160-9

[19] Kin JA, Kim JW, Kang CS, Jin J, Eom TB. Interferometric profile scanning system for measuring large planar mirror surface based on singleinterferogram analysis using Fourier transform. Measurements. 2018;**118**: 113-119. DOI: 10.1016/j. measurement.2018.01.023

[20] Perea J, Libbey B, Nehmetallah G. Multiaxis heterodyne vibrometer for simultaneous observation of 5 degrees of dynamic freedom from a single beam. Optics Letters. 2018;**43**(13):3120-2123. DOI: 10.1364/OL.43.003120

[21] Davoodi M, Mezkin N, Khorasani K. A single dynamic observer-based module for design of simultaneous fault detection, isolation and tracking control scheme. International Journal of Control. 2018;**91**(3):508-523. DOI: 10.1080/00207179.2017.1286041

[22] Ordóñez Hurtado RH, Crisostomi E, Shorten RN. An assessment on the use of stationary vehicles to support cooperative positioning systems.

International Journal of Control. 2018; **91**(3):608-621. DOI: 10.1080/ 00207179.2017.1286537

**Chapter 3**

**Abstract**

Problems

*Mozhgan "Nora" Entekhabi*

problems at a fixed frequency.

areas of inverse scattering theory are either ignored.

Bessel functions

**1. Introduction**

**27**

Inverse Scattering Source

The purpose of this chapter is to discuss some of the highlights of the mathematical theory of direct and inverse scattering and inverse source scattering problem for acoustic, elastic and electromagnetic waves. We also briefly explain the uniqueness of the external source for acoustic, elastic and electromagnetic waves equation. However, we must first issue a caveat to the reader. We will also present the recent results for inverse source problems. The resents results including a logarithmic estimate consists of two parts: the Lipschitz part data discrepancy and the high frequency tail of the source function. In general, it is known that due to the existence of non-radiation source, there is no uniqueness for the inverse source

**Keywords:** scattering theory, inverse scattering theory, Helmholtz equation,

This chapter tries to provide some results and materials on inverse scattering, direct scattering theory and inverse source scattering problems. There have been many scientists who have contributed to the different components of this field, such as linearity or non-linearity of the inverse source problem, computational and numerical solution to the inverse source problem and analytical aspects of the problem, which have their own interests. We obviously cannot give a complete account of inverse scattering here from all angles. Hence, instead of attempting the impossible, we have chosen to present inverse scattering theory from the of our own interests and research program. Particularly, we will focus on inverse source problems for acoustic, elastics and electromagnetic waves. In other words, certain

Scattering theory has played a central role in twentieth century mathematical physics and applied mathematics. Indeed, from Rayleigh's explanation of why the sky is blue, to Rutherford's discovery of the atomic nucleus, through the modern medical and clinical applications of computerized tomography, scattering phenomena have attracted scientists and mathematicians for over a hundred years. Broadly speaking, scattering theory is concerned with the effect an inhomogeneous medium has on an incident particle or wave. In particular, if the total field is viewed as the sum of an incident field *u<sup>i</sup>* and a scattered field *us* then the direct scattering problem is to determine *us* form a knowledge of *u<sup>i</sup>* and the differential equation governing

[23] Xu S, Sun G, Li Z. Finite frequency vibration suppression for space flexible structures in tip position control. International Journal of Control, Automation and Systems. 2018;**16**(3): 1021-1029. DOI: 10.1007/ s12555-0160343-9

[24] Wolovich WA. Automatic Control Systems: Basic Analysis and Design. 1st ed. Sounders College Pub.; 1994. 450 p. ISBN: 0030237734, 9780030237737

[25] Andrade Barbosa FA, Nader G, Tokio Higuti R, Kitano C, Nelli Silva E. A simple interferometric method to measure the calibration factor and displacement amplification in piezoelectric flextensional actuators. Revista Controle & Automação. 2010; **21**(6):577-587

#### **Chapter 3**

simulation. In: Rao SP, editor. Book of Numerical Simulation in Engineering and Science. 1st ed. IntechOpen. DOI:

*Advances in Complex Analysis and Applications*

International Journal of Control. 2018;

[23] Xu S, Sun G, Li Z. Finite frequency vibration suppression for space flexible structures in tip position control. International Journal of Control, Automation and Systems. 2018;**16**(3):

[24] Wolovich WA. Automatic Control Systems: Basic Analysis and Design. 1st ed. Sounders College Pub.; 1994. 450 p. ISBN: 0030237734, 9780030237737

[25] Andrade Barbosa FA, Nader G, Tokio Higuti R, Kitano C, Nelli Silva E. A simple interferometric method to measure the calibration factor and displacement amplification in piezoelectric flextensional actuators. Revista Controle & Automação. 2010;

**91**(3):608-621. DOI: 10.1080/ 00207179.2017.1286537

1021-1029. DOI: 10.1007/

s12555-0160343-9

**21**(6):577-587

[16] de SAuza JC, Oliveira ME, Dos Santos PAM. Brach-cut algorithm for optical phase unwrapping. Optics Letters. 2015;**40**(15):3456-3459. DOI:

[17] Mizuno T, Kitoh T, Oguma M, Inoue Y, Shibata T, Hiroshi T. Mach-Zehnder interferometer with uniform wavelength period. Optics Letters. 2004;**29**(5):454-456. DOI: 10.1364/

[18] Miridonov SV, Shlyagin M, Tentori D. Twin-grating fiber optic sensor demodulation. Optics Communication. 2001;**191**(3–6): 253-362. DOI: 10.1016/S0030-4018(01)

[19] Kin JA, Kim JW, Kang CS, Jin J, Eom TB. Interferometric profile scanning system for measuring large planar mirror surface based on singleinterferogram analysis using Fourier transform. Measurements. 2018;**118**:

[20] Perea J, Libbey B, Nehmetallah G. Multiaxis heterodyne vibrometer for simultaneous observation of 5 degrees of dynamic freedom from a single beam. Optics Letters. 2018;**43**(13):3120-2123.

[21] Davoodi M, Mezkin N, Khorasani K. A single dynamic observer-based module for design of simultaneous fault detection, isolation and tracking control

[22] Ordóñez Hurtado RH, Crisostomi E, Shorten RN. An assessment on the use of stationary vehicles to support cooperative positioning systems.

113-119. DOI: 10.1016/j. measurement.2018.01.023

DOI: 10.1364/OL.43.003120

scheme. International Journal of Control. 2018;**91**(3):508-523. DOI: 10.1080/00207179.2017.1286041

**26**

10.5772/intechopen.75586

10.1364OL.40.003456

OL.29.000454

01160-9

## Inverse Scattering Source Problems

*Mozhgan "Nora" Entekhabi*

#### **Abstract**

The purpose of this chapter is to discuss some of the highlights of the mathematical theory of direct and inverse scattering and inverse source scattering problem for acoustic, elastic and electromagnetic waves. We also briefly explain the uniqueness of the external source for acoustic, elastic and electromagnetic waves equation. However, we must first issue a caveat to the reader. We will also present the recent results for inverse source problems. The resents results including a logarithmic estimate consists of two parts: the Lipschitz part data discrepancy and the high frequency tail of the source function. In general, it is known that due to the existence of non-radiation source, there is no uniqueness for the inverse source problems at a fixed frequency.

**Keywords:** scattering theory, inverse scattering theory, Helmholtz equation, Bessel functions

#### **1. Introduction**

This chapter tries to provide some results and materials on inverse scattering, direct scattering theory and inverse source scattering problems. There have been many scientists who have contributed to the different components of this field, such as linearity or non-linearity of the inverse source problem, computational and numerical solution to the inverse source problem and analytical aspects of the problem, which have their own interests. We obviously cannot give a complete account of inverse scattering here from all angles. Hence, instead of attempting the impossible, we have chosen to present inverse scattering theory from the of our own interests and research program. Particularly, we will focus on inverse source problems for acoustic, elastics and electromagnetic waves. In other words, certain areas of inverse scattering theory are either ignored.

Scattering theory has played a central role in twentieth century mathematical physics and applied mathematics. Indeed, from Rayleigh's explanation of why the sky is blue, to Rutherford's discovery of the atomic nucleus, through the modern medical and clinical applications of computerized tomography, scattering phenomena have attracted scientists and mathematicians for over a hundred years. Broadly speaking, scattering theory is concerned with the effect an inhomogeneous medium has on an incident particle or wave. In particular, if the total field is viewed as the sum of an incident field *u<sup>i</sup>* and a scattered field *us* then the direct scattering problem is to determine *us* form a knowledge of *u<sup>i</sup>* and the differential equation governing

the wave motion. There are even more in the inverse scattering problem of determining the nature of the inhomogeneity from a knowledge of the asymptotic behavior of *u<sup>s</sup>* , i.e., to reconstruct the differential equation or its domain or source functions of definition from the behavior of solutions of the direct problems. In this chapter, we are following this notation; *C* denote generic constants depending on the domain Ω or domain *D*, which is different in different results, and k k*u* ð Þ*<sup>l</sup>* ð Þ Ω

denotes the standard norm in Sobolev space *H<sup>l</sup>* ð Þ Ω .

#### **2. The direct and inverse scattering problem**

The stationary incoming wave *u* of frequency *k* is a solution to the perturbed Helmholtz equation (scattering by medium)

$$Au - k^2 u = 0 \,\text{in} \,\mathbb{R}^3 \tag{1}$$

lim*r*!<sup>∞</sup> *<sup>H</sup><sup>s</sup>* � *<sup>x</sup>* � *rE<sup>s</sup>* <sup>ð</sup> Þ ¼ <sup>0</sup> (11)

*ν* � *curlE* � *iλ ν*ð Þ� � *E ν* ¼ 0, (12)

*u x*ð Þ¼ *<sup>u</sup><sup>i</sup>* <sup>þ</sup> *us* (13)

*g y*ð Þ*K x*ð Þ � *y*; *k d*Γð Þ*y* , (15)

*r*2

� � (14)

*<sup>r</sup>* , *<sup>ξ</sup>*, *<sup>k</sup>* � � with fixed

where (7) are the time-harmonic Maxwell equations and *ν* is again the unit outward normal to *∂D*. As in previous case more general boundary condition can

where *λ* is a positive constant. The mathematical technique used to investigate the direct scattering problems for and electromagnetic waves depends heavily on the frequency of the wave motion. The first question about direct scattering is about uniqueness of a solution. The basic tools used to prove the uniqueness are Green's theorems and the unique continuation property of solutions to elliptic equations. Since Eqs. (2)–(5) for exterior problem have constant coefficients, the uniqueness question is much easier to handle. Similar argument can be applied on the Maxwell equations. The first result being given by Sommerfeld in 1912 for the case of acoustic case [1]. His work was generalized by Rellich [2] and Vekua [3], all under the assumption ℑm*k*≥ 0. The uniqueness of a solution to the exterior scattering problems for acoustic and electromagnetic is more difficult since use must now be made of the unique continuation principle for elliptic equations with non-analytic coefficients. After uniqueness, the most important questions would be the existence and numerical approximation of the solution. The most common technique to existence has been through the method of integral equation. For example for Eqs. (2)–(5), it is easy to see that for all positive values of wave number *k* the field *u*

also be considered, for example the impedance boundary condition

*Inverse Scattering Source Problems*

*DOI: http://dx.doi.org/10.5772/intechopen.92023*

is the unique solution of the Lippman-Schwinger equation

1

*<sup>r</sup> exp ikr* ð ÞA *<sup>x</sup>*

where *r* ¼ ∣*x*∣ and the function A is called the scattering amplitude (or the

The representation (14) follows from the fact that any solution *u<sup>s</sup>* to the Helmholtz equation satisfying the radiation condition (5) has the representation by

*r* , *ξ*, *k* � � <sup>þ</sup> *<sup>O</sup>* <sup>1</sup>

*=*ð4*π*∣*x*∣ and *B* is some large ball (i.e., see [4]).

As showed above, the direct scattering problem has been thoroughly investigated and a considerable amount of information is available concerning its solution. In contrast, the inverse scattering problem has only recently progressed. It is worth to mention that the inverse problem is inherently nonlinear. In areas such as radar, sonar, geophysical exploration, medical imaging and nondestructive testing. As in with direct problem, the first question in inverse scattering problem is, how about uniqueness?. The first result in uniqueness brought up to the attention by Schiffer

¼ *exp ik* ð Þþ *ξ* � *x*

*ui* ð Þ¼ *x* ð *∂B*

[5] who showed for the problem (2)–(5) the far field pattern <sup>A</sup> *<sup>x</sup>*

wave number *k* uniquely determines the scattering obstacle *D*. And result for corresponding exterior problem obtained by Nachman [6], Novikov [7, 8]. Uniqueness theorems for electromagnetic problems were obtained by Colton and Päivärinta [9]. The next step will be the question of existence of the to the inverse

scattering pattern or far field pattern).

a single layer potential

where *K x*ð Þ¼ ; *<sup>k</sup> <sup>e</sup>ik*∣*x*<sup>∣</sup>

**29**

(*A* is the elliptic operator *A* ¼ �**∇**ð Þþ *a*∇ *b:*∇ þ *c* with ℜe*b* ¼ 0,**∇***b* ¼ 0, and ℑm*c*≤ 0, which coincides with the Laplace operator outside a ball *B* and which possesses the uniqueness of continuation property) or to the Helmholtz equation (scattering by an obstacle *D* for acoustic waves)

$$
\Delta u + k^2 u = 0 \,\text{in} \,\mathbb{R}^3 \,\langle \overline{D}, \tag{2}
$$

with the Dirichlet boundary data

$$
u = \mathbf{0} \, on \, \partial D \quad (soft \, observable \, D). \tag{3}$$

or the Neumann boundary data

$$
\partial\_\nu u = 0 \, on \, \partial D \quad (hardo \, bsacle \, D). \tag{4}
$$

The function *u* is assumed to be the sum of the so-called incident plane wave *ui* ð Þ¼ *<sup>x</sup>* exp ð Þ *ik<sup>ξ</sup>* � *<sup>x</sup>* and a scattered wave *<sup>u</sup><sup>s</sup>* satisfying the Sommerfeld radiation condition

$$\lim\_{r \to \infty} r \left( \frac{\partial u^{\epsilon}}{\partial r} - iku^{\epsilon} \right) = 0,\tag{5}$$

where *ξ*∈ <sup>3</sup> , ∣*ξ*∣ ¼ 1, is the so-called incident direction and

$$u(\mathfrak{x}) = \exp\left(ik\mathfrak{f}\cdot\mathfrak{x}\right) + u'(\mathfrak{x}).\tag{6}$$

The electromagnetic scattering problem corresponding to the electric field *E* and magnetic field *H* such as

$$
overline{E} - i \overline{k} H = 0, \quad 
overline{H} + i \overline{k} E = 0 \,\text{in} \,\mathbb{R}^3 \langle \overline{D}, \tag{7}$$

$$E(\boldsymbol{\omega}) = \frac{i}{k} \operatorname{curl} \operatorname{curl} \exp\left(ik\boldsymbol{\xi} \cdot \boldsymbol{\omega}\right) + E'(\boldsymbol{\omega}),\tag{8}$$

$$H(\boldsymbol{\kappa}) = \operatorname{curl} \exp\left(ik\xi \cdot \boldsymbol{\kappa}\right) + H'(\boldsymbol{\kappa}),\tag{9}$$

$$
\nu \times E = \mathbf{0} \quad \text{on} \quad \partial D,\tag{10}
$$

with the Silver- Muller radiation condition;

*Inverse Scattering Source Problems DOI: http://dx.doi.org/10.5772/intechopen.92023*

the wave motion. There are even more in the inverse scattering problem of determining the nature of the inhomogeneity from a knowledge of the asymptotic

functions of definition from the behavior of solutions of the direct problems. In this chapter, we are following this notation; *C* denote generic constants depending on the domain Ω or domain *D*, which is different in different results, and k k*u* ð Þ*<sup>l</sup>* ð Þ Ω

The stationary incoming wave *u* of frequency *k* is a solution to the perturbed

(*A* is the elliptic operator *A* ¼ �**∇**ð Þþ *a*∇ *b:*∇ þ *c* with ℜe*b* ¼ 0,**∇***b* ¼ 0, and ℑm*c*≤ 0, which coincides with the Laplace operator outside a ball *B* and which possesses the uniqueness of continuation property) or to the Helmholtz equation

*<sup>u</sup>* <sup>¼</sup> <sup>0</sup>*in*<sup>3</sup>

The function *u* is assumed to be the sum of the so-called incident plane wave

ð Þ¼ *<sup>x</sup>* exp ð Þ *ik<sup>ξ</sup>* � *<sup>x</sup>* and a scattered wave *<sup>u</sup><sup>s</sup>* satisfying the Sommerfeld radiation

*∂us <sup>∂</sup><sup>r</sup>* � *iku<sup>s</sup>* 

, ∣*ξ*∣ ¼ 1, is the so-called incident direction and

*u x*ð Þ¼ exp ð Þþ *ik<sup>ξ</sup>* � *<sup>x</sup> us*

*curlE* � *ikH* <sup>¼</sup> 0, *curlH* <sup>þ</sup> *ikE* <sup>¼</sup> <sup>0</sup>*in*<sup>3</sup>

*H x*ð Þ¼ *curl* exp ð Þþ *ik<sup>ξ</sup>* � *<sup>x</sup> <sup>H</sup><sup>s</sup>*

The electromagnetic scattering problem corresponding to the electric field *E* and

*curl curl* exp ð Þþ *ik<sup>ξ</sup>* � *<sup>x</sup> Es*

*Au* � *<sup>k</sup>*<sup>2</sup>

<sup>Δ</sup>*<sup>u</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

lim*<sup>r</sup>*!<sup>∞</sup> *<sup>r</sup>*

*E x*ð Þ¼ *<sup>i</sup> k*

with the Silver- Muller radiation condition;

, i.e., to reconstruct the differential equation or its domain or source

ð Þ Ω .

*<sup>u</sup>* <sup>¼</sup> <sup>0</sup>*in*<sup>3</sup> (1)

*<sup>u</sup>* <sup>¼</sup> <sup>0</sup>*on∂D soft* ð Þ *obsacle D :* (3)

*<sup>∂</sup>ν<sup>u</sup>* <sup>¼</sup> <sup>0</sup>*on∂D hardo* ð Þ *bsacleD :* (4)

n*D*, (2)

¼ 0, (5)

ð Þ *x :* (6)

n*D*, (7)

ð Þ *x* , (8)

ð Þ *x* , (9)

*<sup>ν</sup>* � *<sup>E</sup>* <sup>¼</sup> <sup>0</sup> *on <sup>∂</sup>D*, (10)

behavior of *u<sup>s</sup>*

denotes the standard norm in Sobolev space *H<sup>l</sup>*

*Advances in Complex Analysis and Applications*

Helmholtz equation (scattering by medium)

(scattering by an obstacle *D* for acoustic waves)

with the Dirichlet boundary data

or the Neumann boundary data

*ui*

**28**

condition

where *ξ*∈ <sup>3</sup>

magnetic field *H* such as

**2. The direct and inverse scattering problem**

$$\lim\_{r \to \infty} (H^t \times \mathfrak{x} - rE^t) = \mathbf{0} \tag{11}$$

where (7) are the time-harmonic Maxwell equations and *ν* is again the unit outward normal to *∂D*. As in previous case more general boundary condition can also be considered, for example the impedance boundary condition

$$
\nu \times \operatorname{curl} E - i\lambda (\nu \times E) \times \nu = \mathbf{0}, \tag{12}
$$

where *λ* is a positive constant. The mathematical technique used to investigate the direct scattering problems for and electromagnetic waves depends heavily on the frequency of the wave motion. The first question about direct scattering is about uniqueness of a solution. The basic tools used to prove the uniqueness are Green's theorems and the unique continuation property of solutions to elliptic equations. Since Eqs. (2)–(5) for exterior problem have constant coefficients, the uniqueness question is much easier to handle. Similar argument can be applied on the Maxwell equations. The first result being given by Sommerfeld in 1912 for the case of acoustic case [1]. His work was generalized by Rellich [2] and Vekua [3], all under the assumption ℑm*k*≥ 0. The uniqueness of a solution to the exterior scattering problems for acoustic and electromagnetic is more difficult since use must now be made of the unique continuation principle for elliptic equations with non-analytic coefficients. After uniqueness, the most important questions would be the existence and numerical approximation of the solution. The most common technique to existence has been through the method of integral equation. For example for Eqs. (2)–(5), it is easy to see that for all positive values of wave number *k* the field *u* is the unique solution of the Lippman-Schwinger equation

$$u(\mathfrak{x}) = u^i + u^s \tag{13}$$

$$=\exp\left(ik\xi\cdot\mathbf{x}\right)+\frac{1}{r}\exp\left(ikr\right)\mathcal{A}\left(\frac{\mathbf{x}}{r},\xi,k\right)+O\left(\frac{1}{r^2}\right)\tag{14}$$

where *r* ¼ ∣*x*∣ and the function A is called the scattering amplitude (or the scattering pattern or far field pattern).

The representation (14) follows from the fact that any solution *u<sup>s</sup>* to the Helmholtz equation satisfying the radiation condition (5) has the representation by a single layer potential

$$u^i(\mathbf{x}) = \int\_{\partial \mathcal{B}} \mathbf{g}(\mathbf{y}) K(\mathbf{x} - \mathbf{y}; k) d\Gamma(\mathbf{y}), \tag{15}$$

where *K x*ð Þ¼ ; *<sup>k</sup> <sup>e</sup>ik*∣*x*<sup>∣</sup> *=*ð4*π*∣*x*∣ and *B* is some large ball (i.e., see [4]).

As showed above, the direct scattering problem has been thoroughly investigated and a considerable amount of information is available concerning its solution. In contrast, the inverse scattering problem has only recently progressed. It is worth to mention that the inverse problem is inherently nonlinear. In areas such as radar, sonar, geophysical exploration, medical imaging and nondestructive testing. As in with direct problem, the first question in inverse scattering problem is, how about uniqueness?. The first result in uniqueness brought up to the attention by Schiffer [5] who showed for the problem (2)–(5) the far field pattern <sup>A</sup> *<sup>x</sup> <sup>r</sup>* , *<sup>ξ</sup>*, *<sup>k</sup>* � � with fixed wave number *k* uniquely determines the scattering obstacle *D*. And result for corresponding exterior problem obtained by Nachman [6], Novikov [7, 8]. Uniqueness theorems for electromagnetic problems were obtained by Colton and Päivärinta [9]. The next step will be the question of existence of the to the inverse

scattering problem. The mathematically speaking, the solution of the inverse scattering problem does not exist, but we can speak about stabilization and approximation of the solutions. The earliest efforts in this direction attempted to linearize the problem by reducing it to the problem of solving a linear integral equation of the first kind. The initial attempts to treat the inverse scattering problem without linearizing were investigated by Imbriale and Mittra [10]. Their techniques were based on analytic continuation. In 1980's a number of methods were given to solving the inverse scattering problem which explicitly acknowledged the nonlinear and ill-posed nature of the problem. The two-dimensional case can be used as an approximation for the scattering from finitely long cylinders. In the next sections, we will discuss Helmholtz equation and two and three dimensional inverse source scattering problems for acoustic, elastic and electromagnetic waves. The following lemma is establishing the uniqueness for the direct solution(1) in **R**<sup>3</sup> . The following lemma is stated in [4].

**Theorem 1.1.** If *u* solves Eq. (1) in <sup>3</sup> and satisfies the radiation condition (5), then *u* ¼ 0.

**Proof.** There is a weak solution to Eq. (1) in *B* with the test function *ϕ* ¼ *u* we have

$$\int\_{\partial B} \partial\_\nu u \overline{u} = \int\_B \left( a \nabla u \cdot \nabla \overline{u} + b \cdot \nabla u \overline{u} + \left( c - k^2 \right) u \overline{u} \right) \tag{16}$$

$$=\int\_{B} \left( a\nabla u \cdot \nabla \overline{u} + \overline{b \cdot \nabla u u} + (c - k^2) u \overline{u} \right) \tag{17}$$

using the condition <sup>ℜ</sup>e*<sup>b</sup>* <sup>¼</sup> 0,**∇***<sup>b</sup>* <sup>¼</sup> 0 and integration by part over *<sup>∂</sup><sup>B</sup>* the internal is a sum of two part; one involving **∇***u* and another *cuu*. The first term coincides with its complex conjugate, so its imaginary part is zero, and the second one has a non-positive imaginary part due to the condition on *c*, hence

$$\mathfrak{Sym}\int\_{\partial\mathcal{B}}\partial\_{\nu}u\overline{u}\le 0.\tag{18}$$

*u x*ð Þ¼ *g x*ð Þ ð Þ *Dirichlet* , *<sup>∂</sup>u x*ð Þ

*G x*ð Þ¼ � *y*

first kind [12]. It is also can be defined as

ð Þ �<sup>1</sup> *<sup>m</sup>* <sup>1</sup>

<sup>2</sup> *<sup>z</sup>* � �<sup>2</sup>*<sup>m</sup>* ð Þ *<sup>m</sup>*! <sup>2</sup> ,

2 *z* � � � � *<sup>J</sup>*0ð Þ� *<sup>z</sup>* <sup>2</sup>

and *γ* ¼ 0*:*5772157 … is the Euler's constant.

*u x*ð Þ¼ , *k*

*u x*ð Þ¼�<sup>ð</sup>

ð Þ¼ *<sup>x</sup>*, *<sup>k</sup>* <sup>1</sup> 2 *ψ*ð Þ� *x*

where *H*<sup>1</sup>

where

**31**

*<sup>J</sup>*0ð Þ¼ *<sup>z</sup>* <sup>X</sup><sup>∞</sup>

*m*¼0

*<sup>Y</sup>*0ð Þ¼ *<sup>z</sup>* <sup>2</sup> *<sup>γ</sup>* <sup>þ</sup> log <sup>1</sup>

�*ui*

**2.2 Inverse source scattering problem**

<sup>0</sup>ð Þ¼ *<sup>z</sup>* <sup>1</sup> *πi* Ð 1 lim*r*!<sup>∞</sup>*<sup>r</sup> d*�1 2 *∂u*

> *i* 4 *H*1

8 >><

>>:

<sup>1</sup>þ*i*∞*eizs <sup>s</sup>*ð Þ <sup>2</sup> � <sup>1</sup> �1*=*<sup>2</sup>

*H*ð Þ<sup>1</sup>

Let *G x*ð Þ be the Green's function for Helmholtz in *d* dimensions, e.g.

*eik x*ð Þ �*<sup>y</sup>* 4*π*∣*x* � *y*∣

> X∞ *m*¼1

Alternatively, we can solve the double layer potential integral equation

*<sup>∂</sup>G x*ð Þ � *<sup>y</sup> ∂n*

Motivated by the significant applications, the inverse source problems, as an important research subject in inverse scattering theory, have continuously attracted much attention by many researchers. Consequently, a great deal of mathematical and numerical results are available. In general, it is known that there is no uniqueness for the inverse source problem at a fixed frequency due to the existence of nonradiation sources. Hence, additional information is required for the source in order to obtain a unique solution, such as to seek the minimum energy solution. From the

ð *∂D*

Then, we can solve the single layer potential integral equation

ð *∂D*

In this case the scattered solution outside *D* is given by

*∂D*

ð Þ �<sup>1</sup> *<sup>m</sup>* <sup>1</sup>

*G x*ð Þ � *<sup>y</sup> <sup>ψ</sup>*ð Þ*<sup>y</sup> dy*, *<sup>x</sup>*<sup>∈</sup> *<sup>D</sup><sup>c</sup>*

*<sup>ψ</sup>*ð Þ*<sup>y</sup> dy*, *<sup>x</sup>*<sup>∈</sup> *<sup>D</sup><sup>c</sup>*

*<sup>∂</sup>G x*ð Þ � *<sup>y</sup> ∂n*

<sup>2</sup> *<sup>z</sup>* � �<sup>2</sup>*<sup>m</sup>* ð Þ *<sup>m</sup>*! <sup>2</sup> <sup>1</sup> <sup>þ</sup>

and it is a well-posed problem if

*DOI: http://dx.doi.org/10.5772/intechopen.92023*

*Inverse Scattering Source Problems*

*<sup>∂</sup><sup>ν</sup>* <sup>¼</sup> *h x*ð Þ, *<sup>x</sup>*∈*∂D*, ð Þ *Neumann* (20)

*<sup>∂</sup><sup>r</sup>* � *iku* � � <sup>¼</sup> <sup>0</sup>*:* (21)

*ds*, for *Rez* >0, is the Hankel function of the

1 <sup>2</sup> <sup>þ</sup> … <sup>þ</sup>

� �,

*<sup>ψ</sup>*ð Þ*<sup>y</sup> dy*, *<sup>x</sup>*∈*∂D:* (25)

1 *m*

*:* (24)

*:* (26)

(23)

<sup>0</sup>ð Þ *k*j*x* � *y*j , *d* ¼ 2, *x* 6¼ *y*

, *d* ¼ 3, *x* 6¼ *y*

<sup>0</sup> ð Þ¼ *z J*0ð Þþ *z iY*0ð Þ*z* , (22)

since *u* satisfied the Helmholtz equation and the radiation condition, the known results imply that *u* ¼ 0 outside *B*. By uniqueness of the continuation for the elliptic operator *<sup>A</sup>* � *<sup>k</sup>*<sup>2</sup> we obtain that *<sup>u</sup>* <sup>¼</sup> 0 in <sup>3</sup> , so the proof is complete.

#### **2.1 Helmholtz equation**

Studying an inverse problem always requires a solid knowledge of the theory for the corresponding direct problem. Therefore in this section is devoted to presenting the foundations of obstacle scattering problems for time harmonic acoustic waves. The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. Colton and Kress showed that [11] how one can derived the Helmholtz equation from the Euler's equation. Then the domain of the solution is outside a bounded open set *D* ∈ *<sup>d</sup>*, describing the scatterer. The equation is

$$
\Delta u + k^2 u = 0 \tag{19}
$$

where the wave number *k* is given by the positive constant *k* ¼ *ω=c*, with inhomogeneous boundary conditions on *D* of Dirichlet or Neumann type:

*Inverse Scattering Source Problems DOI: http://dx.doi.org/10.5772/intechopen.92023*

$$u(\mathbf{x}) = \mathbf{g}(\mathbf{x}) \quad (Dirichlet), \quad \frac{\partial u(\mathbf{x})}{\partial \nu} = h(\mathbf{x}), \mathbf{x} \in \partial D, \quad (Neumann) \tag{20}$$

and it is a well-posed problem if

$$\lim\_{r \to \infty} r^{\frac{d-1}{2}} \left( \frac{\partial u}{\partial r} - iku \right) = 0. \tag{21}$$

Let *G x*ð Þ be the Green's function for Helmholtz in *d* dimensions, e.g.

$$G(\mathbf{x} - \mathbf{y}) = \begin{cases} \frac{i}{4} H\_0^1(k|\mathbf{x} - \mathbf{y}|), & d = 2, \quad \mathbf{x} \neq \mathbf{y} \\\\ \frac{e^{ik(\mathbf{x} - \mathbf{y})}}{4\pi|\mathbf{x} - \mathbf{y}|}, & d = 3, \quad \mathbf{x} \neq \mathbf{y} \end{cases}$$

where *H*<sup>1</sup> <sup>0</sup>ð Þ¼ *<sup>z</sup>* <sup>1</sup> *πi* Ð 1 <sup>1</sup>þ*i*∞*eizs <sup>s</sup>*ð Þ <sup>2</sup> � <sup>1</sup> �1*=*<sup>2</sup> *ds*, for *Rez* >0, is the Hankel function of the first kind [12]. It is also can be defined as

$$H\_0^{(1)}(\mathbf{z}) = J\_0(\mathbf{z}) + iY\_0(\mathbf{z}),\tag{22}$$

where

scattering problem. The mathematically speaking, the solution of the inverse scattering problem does not exist, but we can speak about stabilization and approximation of the solutions. The earliest efforts in this direction attempted to linearize the problem by reducing it to the problem of solving a linear integral equation of the first kind. The initial attempts to treat the inverse scattering problem without linearizing were investigated by Imbriale and Mittra [10]. Their techniques were based on analytic continuation. In 1980's a number of methods were given to solving the inverse scattering problem which explicitly acknowledged the nonlinear and ill-posed nature of the problem. The two-dimensional case can be used as an approximation for the scattering from finitely long cylinders. In the next sections, we will discuss Helmholtz equation and two and three dimensional inverse source scattering problems for acoustic, elastic and electromagnetic waves. The following

**Theorem 1.1.** If *u* solves Eq. (1) in <sup>3</sup> and satisfies the radiation condition (5),

**Proof.** There is a weak solution to Eq. (1) in *B* with the test function *ϕ* ¼ *u* we have

using the condition <sup>ℜ</sup>e*<sup>b</sup>* <sup>¼</sup> 0,**∇***<sup>b</sup>* <sup>¼</sup> 0 and integration by part over *<sup>∂</sup><sup>B</sup>* the internal is a sum of two part; one involving **∇***u* and another *cuu*. The first term coincides with its complex conjugate, so its imaginary part is zero, and the second one has a

since *u* satisfied the Helmholtz equation and the radiation condition, the known results imply that *u* ¼ 0 outside *B*. By uniqueness of the continuation for the elliptic

Studying an inverse problem always requires a solid knowledge of the theory for the corresponding direct problem. Therefore in this section is devoted to presenting the foundations of obstacle scattering problems for time harmonic acoustic waves. The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. Colton and Kress showed that [11] how one can derived the Helmholtz equation from the Euler's equation. Then the domain of the solution is outside a

*<sup>a</sup>*∇*<sup>u</sup>* � <sup>∇</sup>*<sup>u</sup>* <sup>þ</sup> *<sup>b</sup>* � <sup>∇</sup>*uu* <sup>þ</sup> *<sup>c</sup>* � *<sup>k</sup>*<sup>2</sup> � �*uu* � � (16)

*<sup>a</sup>*∇*<sup>u</sup>* � <sup>∇</sup>*<sup>u</sup>* <sup>þ</sup> *<sup>b</sup>* � <sup>∇</sup>*uu* <sup>þ</sup> *<sup>c</sup>* � *<sup>k</sup>*<sup>2</sup> � �*uu* � � (17)

, so the proof is complete.

*∂νuu*≤ 0*:* (18)

*u* ¼ 0 (19)

. The following

lemma is establishing the uniqueness for the direct solution(1) in **R**<sup>3</sup>

lemma is stated in [4].

ð *∂B*

*Advances in Complex Analysis and Applications*

operator *<sup>A</sup>* � *<sup>k</sup>*<sup>2</sup> we obtain that *<sup>u</sup>* <sup>¼</sup> 0 in <sup>3</sup>

**2.1 Helmholtz equation**

**30**

*<sup>∂</sup>νuu* <sup>¼</sup>

ð *B*

¼ ð *B*

non-positive imaginary part due to the condition on *c*, hence

ℑm ð *∂B*

bounded open set *D* ∈ *<sup>d</sup>*, describing the scatterer. The equation is

<sup>Δ</sup>*<sup>u</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

where the wave number *k* is given by the positive constant *k* ¼ *ω=c*, with inhomogeneous boundary conditions on *D* of Dirichlet or Neumann type:

then *u* ¼ 0.

$$\begin{aligned} J\_0(\mathbf{z}) &= \sum\_{m=0}^{\infty} \frac{(-1)^m \left(\frac{1}{2}\mathbf{z}\right)^{2m}}{\left(m!\right)^2}, \\ Y\_0(\mathbf{z}) &= 2\left\{\chi + \log\left(\frac{1}{2}\mathbf{z}\right)\right\} J\_0(\mathbf{z}) - 2\sum\_{m=1}^{\infty} \frac{(-1)^m \left(\frac{1}{2}\mathbf{z}\right)^{2m}}{\left(m!\right)^2} \left\{1 + \frac{1}{2} + \dots + \frac{1}{m}\right\}, \end{aligned} \tag{23}$$

and *γ* ¼ 0*:*5772157 … is the Euler's constant.

Then, we can solve the single layer potential integral equation

$$\mu(\varkappa,k) = \int\_{\partial \mathcal{D}} G(\varkappa - \jmath)\nu(\jmath)d\jmath, \qquad \varkappa \in \overline{\mathcal{D}}^{\uparrow}. \tag{24}$$

Alternatively, we can solve the double layer potential integral equation

$$-u^i(\mathbf{x},k) = \frac{1}{2}\boldsymbol{\nu}(\mathbf{x}) - \int\_{\partial\mathcal{D}} \frac{\partial G(\mathbf{x}-\boldsymbol{\mathcal{y}})}{\partial \boldsymbol{n}} \boldsymbol{\nu}(\mathbf{y})d\boldsymbol{\mathcal{y}}, \qquad \boldsymbol{x} \in \partial\mathcal{D}.\tag{25}$$

In this case the scattered solution outside *D* is given by

$$\mu(\varkappa) = -\int\_{\partial D} \frac{\partial G(\varkappa - y)}{\partial n} \varphi(y) dy, \qquad \varkappa \in \overline{D}^{\varepsilon}. \tag{26}$$

#### **2.2 Inverse source scattering problem**

Motivated by the significant applications, the inverse source problems, as an important research subject in inverse scattering theory, have continuously attracted much attention by many researchers. Consequently, a great deal of mathematical and numerical results are available. In general, it is known that there is no uniqueness for the inverse source problem at a fixed frequency due to the existence of nonradiation sources. Hence, additional information is required for the source in order to obtain a unique solution, such as to seek the minimum energy solution. From the numerical and computational point of view, a more challenging issue is lack of stability. A small variation of the data might lead to a huge error in the reconstruction. Recently, it has been realized that the use of multi-frequency data is an effective approach to overcome the difficulties of non- uniqueness and instability which are encountered at a single frequency. An attempt was made in [13] to extend the stability results to the inverse random source of the one-dimensional stochastic Helmholtz equation. The inverse source problem seeks for the right hand side of a partial differential equation from boundary data. The inverse source problems are also considered as a basic mathematical tool for solving many imaging problems including reflection tomography, diffusion-based optical tomography, lidar imaging for chemical and biological threat detection, and fluorescence microscopy. In general, a feature of inverse problems for elliptic equations is a logarithmic type stability estimate which results in a robust recovery of only few parameters describing the source and hence yields very low resolution numerically.

where Γ is an non empty open subset of ∂Ω with outer unit normal *ν* and

ð Þ<sup>1</sup> ð Þ <sup>Ω</sup> <sup>≤</sup>*<sup>C</sup> <sup>ε</sup>*<sup>2</sup> <sup>þ</sup>

ð Þ <sup>0</sup> ð Þ ∂Ω � �*dω*, 0 <sup>&</sup>lt;*<sup>E</sup>* ¼ � ln <sup>ϵ</sup>*:* (31)

<sup>Δ</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *ikb* � �*<sup>u</sup>* ¼ �*f*<sup>1</sup> � *bf* <sup>0</sup> <sup>þ</sup> *ikf* <sup>0</sup> (32)

*<sup>ε</sup>*<sup>2</sup> <sup>þ</sup>

ð Þ <sup>Ω</sup> solving (1), with 1<*<sup>K</sup>* and *<sup>M</sup>*<sup>3</sup> <sup>¼</sup> max <sup>∥</sup> *<sup>f</sup>*0∥ð Þ <sup>4</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup> *<sup>f</sup>*1∥ð Þ<sup>3</sup> ð Þ <sup>Ω</sup> , 1 n o.

*<sup>b</sup>*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �*M*<sup>2</sup>

**<sup>u</sup>** ¼ �**<sup>f</sup>** <sup>1</sup> � <sup>i</sup>*k***f**<sup>0</sup> *in <sup>n</sup>*, (34) **u** ¼ **u**<sup>0</sup> in Γ, (35)

ð Þ Ω are the external force are assumed to be compactly

<sup>1</sup> <sup>þ</sup> *<sup>K</sup>*<sup>2</sup> 3*E* 1 <sup>4</sup> þ *b*

!

3

(33)

While Bao, Li and Lu used Dirichlet to Neumann map to simplify the boundary conditions for two dimensional and three dimensional domains (disks and balls), Isakov, Lu, Chang and Entekhabi used the Fourier transform and observability bound for corresponding hyperbolic initial value boundary problem (wave equa-

authors considered inverse source scattering problems with damping factor for two and three dimensional domains, that is, they considered the following equation:

where *b*>0 is the damping factor. In particular attenuation can have various reasons and in application, one of the fundamental reasons of poor resolution in inverse problems is a spatial decay of the signal due in part to the damping factor.

**Theorem 1.3.** There exists a generic constant *C* depending on the domain Ω such

ð Þ<sup>1</sup> ð Þ <sup>Ω</sup> <sup>≤</sup>*CeCb*<sup>2</sup>

In papers [21, 22], authors considered inverse source scattering problems for double layers medium. The results in the papers [23, 24] showed an stability estimate for elastic and electromagnetic waves. Also authors in [23] proved a stability estimate using just Dirichlet data. Increasing stability for the Schrodinger potential from the complete set of the boundary data (the Dirichlet-to Neumann map) was demonstrated in [25, 26]. They showed that the boundary condition for elastic

where *σ* ¼ ð Þ *μ*Δ þ ð Þ *μ* þ *λ* ∇ � ∇ , where *μ*, *λ* are Lame constants satisfying *μ*>0 and


As you can see, the results showed a deterioration of stability with growing

ð Þ<sup>3</sup> ð Þ <sup>Ω</sup> <sup>≤</sup> *<sup>M</sup>*, 1<sup>≤</sup> *<sup>M</sup>*, and *<sup>δ</sup>*<sup>&</sup>lt; <sup>∣</sup>*<sup>x</sup>* � *<sup>y</sup>*∣, *<sup>x</sup>*∈ ∂Ω,

*M*<sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>K</sup>*<sup>2</sup> 3*E* 1 4 � � (30)


ð Þ <sup>4</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup> *<sup>f</sup>*1∥<sup>2</sup>

ð Þ <sup>0</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup> *<sup>f</sup>*0∥<sup>2</sup>

ð Þ Ω solving (27) and (28) with 1<*K*. Here

ð Þ <sup>0</sup> ð Þþ <sup>∂</sup><sup>Ω</sup> ∥∇*u*ð Þ , *<sup>ω</sup>* <sup>∥</sup><sup>2</sup>

Then there exist a constant *C* ¼ *C*ð Þ Ω, *δ* such that

0<*K* <sup>∗</sup> <*K*, was the following theorem;

*DOI: http://dx.doi.org/10.5772/intechopen.92023*

*y*∈*suppf* <sup>0</sup> ∪ *suppf* <sup>1</sup> for some positive *δ*.

∥ *f*1∥<sup>2</sup>

<sup>∥</sup>*u*ð Þ ,*<sup>ω</sup>* <sup>∥</sup><sup>2</sup>

The results was the following theorem:

∥ *f*0∥<sup>2</sup>

attenuation/damping constant *b*.

*<sup>μ</sup>* <sup>þ</sup> *<sup>λ</sup>*>0, functions **<sup>f</sup> <sup>1</sup>**,**f0** <sup>∈</sup>*L*<sup>2</sup>

supported in a *C*<sup>2</sup>

**33**

waves they considered the following equation

*<sup>σ</sup>*ð Þþ **<sup>u</sup>** *<sup>k</sup>*<sup>2</sup>

the Helmholtz decomposition, the displacement filed **u** can be written as

ð Þ<sup>1</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup> *<sup>f</sup>*1∥<sup>2</sup>

tion) for two and three dimensional domain with *C*<sup>2</sup>

**Theorem 1.2.** Let ∥ *f*0∥<sup>2</sup>

*Inverse Scattering Source Problems*

for all *u* ∈ *H*<sup>2</sup>

<sup>ϵ</sup><sup>2</sup> <sup>¼</sup> ð*K* 0 *ω*2

that

for all *u* ∈ *H*<sup>2</sup>

For the Helmholtz equations, the results have shown increasing (getting nearly Lipschitz) stability when the Dirichlet data or Cauchy data are given on the whole boundary and K is getting large. Similar results are obtained for the time periodic solutions of the more complicated dynamical elasticity system. For elastic waves, the inverse source problem is to determine the external force that produces the measured displacement. The inverse source scattering problem for Maxwell equation arises in many scientific areas such as medical imaging. More specifically, Magnetoencephalography (MEG), the imaging modality is a non-invasive neurophysiological technique that measures the electric or magnetic fields generated by neuronal activity of the brain. For electromagnetic waves, the inverse source problem is to reconstruct the electric current density from tangential trace of electric field. As we know in [14], the inverse source problem does not have a unique solution at a single or at finitely many wave numbers. On the other hand, if we use all wave numbers in 0, ð Þ *K* one can regain uniqueness. Another purpose of this chapter is to establish uniqueness for the source from the Cauchy data on any open non empty part of the boundary for arbitrary positive *K*. For uniqueness, we will show two different techniques. The first technique is to use the stability estimate for the source functions and the second technique is a direct proof.

First increasing stability results were obtained in [15] by using the spatial Fourier transform. In [16, 17] more general and sharp results were obtained in sub-domain of <sup>3</sup> *and* <sup>2</sup> in an arbitrary domains with *C*<sup>2</sup> boundary by the temporal Fourier transform, with a possibility of handling spatially variable coefficients. The recent results showed that the estimate for source functions is a logarithmic type. The right handside of the estimate consists of two parts: data discrepancy and the high frequency tail. In the papers [15, 18], Li, Bao and others showed the similar results for disc and ball. For instance, the results by Entekhabi and Isakov are as follows;

Let the radiated wave field *u x*ð Þ , *<sup>k</sup>* solve the scattering problem in <sup>2</sup> with the source term �*f*<sup>1</sup> � *ikf* <sup>0</sup> and the radiation condition

$$(\Delta + k^2)u = -f\_1 - ikf\_0 \text{ in } \mathbb{R}^2,\tag{27}$$

$$\lim r^{1/2} (\partial\_r u - iku) = 0 \text{ as } r = |\mathfrak{x}| \to +\infty. \tag{28}$$

Both *f*0, *f*<sup>1</sup> ∈*L*<sup>2</sup> ð Þ Ω are assumed having *suppf* <sup>0</sup>, *suppf* <sup>1</sup> ⊂ Ω where Ω is a bounded domain with the boundary ∂Ω ∈*C*<sup>2</sup> .

The stability of functions *f*0, *f*<sup>1</sup> from the data

$$
\mu = u\_0, \partial\_\nu u = u\_1 \text{on } \Gamma \text{, when } K\_\* < k < K,\tag{29}
$$

numerical and computational point of view, a more challenging issue is lack of stability. A small variation of the data might lead to a huge error in the reconstruction. Recently, it has been realized that the use of multi-frequency data is an effective approach to overcome the difficulties of non- uniqueness and instability which are encountered at a single frequency. An attempt was made in [13] to extend the stability results to the inverse random source of the one-dimensional stochastic Helmholtz equation. The inverse source problem seeks for the right hand side of a partial differential equation from boundary data. The inverse source problems are also considered as a basic mathematical tool for solving many imaging problems including reflection tomography, diffusion-based optical tomography, lidar imaging for chemical and biological threat detection, and fluorescence microscopy. In general, a feature of inverse problems for elliptic equations is a logarithmic type stability estimate which results in a robust recovery of only few parameters describing the source and hence yields very low resolution numerically.

*Advances in Complex Analysis and Applications*

For the Helmholtz equations, the results have shown increasing (getting nearly Lipschitz) stability when the Dirichlet data or Cauchy data are given on the whole boundary and K is getting large. Similar results are obtained for the time periodic solutions of the more complicated dynamical elasticity system. For elastic waves, the inverse source problem is to determine the external force that produces the measured displacement. The inverse source scattering problem for Maxwell equation arises in many scientific areas such as medical imaging. More specifically, Magnetoencephalography (MEG), the imaging modality is a non-invasive neurophysiological technique that measures the electric or magnetic fields generated by neuronal activity of the brain. For electromagnetic waves, the inverse source problem is to reconstruct the electric current density from tangential trace of electric field. As we know in [14], the inverse source problem does not have a unique solution at a single or at finitely many wave numbers. On the other hand, if we use all wave numbers in 0, ð Þ *K* one can regain uniqueness. Another purpose of this chapter is to establish uniqueness for the source from the Cauchy data on any open non empty part of the boundary for arbitrary positive *K*. For uniqueness, we will show two different techniques. The first technique is to use the stability estimate for

First increasing stability results were obtained in [15] by using the spatial Fourier transform. In [16, 17] more general and sharp results were obtained in sub-domain of <sup>3</sup> *and* <sup>2</sup> in an arbitrary domains with *C*<sup>2</sup> boundary by the temporal Fourier transform, with a possibility of handling spatially variable coefficients. The recent results showed that the estimate for source functions is a logarithmic type. The right handside of the estimate consists of two parts: data discrepancy and the high frequency tail. In the papers [15, 18], Li, Bao and others showed the similar results for disc and

Let the radiated wave field *u x*ð Þ , *<sup>k</sup>* solve the scattering problem in <sup>2</sup> with the

<sup>Δ</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *<sup>u</sup>* ¼ �*f*<sup>1</sup> � *ikf* <sup>0</sup> in <sup>2</sup>

, (27)

ð Þ¼ *<sup>∂</sup>ru* � *iku* 0 as *<sup>r</sup>* <sup>¼</sup> <sup>∣</sup>*x*<sup>∣</sup> ! þ∞*:* (28)

ð Þ Ω are assumed having *suppf* <sup>0</sup>, *suppf* <sup>1</sup> ⊂ Ω where Ω is a bounded

*<sup>u</sup>* <sup>¼</sup> *<sup>u</sup>*0, *<sup>∂</sup>ν<sup>u</sup>* <sup>¼</sup> *<sup>u</sup>*<sup>1</sup> on <sup>Γ</sup>, when *<sup>K</sup>* <sup>∗</sup> <sup>&</sup>lt;*k*<*K*, (29)

the source functions and the second technique is a direct proof.

ball. For instance, the results by Entekhabi and Isakov are as follows;

.

source term �*f*<sup>1</sup> � *ikf* <sup>0</sup> and the radiation condition

lim *r* 1*=*2

The stability of functions *f*0, *f*<sup>1</sup> from the data

Both *f*0, *f*<sup>1</sup> ∈*L*<sup>2</sup>

**32**

domain with the boundary ∂Ω ∈*C*<sup>2</sup>

where Γ is an non empty open subset of ∂Ω with outer unit normal *ν* and 0<*K* <sup>∗</sup> <*K*, was the following theorem;

**Theorem 1.2.** Let ∥ *f*0∥<sup>2</sup> ð Þ <sup>4</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup> *<sup>f</sup>*1∥<sup>2</sup> ð Þ<sup>3</sup> ð Þ <sup>Ω</sup> <sup>≤</sup> *<sup>M</sup>*, 1<sup>≤</sup> *<sup>M</sup>*, and *<sup>δ</sup>*<sup>&</sup>lt; <sup>∣</sup>*<sup>x</sup>* � *<sup>y</sup>*∣, *<sup>x</sup>*∈ ∂Ω, *y*∈*suppf* <sup>0</sup> ∪ *suppf* <sup>1</sup> for some positive *δ*.

Then there exist a constant *C* ¼ *C*ð Þ Ω, *δ* such that

$$\left\|\|f\_1\|\right\|\_{\left(0\right)}^2(\Omega) + \left\|f\_0\right\|\_{\left(1\right)}^2(\Omega) \le C\left(\varepsilon^2 + \frac{M^2}{\mathbf{1} + K^\ddagger E^\ddagger}\right) \tag{30}$$

for all *u* ∈ *H*<sup>2</sup> ð Þ Ω solving (27) and (28) with 1<*K*. Here

$$\epsilon^2 = \int\_0^\mathbb{K} \left( \alpha^2 \|\mu(\cdot, \alpha)\|^2\_{(0)}(\partial \Omega) + \|\nabla u(\cdot, \alpha)\|^2\_{(0)}(\partial \Omega) \right) d\alpha, \quad 0 < E = -\ln \epsilon. \tag{31}$$

While Bao, Li and Lu used Dirichlet to Neumann map to simplify the boundary conditions for two dimensional and three dimensional domains (disks and balls), Isakov, Lu, Chang and Entekhabi used the Fourier transform and observability bound for corresponding hyperbolic initial value boundary problem (wave equation) for two and three dimensional domain with *C*<sup>2</sup> -boundary. In papers [19, 20], authors considered inverse source scattering problems with damping factor for two and three dimensional domains, that is, they considered the following equation:

$$(\Delta + k^2 + ikb)u = -f\_1 - bf\_0 + ikf\_0 \tag{32}$$

where *b*>0 is the damping factor. In particular attenuation can have various reasons and in application, one of the fundamental reasons of poor resolution in inverse problems is a spatial decay of the signal due in part to the damping factor. The results was the following theorem:

**Theorem 1.3.** There exists a generic constant *C* depending on the domain Ω such that

$$\|f\_0\|\_{\left(1\right)}^2(\Omega) + \|f\_1\|\_{\left(1\right)}^2(\Omega) \le C e^{Cb^2} \left(e^2 + \frac{\left(b^2 + 1\right)M\_3^2}{1 + K^\sharp E^\ddagger + b}\right) \tag{33}$$

for all *u* ∈ *H*<sup>2</sup> ð Þ <sup>Ω</sup> solving (1), with 1<*<sup>K</sup>* and *<sup>M</sup>*<sup>3</sup> <sup>¼</sup> max <sup>∥</sup> *<sup>f</sup>*0∥ð Þ <sup>4</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup> *<sup>f</sup>*1∥ð Þ<sup>3</sup> ð Þ <sup>Ω</sup> , 1 n o. As you can see, the results showed a deterioration of stability with growing

attenuation/damping constant *b*.

In papers [21, 22], authors considered inverse source scattering problems for double layers medium. The results in the papers [23, 24] showed an stability estimate for elastic and electromagnetic waves. Also authors in [23] proved a stability estimate using just Dirichlet data. Increasing stability for the Schrodinger potential from the complete set of the boundary data (the Dirichlet-to Neumann map) was demonstrated in [25, 26]. They showed that the boundary condition for elastic waves they considered the following equation

$$
\sigma(\mathbf{u}) + k^2 \mathbf{u} = -\mathbf{f}\_1 - \mathrm{i}k \mathbf{f}\_0 \text{ in } \mathbb{R}^n,\tag{34}
$$

$$\mathbf{u} = \mathbf{u}\_0 \text{ in } \Gamma,\tag{35}$$

where *σ* ¼ ð Þ *μ*Δ þ ð Þ *μ* þ *λ* ∇ � ∇ , where *μ*, *λ* are Lame constants satisfying *μ*>0 and *<sup>μ</sup>* <sup>þ</sup> *<sup>λ</sup>*>0, functions **<sup>f</sup> <sup>1</sup>**,**f0** <sup>∈</sup>*L*<sup>2</sup> ð Þ Ω are the external force are assumed to be compactly supported in a *C*<sup>2</sup> -boundary domain Ω ⊂ *<sup>n</sup>* and Γ⊂ ∂Ω is an open non-void set. By the Helmholtz decomposition, the displacement filed **u** can be written as

$$\mathbf{u} = \mathbf{u}\_p + \mathbf{u}, \text{ in } \mathbb{R}^n \backslash \overline{\Omega}, \tag{36}$$

*∂*2

*DOI: http://dx.doi.org/10.5772/intechopen.92023*

*Inverse Scattering Source Problems*

Laplace transform we conclude that

Now let

will obtain

**3. Conclusions**

**35**

*<sup>t</sup> U*<sup>0</sup> � Δ*U*<sup>0</sup> ¼ 0 on Ω � ð Þ 0, ∞ ,

<sup>∥</sup>*U*0ð Þ , *<sup>t</sup>* <sup>∥</sup>ð Þ<sup>1</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup>*∂tU*0ð Þ , *<sup>t</sup>* <sup>∥</sup>ð Þ <sup>0</sup> ð Þ <sup>Ω</sup> <sup>≤</sup>*<sup>C</sup>* <sup>∥</sup>*f*0∥ð Þ<sup>1</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup>*f*1∥ð Þ <sup>0</sup> ð Þ <sup>Ω</sup>

*<sup>u</sup>*<sup>∗</sup> ð Þ¼ *<sup>x</sup>*, *<sup>k</sup>* <sup>1</sup>

*<sup>U</sup>*<sup>0</sup> ¼ �*f*0, *<sup>∂</sup>tU*<sup>0</sup> <sup>¼</sup> *<sup>f</sup>*<sup>1</sup> on <sup>Ω</sup> � f g <sup>0</sup> , *<sup>U</sup>*<sup>0</sup> <sup>¼</sup> 0 on <sup>∂</sup><sup>Ω</sup> � ð Þ 0, <sup>þ</sup><sup>∞</sup> *:* (40)

� �.

*U*0ð Þ *x*, *t e*

<sup>Δ</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> � �*u*<sup>∗</sup> ¼ �*f*<sup>1</sup> � *ikf* <sup>0</sup> in <sup>Ω</sup>, *<sup>u</sup>*<sup>∗</sup> <sup>¼</sup> 0 on <sup>∂</sup>Ω*:* (41)

*iktdt*

Under their assumptions, there is a unique solution to the problem (40) with

ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup>

ð<sup>∞</sup> 0

Due to the properties of *U*0, in particular to the conservation of the energy, the function *<sup>u</sup>*<sup>∗</sup> ð Þ *<sup>x</sup>*, *<sup>k</sup>* is well defined and analytic with respect to *<sup>k</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>1</sup> <sup>þ</sup> *ik*2, *<sup>k</sup>*<sup>2</sup> <sup>&</sup>gt;0. Applying the integration by parts and using standard properties of the Fourier-

Due to the assumption, the function *u* solves the same Dirichlet problem for <sup>Δ</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> when 0<sup>&</sup>lt; *<sup>k</sup>*1, 0 <sup>&</sup>lt;*k*2. Indeed, *<sup>u</sup>* solves the homogeneous Helmholtz equation in *<sup>n</sup>*n<sup>Ω</sup> and has zero Cauchy data on <sup>Γ</sup>. By the uniqueness in the Cauchy problem for elliptic equations, *<sup>u</sup>* <sup>¼</sup> 0 on *<sup>n</sup>*n<sup>Ω</sup> and hence on <sup>∂</sup><sup>Ω</sup> provided *<sup>K</sup>*<sup>∗</sup> <sup>&</sup>lt;*k*<sup>&</sup>lt; *<sup>K</sup>*. As follows from the integral representation of solution (27), the function *u*ð Þ ; *k* is (complex) analytic when 0 < ℜe*k*, hence *u*ð Þ¼ ; *k* 0 on ∂Ω provided 0< ℜe*k*. Since *<sup>k</sup>*<sup>2</sup> <sup>&</sup>gt;0, the solution of (41) is unique, hence *<sup>u</sup>* <sup>¼</sup> *<sup>u</sup>*<sup>∗</sup> on <sup>Ω</sup> (see Section 4). Consequently, we obtain *<sup>u</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>u</sup>* <sup>¼</sup> 0, *<sup>∂</sup>νu*<sup>∗</sup> <sup>¼</sup> *<sup>∂</sup>ν<sup>u</sup>* <sup>¼</sup> 0 on <sup>Γ</sup>. Since *<sup>u</sup>*<sup>∗</sup> is an analytic function, we can conclude that *<sup>u</sup>*<sup>∗</sup> <sup>¼</sup> 0, *<sup>∂</sup>νu*<sup>∗</sup> <sup>¼</sup> 0 on <sup>Γ</sup> for all *<sup>k</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>1</sup> <sup>þ</sup> *ik*<sup>2</sup> with *k*<sup>2</sup> >0. Due to the uniqueness of the inversion of the Fourier-Laplace transform we

*<sup>∂</sup>νU*<sup>0</sup> <sup>¼</sup> 0 on <sup>Γ</sup> � ð Þ 0, <sup>∞</sup> *:*

at Ω � f g *T* we conclude that *U*<sup>0</sup> ¼ 0 on Ω � ð Þ 0, *T* . So �*U*0ð Þ¼ , 0 *f*<sup>0</sup> ¼ 0, *<sup>∂</sup>tU*ð Þ¼ , 0 *<sup>f</sup>*<sup>1</sup> <sup>¼</sup> 0 on <sup>Ω</sup> which finishes the proof of uniqueness.

Due to the uniqueness in the lateral Cauchy problem for the wave equation (40) with the Cauchy data on Γ � ð Þ 0, þ∞ [Holmgren-John theorem ([28], Section 3.4)], we can conclude that *U*<sup>0</sup> ¼ 0 on Ω � ð Þ *T*, þ∞ for some positive *T*. Hence from the uniqueness in the backward initial boundary value problem for the hyperbolic equation (39) in Ω � ð Þ 0, *T* with zero boundary data on ∂Ω � ð Þ 0, *T* and initial data

In this section, the scattering and inverse scattering theory, inverse source scattering problem were considered briefly. The recent results such stability estimates for external source and electric current density from boundary measurements of radiated wave field and uniqueness for source function for Helmholtz equation, Elasticity and Maxwell system have showed. We also show some result for discrete data. In addition, we also showed some results of using just Dirichlet data for improving stability which was a big improvement. There are still many challenges remain in this field. For instance, studying the stability in the inverse source problems for inhomogeneous media where the analytical Green tensors are not available and the present method may not be directly applicable. Another interesting topic in

where **u***<sup>p</sup>* and the sheer part **u***<sup>s</sup>* which satisfy Sommerfeld radiation conditions

$$\lim\_{r \to \infty} r \left(\partial\_r \mathbf{u}\_p - \mathbf{i} k\_p \mathbf{u}\_p\right) = \mathbf{0}, \quad \lim\_{r \to \infty} r (\partial\_r \mathbf{u}\_t - \mathbf{i} k\_t \mathbf{u}\_t) = \mathbf{0}, \quad r = |\mathbf{x}|, \tag{37}$$

To achieve the result, authors used Helmholtz decomposition. The decomposition was allowed them to break the Navier-Lame equation to two elliptic equations.

The results for discrete data for inverse source problem which was obtained in [27] are as follows:

**Theorem 1.4.** Let **u** be the solution of the following scattering problem corresponding source **f** ∈*FM*ð Þ ,

$$(\mu \Delta + (\mu + \lambda) \nabla \cdot \nabla)(\mathbf{u}) + \rho^2 \mathbf{u} = \mathbf{f} \text{ in } \mathbb{R}^n,\tag{38}$$

with radiation condition (37),

Then

$$\left\|\mathbf{f}\right\|\_{\left(0\right)}^2(B\_R) \le C \left(\epsilon\_2^2 + \frac{M^2}{\left(\frac{N^{\frac{\tilde{\mathbf{S}}}{\left(\ln e\_3\right)^{\tilde{\mathbf{S}}}\right)^{\tilde{\mathbf{S}}}}}{\left(6m-3n+3\right)^3}}\right)^2\tag{39}$$

where

$$\epsilon\_2 = \left(\sum\_{n=1}^N \|\mathbf{u}\left(., a\_{p,n}\right)\|\_0^2 (\Gamma\_R) + \|\mathbf{u}(., a\_{\mathfrak{s},n})\|\_0^2 (\Gamma\_R)\right)^{\frac{1}{2}},$$

$$\mathfrak{c}\_3 = \sup\_{\boldsymbol{\alpha} \in \left(\mathbb{G}, \frac{\boldsymbol{\zeta}}{\gamma^{\mathfrak{s}}}\right)} \|\mathbf{u}(., \boldsymbol{\alpha})\|\_0^2 (\Gamma\_R),$$

and

$$F\_M(\mathbb{B}\_{\mathbb{R}}) = \left\{ \mathbf{f} \in H^{m+1}(B\_R) \, : \, \|\mathbf{f}\|\_{\left(m+1\right)}(B\_R) \le M \right\}.$$

The stability increases as *N* increases, i.e., the inverse problem is more stable when higher frequency data is used.

#### *2.2.1 Uniqueness of source function*

To achieve the uniqueness, we introduced two different approaches. The first approach is using the estimate for the source function. Letting the norm of the boundary data goes to zero, then the proof is complete. For instance, consider Theorem 1.2 and let ϵ ! 0. The second approach is the result has proved by Isakov, Chang and Lu. They used classical result of the hyperbolic initial value boundary problem indirectly. The following theorem is the result of [16]. In the following theorem <sup>Ω</sup> <sup>⊂</sup> *<sup>n</sup>* with *<sup>n</sup>* <sup>¼</sup> 2, 3 and <sup>Γ</sup>⊂ ∂Ω.

**Theorem 1.5.** Let *u* be a solution to the scattering problem (27) and (28) with *f*<sup>0</sup> ∈ *H*<sup>1</sup> ð Þ <sup>Ω</sup> , *<sup>f</sup>*<sup>1</sup> <sup>∈</sup>*L*<sup>2</sup> ð Þ Ω . If the Cauchy data *u*<sup>0</sup> ¼ *u*<sup>1</sup> ¼ 0 on Γ when *k*∈ ð Þ *K* <sup>∗</sup> , *K* , then *f*<sup>0</sup> ¼ *f*<sup>1</sup> ¼ 0 in Ω.

**Proof.** Denote by *U*<sup>0</sup> the solution to the following hyperbolic problem

*Inverse Scattering Source Problems DOI: http://dx.doi.org/10.5772/intechopen.92023*

$$\partial\_t^2 U\_0 - \Delta U\_0 = 0 \quad \text{on} \quad \mathfrak{Q} \times (0, \infty),$$

$$U\_0 = -f\_0, \partial\_t U\_0 = f\_1 \quad \text{on} \quad \mathfrak{Q} \times \{0\}, \\ U\_0 = 0 \quad \text{on} \quad \partial \mathfrak{Q} \times (0, +\infty).$$

Under their assumptions, there is a unique solution to the problem (40) with <sup>∥</sup>*U*0ð Þ , *<sup>t</sup>* <sup>∥</sup>ð Þ<sup>1</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup>*∂tU*0ð Þ , *<sup>t</sup>* <sup>∥</sup>ð Þ <sup>0</sup> ð Þ <sup>Ω</sup> <sup>≤</sup>*<sup>C</sup>* <sup>∥</sup>*f*0∥ð Þ<sup>1</sup> ð Þþ <sup>Ω</sup> <sup>∥</sup>*f*1∥ð Þ <sup>0</sup> ð Þ <sup>Ω</sup> � �.

Now let

**<sup>u</sup>** <sup>¼</sup> **<sup>u</sup>***<sup>p</sup>* <sup>þ</sup> **<sup>u</sup>***<sup>s</sup>* in *<sup>n</sup>*nΩ, (36)

**<sup>u</sup>** <sup>¼</sup> **<sup>f</sup>** *in <sup>n</sup>*, (38)

(39)

1

CCCA

<sup>0</sup>ð Þ Γ*<sup>R</sup>*

2 ,

*:*

where **u***<sup>p</sup>* and the sheer part **u***<sup>s</sup>* which satisfy Sommerfeld radiation conditions

To achieve the result, authors used Helmholtz decomposition. The decomposition was allowed them to break the Navier-Lame equation to two elliptic equations. The results for discrete data for inverse source problem which was obtained in

**Theorem 1.4.** Let **u** be the solution of the following scattering problem

<sup>2</sup> þ

*N*5 <sup>8</sup>j j *ln* ϵ<sup>3</sup> 1 9 ð Þ <sup>6</sup>*m*�3*n*þ<sup>3</sup> <sup>3</sup> � �<sup>2</sup>*m*�*n*þ<sup>1</sup>

!<sup>1</sup>

<sup>∥</sup>**u**ð Þ *:*,*<sup>ω</sup>* <sup>∥</sup><sup>2</sup>

n o

The stability increases as *N* increases, i.e., the inverse problem is more stable

To achieve the uniqueness, we introduced two different approaches. The first approach is using the estimate for the source function. Letting the norm of the boundary data goes to zero, then the proof is complete. For instance, consider Theorem 1.2 and let ϵ ! 0. The second approach is the result has proved by Isakov, Chang and Lu. They used classical result of the hyperbolic initial value boundary problem indirectly. The following theorem is the result of [16]. In the following

**Theorem 1.5.** Let *u* be a solution to the scattering problem (27) and (28) with

**Proof.** Denote by *U*<sup>0</sup> the solution to the following hyperbolic problem

ð Þ Ω . If the Cauchy data *u*<sup>0</sup> ¼ *u*<sup>1</sup> ¼ 0 on Γ when *k*∈ ð Þ *K* <sup>∗</sup> , *K* , then

<sup>0</sup>ð Þþ <sup>Γ</sup>*<sup>R</sup>* <sup>∥</sup>**u**ð Þ *:*, *<sup>ω</sup><sup>s</sup>*,*<sup>n</sup>* <sup>∥</sup><sup>2</sup>

<sup>0</sup>ð Þ Γ*<sup>R</sup>* ,

ð Þ *BR* : ∥**f**∥ð Þ *<sup>m</sup>*þ<sup>1</sup> ð Þ *BR* ≤ *M*

*M*<sup>2</sup>

ð Þ *<sup>μ</sup>*<sup>Δ</sup> <sup>þ</sup> ð Þ *<sup>μ</sup>* <sup>þ</sup> *<sup>λ</sup>* <sup>∇</sup> � <sup>∇</sup> ð Þþ **<sup>u</sup>** *<sup>ω</sup>*<sup>2</sup>

0

BBB@

ð Þ <sup>0</sup> ð Þ *BR* <sup>≤</sup>*<sup>C</sup>* <sup>ϵ</sup><sup>2</sup>

∥**u** *:*, *ω<sup>p</sup>*,*<sup>n</sup>* � �∥<sup>2</sup>

ϵ<sup>3</sup> ¼ sup *ω*∈ 0, *<sup>π</sup> cpR*, � �i

*FM*ð Þ¼ **<sup>f</sup>** <sup>∈</sup> *<sup>H</sup><sup>m</sup>*þ<sup>1</sup>

� � <sup>¼</sup> 0, lim*r*!<sup>∞</sup> *<sup>r</sup>*ð Þ¼ *<sup>∂</sup>r***u***<sup>s</sup>* � <sup>i</sup>*ks***u***<sup>s</sup>* 0, *<sup>r</sup>* <sup>¼</sup> <sup>∣</sup>**x**∣, (37)

lim*r*!<sup>∞</sup> *<sup>r</sup> <sup>∂</sup>r***u***<sup>p</sup>* � <sup>i</sup>*kp***u***<sup>p</sup>*

*Advances in Complex Analysis and Applications*

corresponding source **f** ∈*FM*ð Þ ,

with radiation condition (37),

∥**f**∥<sup>2</sup>

<sup>ϵ</sup><sup>2</sup> <sup>¼</sup> <sup>X</sup> *N*

when higher frequency data is used.

theorem <sup>Ω</sup> <sup>⊂</sup> *<sup>n</sup>* with *<sup>n</sup>* <sup>¼</sup> 2, 3 and <sup>Γ</sup>⊂ ∂Ω.

*2.2.1 Uniqueness of source function*

*n*¼1

[27] are as follows:

Then

where

and

*f*<sup>0</sup> ∈ *H*<sup>1</sup>

**34**

ð Þ <sup>Ω</sup> , *<sup>f</sup>*<sup>1</sup> <sup>∈</sup>*L*<sup>2</sup>

*f*<sup>0</sup> ¼ *f*<sup>1</sup> ¼ 0 in Ω.

$$u^\*\left(\varkappa,k\right) = \frac{1}{\sqrt{2\pi}} \int\_0^\infty U\_0\left(\varkappa,t\right)e^{ikt}dt$$

Due to the properties of *U*0, in particular to the conservation of the energy, the function *<sup>u</sup>*<sup>∗</sup> ð Þ *<sup>x</sup>*, *<sup>k</sup>* is well defined and analytic with respect to *<sup>k</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>1</sup> <sup>þ</sup> *ik*2, *<sup>k</sup>*<sup>2</sup> <sup>&</sup>gt;0. Applying the integration by parts and using standard properties of the Fourier-Laplace transform we conclude that

$$(\Delta + k^2)u^\* = -f\_1 - ikf\_0 \quad \text{in} \quad \Omega, \quad u^\* = 0 \quad \text{on} \quad \partial\Omega. \tag{41}$$

Due to the assumption, the function *u* solves the same Dirichlet problem for <sup>Δ</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> when 0<sup>&</sup>lt; *<sup>k</sup>*1, 0 <sup>&</sup>lt;*k*2. Indeed, *<sup>u</sup>* solves the homogeneous Helmholtz equation in *<sup>n</sup>*n<sup>Ω</sup> and has zero Cauchy data on <sup>Γ</sup>. By the uniqueness in the Cauchy problem for elliptic equations, *<sup>u</sup>* <sup>¼</sup> 0 on *<sup>n</sup>*n<sup>Ω</sup> and hence on <sup>∂</sup><sup>Ω</sup> provided *<sup>K</sup>*<sup>∗</sup> <sup>&</sup>lt;*k*<sup>&</sup>lt; *<sup>K</sup>*. As follows from the integral representation of solution (27), the function *u*ð Þ ; *k* is (complex) analytic when 0 < ℜe*k*, hence *u*ð Þ¼ ; *k* 0 on ∂Ω provided 0< ℜe*k*. Since *<sup>k</sup>*<sup>2</sup> <sup>&</sup>gt;0, the solution of (41) is unique, hence *<sup>u</sup>* <sup>¼</sup> *<sup>u</sup>*<sup>∗</sup> on <sup>Ω</sup> (see Section 4). Consequently, we obtain *<sup>u</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>u</sup>* <sup>¼</sup> 0, *<sup>∂</sup>νu*<sup>∗</sup> <sup>¼</sup> *<sup>∂</sup>ν<sup>u</sup>* <sup>¼</sup> 0 on <sup>Γ</sup>. Since *<sup>u</sup>*<sup>∗</sup> is an analytic function, we can conclude that *<sup>u</sup>*<sup>∗</sup> <sup>¼</sup> 0, *<sup>∂</sup>νu*<sup>∗</sup> <sup>¼</sup> 0 on <sup>Γ</sup> for all *<sup>k</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>1</sup> <sup>þ</sup> *ik*<sup>2</sup> with *k*<sup>2</sup> >0. Due to the uniqueness of the inversion of the Fourier-Laplace transform we will obtain

$$
\partial\_\nu U\_0 = \mathbf{0} \quad \text{on} \quad \Gamma \times (\mathbf{0}, \infty).
$$

Due to the uniqueness in the lateral Cauchy problem for the wave equation (40) with the Cauchy data on Γ � ð Þ 0, þ∞ [Holmgren-John theorem ([28], Section 3.4)], we can conclude that *U*<sup>0</sup> ¼ 0 on Ω � ð Þ *T*, þ∞ for some positive *T*. Hence from the uniqueness in the backward initial boundary value problem for the hyperbolic equation (39) in Ω � ð Þ 0, *T* with zero boundary data on ∂Ω � ð Þ 0, *T* and initial data at Ω � f g *T* we conclude that *U*<sup>0</sup> ¼ 0 on Ω � ð Þ 0, *T* . So �*U*0ð Þ¼ , 0 *f*<sup>0</sup> ¼ 0, *<sup>∂</sup>tU*ð Þ¼ , 0 *<sup>f</sup>*<sup>1</sup> <sup>¼</sup> 0 on <sup>Ω</sup> which finishes the proof of uniqueness.

#### **3. Conclusions**

In this section, the scattering and inverse scattering theory, inverse source scattering problem were considered briefly. The recent results such stability estimates for external source and electric current density from boundary measurements of radiated wave field and uniqueness for source function for Helmholtz equation, Elasticity and Maxwell system have showed. We also show some result for discrete data. In addition, we also showed some results of using just Dirichlet data for improving stability which was a big improvement. There are still many challenges remain in this field. For instance, studying the stability in the inverse source problems for inhomogeneous media where the analytical Green tensors are not available and the present method may not be directly applicable. Another interesting topic in stability of the external source is to consider the governing equation in the time domain. The non-linear case is also is a very challenging problem. The direct and inverse scattering problems when both the source and the linear load are random is also an open problem. Another challenging problem is to study the random source scattering problem for three dimensional elastic wave equation. As I mentioned before, there are many scientist and researcher have been working on inverse scattering and more specifically on inverse source problems. To expand your knowledge and further mathematical development in this field of research, please see the result authors in [29–41], which were discussed different aspects of the problems.

**References**

309-353

57-65

263-272

633-642

**37**

[1] Sommerfeld A. Die Greensche Funktion der Schwingungsgleichung.

*DOI: http://dx.doi.org/10.5772/intechopen.92023*

[11] Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. New York: Springer Verlag; 2013. ISBN-

[12] Watson GN. A Treatise on the Theory of Bessel Functions. USA: Cambridge University Press; 1922.

[13] Li P, Bao G, Chen C. Inverse random source scattering for elastic waves. SIAM Journal on Numerical Analysis.

[14] Eller M, Valdivia N. Acoustic source identification using multiple frequency information. Inverse Problems. 2009;**25**:

[15] Bao G, Lin J, Triki F. A multifrequency inverse source problem. Journal of Difference Equations. 2010;

[16] Cheng J, Isakov V, Lu S. Increasing stability in the inverse source problem with many frequencies. Journal of Difference Equations. 2016;**260**:

[17] Entekhabi MN, Isakov V. On increasing stability in the two

[18] Bao G, Li P. Inverse medium scattering problems in near-field optics. Journal of Computational Mathematics.

Problems. 2017;**34**:055005

2007;**25**(3):252-265

2018;**18**:1-18

dimensional inverse source scattering problem with many frequencies. Inverse

[19] Isakov V, Lu S. Increasing stability in the inverse source problem with attenuation and many frequencies. SIAM Journal on Applied Mathematics.

[20] Entekhabi MN. Increasing stability in the two dimensional inverse source

10: 1461449413

ISBN: 9780521483919

2017;**55**:2616-2643

115005

**249**:3443-3465

4786-4804

Mathematiker-Vereinigung. 1912;**21**:

[2] Rellich F. Uber des asymptotiche Verhalten der Losungen von Δ*u* þ *λu* ¼ 0 in unendlichen Gebieten. Deutsche Mathematiker-Vereinigung. 1943;**5**:

[3] Vekua IN. Metaharmonic functions. Trudy Belinskogo Matematicheskoe

[4] Isakov V. Inverse Problems for Partial Differential Equations. 2nd ed. New York: Springer International Publishing, Verlag; 2017. pp. 173-177. DOI: 10.1007/978-3-319-51658-5

[5] Lax PD, Phillips RS. Scattering Theory. New York: Academic Press;

[6] Nachman A. Reconstructions from boundary measurements. Annals of Mathematics. 1988;**128**:531-576

[7] Novikov R. Multidimensional inverse spectral problems for the equation �Δ*ψ* þ ð Þ *v x*ð Þ� *Eu x*ð Þ *ψ* ¼ 0. Functional Analysis and Its Applications. 1988;**22**:

[8] Ramm AG. Recovery of the potential from fixed energy scattering data. Inverse Problems. 1988;**4**:877-886

[9] Colton D, Päivärinta L. The uniqueness of a solution to an inverse scattering problem for electromagnetic waves. Archive for Rational Mechanics

and Analysis. 1992;**119**:59-70

[10] Imbriale WA, Mittra R. The two-dimensional inverse scattering problem. IEEE Transactions on Antennas and Propagation. 1970;**18**:

1967. ISBN 10: 0124400507

Instituta. 1943;**12**:105-174

Jahresbericht der Deutschen

*Inverse Scattering Source Problems*

#### **Acknowledgements**

Without doubt, our world is a beautiful place full of questions and challenges thanks to people who want to develop and overcome these challenges who share the gift of their time and passion to mentor future generation. Thank you to everyone who strives to grow and help others grow. To all the individuals I have had the opportunity to lead, be led by, or watch their leadership and mentoring from afar, I want to say thank you for being the inspiration.

I am grateful to all of those with whom I have had the pleasure to work and those who help me to grow and learn. I would especially like to thank Dr. Victor Isakov, my PhD adviser who has provided me extensive personal and professional guidance and taught me a great deal about both scientific research and life in general. He has taught me more than I could ever give him credit for here. He has shown me, by his example, what a good scientist (and person) should be. I also would like to thank Professor Alexander Bukhgeym and Professor Thomas K. DeLillo for their help and advice.

I would like to thank my mother and belated father, whose love and guidance are with me in whatever I pursue. Many thanks to my sisters Marjan, Mona and Mina and my brother Jamshid for their constant support and unending inspiration.

This chapter is supported in part by NSF Award HRD-1824267.

#### **Author details**

Mozhgan "Nora" Entekhabi Florida Agricultural and Mechanical University, Tallahassee, USA

\*Address all correspondence to: mozhgan.entekhabi@famu.edu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Inverse Scattering Source Problems DOI: http://dx.doi.org/10.5772/intechopen.92023*

#### **References**

stability of the external source is to consider the governing equation in the time domain. The non-linear case is also is a very challenging problem. The direct and inverse scattering problems when both the source and the linear load are random is also an open problem. Another challenging problem is to study the random source scattering problem for three dimensional elastic wave equation. As I mentioned before, there are many scientist and researcher have been working on inverse scattering and more specifically on inverse source problems. To expand your knowledge and further mathematical development in this field of research, please see the result authors in [29–41], which were discussed different aspects of the

Without doubt, our world is a beautiful place full of questions and challenges thanks to people who want to develop and overcome these challenges who share the gift of their time and passion to mentor future generation. Thank you to everyone who strives to grow and help others grow. To all the individuals I have had the opportunity to lead, be led by, or watch their leadership and mentoring from afar,

I am grateful to all of those with whom I have had the pleasure to work and those who help me to grow and learn. I would especially like to thank Dr. Victor Isakov, my PhD adviser who has provided me extensive personal and professional guidance and taught me a great deal about both scientific research and life in general. He has taught me more than I could ever give him credit for here. He has shown me, by his example, what a good scientist (and person) should be. I also would like to thank Professor Alexander Bukhgeym and Professor Thomas K. DeLillo for their help and

I would like to thank my mother and belated father, whose love and guidance are with me in whatever I pursue. Many thanks to my sisters Marjan, Mona and Mina and my brother Jamshid for their constant support and unending inspiration.

This chapter is supported in part by NSF Award HRD-1824267.

Florida Agricultural and Mechanical University, Tallahassee, USA

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: mozhgan.entekhabi@famu.edu

problems.

advice.

**Author details**

**36**

Mozhgan "Nora" Entekhabi

provided the original work is properly cited.

**Acknowledgements**

I want to say thank you for being the inspiration.

*Advances in Complex Analysis and Applications*

[1] Sommerfeld A. Die Greensche Funktion der Schwingungsgleichung. Jahresbericht der Deutschen Mathematiker-Vereinigung. 1912;**21**: 309-353

[2] Rellich F. Uber des asymptotiche Verhalten der Losungen von Δ*u* þ *λu* ¼ 0 in unendlichen Gebieten. Deutsche Mathematiker-Vereinigung. 1943;**5**: 57-65

[3] Vekua IN. Metaharmonic functions. Trudy Belinskogo Matematicheskoe Instituta. 1943;**12**:105-174

[4] Isakov V. Inverse Problems for Partial Differential Equations. 2nd ed. New York: Springer International Publishing, Verlag; 2017. pp. 173-177. DOI: 10.1007/978-3-319-51658-5

[5] Lax PD, Phillips RS. Scattering Theory. New York: Academic Press; 1967. ISBN 10: 0124400507

[6] Nachman A. Reconstructions from boundary measurements. Annals of Mathematics. 1988;**128**:531-576

[7] Novikov R. Multidimensional inverse spectral problems for the equation �Δ*ψ* þ ð Þ *v x*ð Þ� *Eu x*ð Þ *ψ* ¼ 0. Functional Analysis and Its Applications. 1988;**22**: 263-272

[8] Ramm AG. Recovery of the potential from fixed energy scattering data. Inverse Problems. 1988;**4**:877-886

[9] Colton D, Päivärinta L. The uniqueness of a solution to an inverse scattering problem for electromagnetic waves. Archive for Rational Mechanics and Analysis. 1992;**119**:59-70

[10] Imbriale WA, Mittra R. The two-dimensional inverse scattering problem. IEEE Transactions on Antennas and Propagation. 1970;**18**: 633-642

[11] Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. New York: Springer Verlag; 2013. ISBN-10: 1461449413

[12] Watson GN. A Treatise on the Theory of Bessel Functions. USA: Cambridge University Press; 1922. ISBN: 9780521483919

[13] Li P, Bao G, Chen C. Inverse random source scattering for elastic waves. SIAM Journal on Numerical Analysis. 2017;**55**:2616-2643

[14] Eller M, Valdivia N. Acoustic source identification using multiple frequency information. Inverse Problems. 2009;**25**: 115005

[15] Bao G, Lin J, Triki F. A multifrequency inverse source problem. Journal of Difference Equations. 2010; **249**:3443-3465

[16] Cheng J, Isakov V, Lu S. Increasing stability in the inverse source problem with many frequencies. Journal of Difference Equations. 2016;**260**: 4786-4804

[17] Entekhabi MN, Isakov V. On increasing stability in the two dimensional inverse source scattering problem with many frequencies. Inverse Problems. 2017;**34**:055005

[18] Bao G, Li P. Inverse medium scattering problems in near-field optics. Journal of Computational Mathematics. 2007;**25**(3):252-265

[19] Isakov V, Lu S. Increasing stability in the inverse source problem with attenuation and many frequencies. SIAM Journal on Applied Mathematics. 2018;**18**:1-18

[20] Entekhabi MN. Increasing stability in the two dimensional inverse source

scattering problem with attenuation and many frequencies. Inverse Problems. 2018;**34**:115001

[21] Entekhabi MN, Gunaratne A. A Logarithmic Estimate for Inverse Source Scattering Problem with Attenuation in a Two-Layered Medium, to be Appeared in Journal of Inverse and Ill-Posed Problems. 2019;. Available from: https://arxiv.org/pdf/ 1903.03475.pdf

[22] Zhao Y, Li P. Stability on the onedimensional inverse source scattering problem in a two-layered medium. Applicable Analysis. 2017;**98**(4): 682-692. DOI: 10.1080/ 00036811.2017.1399365

[23] Entekhabi MN, Isakov V. Increasing stability in acoustic and elastic inverse source problems, SIAM Journal on Mathematical Analysis. 2018. Available from: https://arxiv.org/abs/ 1808.10528

[24] Li P, Helin T. Inverse Random Source Problems for Time-Harmonic Acoustic and Elastic Waves, Submitted. 2018

[25] Isakov V, Lai R-Y, Wang J-N. Increasing stability for the attenuation and conductivity coefficients. SIAM Journal on Mathematical Analysis. 2016; **48**:1-18

[26] Isakov V. Increasing stability for the Schrodinger potential from the Dirichlet-to Neumann map. Discrete and Continuous Dynamical Systems. 2011;**4**:631-641

[27] Bao G, Li P, Zhao Y. Stability for the inverse source problems in elastic and electromagnetic waves. Journal de Mathématiques Pures et Appliquées. 2018

[28] John F. Partial Differential Equations, Applied Mathematical Sciences. New York/Berlin: Springer-Verlag; 1982

[29] Isakov V, Lu S. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging. 2018;**12**:955-970

Numerical Mathematics: Theory, Methods and Applications. 2011;**4**:

*Inverse Scattering Source Problems*

[39] Bao G, Liu J. Numerical solution of inverse scattering problems with multiexperimental limited aperture data. SIAM Journal on Scientific Computing.

*DOI: http://dx.doi.org/10.5772/intechopen.92023*

[40] Isakov V. Inverse Source Problems. Vol. 34. USA: American Mathematical Society; 1990. ISBN: 978-0-8218-1532-8

[41] Bukhgeim AL. Introduction to the Theory of Inverse Problems. 2000

419-442

**39**

2003;**25**(3):1102-1117

[30] Isakov V, Lu S, Xu B. Linearized inverse Schrödinger potential problem at a large wave number. SIAM Journal on Applied Mathematics. arXiv: 1812.05011

[31] Isakov V. On increasing stability in the continuation for elliptic equations of second order without (pseudo) convexity assumptions. Inverse Problems & Imaging. 2019;**13**:983-1006

[32] Isakov V, Wang J-N. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to Neumann map. Inverse Problems & Imaging. 2014;**8**:1139-1150

[33] Isakov V, Lai R-Y, Wang J-N. Increasing stability for conductivity and attenuation coefficients. SIAM Journal on Mathematical Analysis. 2016;**48**: 569-594

[34] Nakamura G, Saitoh S, Seo JK. Inverse Problems and Related Topics. New York; 2000. ISBN-13: 978- 15848819192000

[35] Ivanov VK, Vasin VV, Tanana VP. Theory of Linear Ill-Posed Problems and its Applications. 2002. ISBN: 978-3- 11-094482-2

[36] Lavrentiev R, Vasiliev. Inverse problems for second-order elliptic equations. In: Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Mathematics. Vol. 167. Berlin/Heidelberg: Springer; 1970

[37] Tikhonov AN. On the regularization of ill-posed problems. SSSS Doklady Akademii Nauk SSSR. 1963;**153**:49-52

[38] Bao G, Gao J, Li P. Analysis of direct and inverse cavity scattering problems.

*Inverse Scattering Source Problems DOI: http://dx.doi.org/10.5772/intechopen.92023*

Numerical Mathematics: Theory, Methods and Applications. 2011;**4**: 419-442

scattering problem with attenuation and many frequencies. Inverse Problems.

*Advances in Complex Analysis and Applications*

[29] Isakov V, Lu S. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems &

[30] Isakov V, Lu S, Xu B. Linearized inverse Schrödinger potential problem at a large wave number. SIAM Journal on Applied Mathematics. arXiv:

[31] Isakov V. On increasing stability in the continuation for elliptic equations of

Problems & Imaging. 2019;**13**:983-1006

[32] Isakov V, Wang J-N. Increasing stability for determining the potential in

second order without (pseudo) convexity assumptions. Inverse

the Schrödinger equation with attenuation from the Dirichlet-to Neumann map. Inverse Problems &

[33] Isakov V, Lai R-Y, Wang J-N. Increasing stability for conductivity and attenuation coefficients. SIAM Journal on Mathematical Analysis. 2016;**48**:

[34] Nakamura G, Saitoh S, Seo JK. Inverse Problems and Related Topics. New York; 2000. ISBN-13: 978-

[35] Ivanov VK, Vasin VV, Tanana VP. Theory of Linear Ill-Posed Problems and its Applications. 2002. ISBN: 978-3-

[36] Lavrentiev R, Vasiliev. Inverse problems for second-order elliptic equations. In: Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Mathematics. Vol. 167. Berlin/Heidelberg: Springer; 1970

[37] Tikhonov AN. On the regularization of ill-posed problems. SSSS Doklady Akademii Nauk SSSR. 1963;**153**:49-52

[38] Bao G, Gao J, Li P. Analysis of direct and inverse cavity scattering problems.

Imaging. 2014;**8**:1139-1150

Imaging. 2018;**12**:955-970

1812.05011

569-594

15848819192000

11-094482-2

Attenuation in a Two-Layered Medium, to be Appeared in Journal of Inverse and Ill-Posed Problems. 2019;. Available from: https://arxiv.org/pdf/

[22] Zhao Y, Li P. Stability on the onedimensional inverse source scattering problem in a two-layered medium. Applicable Analysis. 2017;**98**(4):

[23] Entekhabi MN, Isakov V. Increasing

inverse source problems, SIAM Journal on Mathematical Analysis. 2018. Available from: https://arxiv.org/abs/

[24] Li P, Helin T. Inverse Random Source Problems for Time-Harmonic Acoustic and Elastic Waves, Submitted.

[25] Isakov V, Lai R-Y, Wang J-N. Increasing stability for the attenuation and conductivity coefficients. SIAM Journal on Mathematical Analysis. 2016;

Schrodinger potential from the Dirichlet-to Neumann map. Discrete and Continuous Dynamical Systems.

[28] John F. Partial Differential Equations, Applied Mathematical Sciences. New York/Berlin: Springer-

[26] Isakov V. Increasing stability for the

[27] Bao G, Li P, Zhao Y. Stability for the inverse source problems in elastic and electromagnetic waves. Journal de Mathématiques Pures et Appliquées.

stability in acoustic and elastic

[21] Entekhabi MN, Gunaratne A. A Logarithmic Estimate for Inverse Source Scattering Problem with

2018;**34**:115001

1903.03475.pdf

1808.10528

2018

**48**:1-18

2018

**38**

Verlag; 1982

2011;**4**:631-641

682-692. DOI: 10.1080/ 00036811.2017.1399365 [39] Bao G, Liu J. Numerical solution of inverse scattering problems with multiexperimental limited aperture data. SIAM Journal on Scientific Computing. 2003;**25**(3):1102-1117

[40] Isakov V. Inverse Source Problems. Vol. 34. USA: American Mathematical Society; 1990. ISBN: 978-0-8218-1532-8

[41] Bukhgeim AL. Introduction to the Theory of Inverse Problems. 2000

**Chapter 4**

*Robson Pires*

**Abstract**

Solution Methods of Large

System of Equations

rotations method in complex plane.

Cartesian coordinates

**1. Introduction**

**41**

Complex-Valued Nonlinear

Nonlinear systems of equations in complex plane are frequently encountered in applied mathematics, e.g., power systems, signal processing, control theory, neural networks, and biomedicine, to name a few. The solution of these problems often requires a first- or second-order approximation of nonlinear functions to generate a new step or descent direction to meet the solution iteratively. However, such methods cannot be applied to functions of complex and complex conjugate variables because they are necessarily nonanalytic. To overcome this problem, the Wirtinger calculus allows an expansion of nonlinear functions in its original complex and complex conjugate variables once they are analytic in their argument as a whole. Thus, the goal is to apply this methodology for solving nonlinear systems of equations emerged from applications in the industry. For instances, the complexvalued Jacobian matrix emerged from the power flow analysis model which is solved by Newton-Raphson method can be exactly determined. Similarly, overdetermined Jacobian matrices can be dealt, e.g., through the Gauss-Newton method in complex plane aimed to solve power system state estimation problems. Finally, the factorization method of the aforementioned Jacobian matrices is addressed through the *fast Givens transformation* algorithm which means the square root-free Givens

**Keywords:** large nonlinear system of equation solution in complex plane, complex-valued Newton-Raphson and gauss-Newton iterative algorithms,

This work is a tribute to Steinmetz's contribution [1]. The reasons and motivations are stated throughout the whole document once the numerical solutions for solving power system applications are typically carried out in the real domain. For instance, the power flow analysis and power system state estimation are wellknown tools, among others. It turns out that these solutions are not well suited for modeling voltage and current phasor. To overcome this difficulty, the proposal described in this chapter aims to model the aforementioned applications in a unified system of coordinates, e.g., complex domain. Nonetheless, the solution methods of these problems often require a first- or second-order approximation of the set of power flow equations; such methods cannot be applied to nonlinear functions of

#### **Chapter 4**

## Solution Methods of Large Complex-Valued Nonlinear System of Equations

*Robson Pires*

#### **Abstract**

Nonlinear systems of equations in complex plane are frequently encountered in applied mathematics, e.g., power systems, signal processing, control theory, neural networks, and biomedicine, to name a few. The solution of these problems often requires a first- or second-order approximation of nonlinear functions to generate a new step or descent direction to meet the solution iteratively. However, such methods cannot be applied to functions of complex and complex conjugate variables because they are necessarily nonanalytic. To overcome this problem, the Wirtinger calculus allows an expansion of nonlinear functions in its original complex and complex conjugate variables once they are analytic in their argument as a whole. Thus, the goal is to apply this methodology for solving nonlinear systems of equations emerged from applications in the industry. For instances, the complexvalued Jacobian matrix emerged from the power flow analysis model which is solved by Newton-Raphson method can be exactly determined. Similarly, overdetermined Jacobian matrices can be dealt, e.g., through the Gauss-Newton method in complex plane aimed to solve power system state estimation problems. Finally, the factorization method of the aforementioned Jacobian matrices is addressed through the *fast Givens transformation* algorithm which means the square root-free Givens rotations method in complex plane.

**Keywords:** large nonlinear system of equation solution in complex plane, complex-valued Newton-Raphson and gauss-Newton iterative algorithms, Cartesian coordinates

#### **1. Introduction**

This work is a tribute to Steinmetz's contribution [1]. The reasons and motivations are stated throughout the whole document once the numerical solutions for solving power system applications are typically carried out in the real domain. For instance, the power flow analysis and power system state estimation are wellknown tools, among others. It turns out that these solutions are not well suited for modeling voltage and current phasor. To overcome this difficulty, the proposal described in this chapter aims to model the aforementioned applications in a unified system of coordinates, e.g., complex domain. Nonetheless, the solution methods of these problems often require a first- or second-order approximation of the set of power flow equations; such methods cannot be applied to nonlinear functions of

complex variables because they are nonanalytic in their arguments. Consequently, for these functions Taylor series expansions do not exist. Hence, for many decades this problem has been solved redefining the nonlinear functions as separate functions of the real and imaginary parts of their complex arguments so that standard methods can be applied. Although not widely known, it is also possible to construct an extended nonlinear function that includes not only the original complex state variables but also their complex conjugates, and then the Wirtinger calculus can be applied [2]. This property lies on the fact that if a function is analytic in the space spanned by ℜf g*x* and f g*x* in , it is also analytic in the space spanned by *x* and *x*\* in . In complex analysis of one and several complex variables, *Wirtinger operators* are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to *holomorphic* functions, *non-holomorphic* functions, or simply differentiable functions on complex domain. These operators allow the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables [2, 3]. Then, taken into account the Wirtinger calculus, this chapter shows how the Jacobian matrix patterns emerge in complex plane corresponding to the steady-state models of power flow analysis and power system state estimation, respectively.

*∂u <sup>∂</sup><sup>a</sup>* <sup>¼</sup> *<sup>∂</sup><sup>v</sup>*

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

with *x* ¼ *a* þ *j b*. Then,

and *b*, yielding

Since we have

and by setting *<sup>∂</sup>x*<sup>∗</sup>

By setting *<sup>∂</sup><sup>x</sup>*

**43**

*<sup>∂</sup><sup>b</sup>* , *<sup>∂</sup><sup>v</sup>*

These conditions are necessary for *f x*ð Þ to be complex differentiable. If the partial derivatives of *u a*ð Þ , *b* and *v a*ð Þ , *b* are continuous on their entire domain, then they are sufficient as well. Therefore, the complex function *f x*ð Þ is called an analytic or holomorphic function [2]. As an example, let *f x*ð Þ¼ *<sup>x</sup>*<sup>2</sup> be a complex function


*∂u*

Introduced by Wilhelm Wirtinger in 1927 [2], the *CR-Calculus*, also known as the Wirtinger calculus, provides a way to differentiate nonanalytic functions of complex variables. Specifically, this calculus is applicable to a function *f x*ð Þ given by Eq. (1) if *u a*ð Þ , *b* and *v a*ð Þ , *b* have continuous partial derivatives with respect to *a*

<sup>2</sup> , *<sup>∂</sup><sup>a</sup>* <sup>¼</sup> *<sup>∂</sup><sup>x</sup>* <sup>þ</sup> *<sup>∂</sup>x*<sup>∗</sup> ð Þ

<sup>2</sup> , *<sup>∂</sup><sup>b</sup>* <sup>¼</sup> *<sup>j</sup> <sup>∂</sup><sup>x</sup>* ð Þ <sup>∗</sup> � *<sup>∂</sup><sup>x</sup>*

*∂f <sup>∂</sup><sup>a</sup>* � *<sup>j</sup> <sup>∂</sup><sup>f</sup> ∂b*

Note that the Cauchy-Riemann conditions for *f*ð Þ� to be analytic in *x* can be

*∂a ∂x*<sup>∗</sup> þ *∂f ∂b* *∂b*

Similarly, if we take the derivative of *<sup>f</sup>*ð Þ� with respect to *<sup>x</sup>*<sup>∗</sup> , that is,

These results show that the Cauchy-Riemann equations hold, and hence

*f x*ð Þ¼ *<sup>x</sup>*<sup>2</sup> <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> � *<sup>b</sup>*<sup>2</sup>

*<sup>∂</sup><sup>b</sup>* <sup>¼</sup> <sup>2</sup>*a*;

*∂f <sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup><sup>f</sup> ∂a ∂a ∂x* þ *∂f ∂b ∂b*

*<sup>a</sup>* <sup>¼</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>∗</sup> ð Þ

*<sup>b</sup>* <sup>¼</sup> *<sup>j</sup> <sup>x</sup>*ð Þ <sup>∗</sup> � *<sup>x</sup>*

expressed compactly using the gradient as *<sup>∂</sup><sup>f</sup>*

*<sup>∂</sup>x*<sup>∗</sup> to zero, we get

*<sup>∂</sup><sup>x</sup>* to zero, it follows that

*∂f <sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>1</sup> 2

*∂f <sup>∂</sup>x*<sup>∗</sup> <sup>¼</sup> *<sup>∂</sup><sup>f</sup> ∂a*

which under differentiation rule leads to

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

*<sup>∂</sup><sup>a</sup>* <sup>¼</sup> <sup>2</sup>*<sup>a</sup>* <sup>¼</sup> *<sup>∂</sup><sup>v</sup>*

*f x*ð Þ¼ *<sup>y</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> is a holomorphic function.

**2.2** *CR-Calculus* **or Wirtinger calculus**

*∂u*

*<sup>∂</sup><sup>a</sup>* ¼ � *<sup>∂</sup><sup>u</sup>*

þ *j* 2*ab* |{z} ¼*v*

*<sup>∂</sup><sup>b</sup>* ¼ �2*<sup>b</sup>* ¼ � *<sup>∂</sup><sup>v</sup>*

*<sup>∂</sup><sup>b</sup> :* (3)

¼ *y*, (4)

� �*:* (5)

*<sup>∂</sup><sup>a</sup>* <sup>¼</sup> <sup>2</sup>*<sup>b</sup>*

*<sup>∂</sup><sup>x</sup> :* (6)

<sup>2</sup> , (7)

<sup>2</sup> , (8)

� �*:* (9)

*<sup>∂</sup>x*<sup>∗</sup> ¼ 0, i.e., *f*ð Þ� is a function of only *x*.

*<sup>∂</sup>x*<sup>∗</sup> *:* (10)

In this chapter the classical Newton-Raphson and Gauss-Newton methods in complex plane aiming the numerical solution of the power flow analysis and power system state estimation are derived, respectively. Moreover, the factorization methods addressed to deal with the Jacobian matrices emerged from these approaches are included [4].

This chapter is organized as follows. The theoretical foundation which is based on *Wirtinger calculus* is summed up in Section 2. Section 3 describes two algorithms suggested to factorize Jacobian matrix in complex plane regardless if it is exactly determined or overdetermined. In Section 4, the complex-valued static model solution by using Newton-Raphson method is derived, whereas in Section 5, the Gauss-Newton method developed in complex plane is equally presented. Finally, in Section 6 some conclusions are gathered and stated the next issues to be investigated in the near future.

#### **2. Theoretical foundation**

#### **2.1 Complex differentiability**

A complex function is defined as

$$f(\mathbf{x}) = \mathfrak{u}(a, b) + j \,\, \mathfrak{v}(a, b), \tag{1}$$

where *<sup>x</sup>* <sup>¼</sup> *<sup>a</sup>* <sup>þ</sup> *j b* and *u a*ð Þ , *<sup>b</sup>* , *v a*ð Þ , *<sup>b</sup>* are real functions, *<sup>u</sup>*, *<sup>v</sup>*: <sup>2</sup> ! . Functions like Eq. (1) are in general complex but may be real-valued in special cases, e.g., squared error cost function <sup>J</sup> **<sup>e</sup>**<sup>2</sup> . The definition of complex differentiability requires that the derivatives defined as the limit be independent of the direction in which Δ*x* approaches 0 in complex plane:

$$f'(\mathbf{x}\_0) = \lim\_{\Delta \mathbf{x} \to 0} \frac{f(\mathbf{x} + \Delta \mathbf{x}) - f(\mathbf{x})}{\Delta \mathbf{x}}.\tag{2}$$

This requires that the Cauchy-Riemann equations be satisfied, i.e.,

*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*

$$\frac{\partial u}{\partial a} = \frac{\partial v}{\partial b}, \qquad \frac{\partial v}{\partial a} = -\frac{\partial u}{\partial b}. \tag{3}$$

These conditions are necessary for *f x*ð Þ to be complex differentiable. If the partial derivatives of *u a*ð Þ , *b* and *v a*ð Þ , *b* are continuous on their entire domain, then they are sufficient as well. Therefore, the complex function *f x*ð Þ is called an analytic or holomorphic function [2]. As an example, let *f x*ð Þ¼ *<sup>x</sup>*<sup>2</sup> be a complex function with *x* ¼ *a* þ *j b*. Then,

$$f(\mathbf{x}) = \mathbf{x}^2 = \underbrace{\mathbf{a}^2 - \mathbf{b}^2}\_{=\mathbf{u}} + j\underbrace{\mathbf{2}ab}\_{=\mathbf{v}} = \mathbf{y},\tag{4}$$

which under differentiation rule leads to

$$\frac{\partial u}{\partial a} = 2a = \frac{\partial v}{\partial b} = 2a; \qquad \frac{\partial u}{\partial b} = -2b = -\left(\frac{\partial v}{\partial a} = 2b\right). \tag{5}$$

These results show that the Cauchy-Riemann equations hold, and hence *f x*ð Þ¼ *<sup>y</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> is a holomorphic function.

#### **2.2** *CR-Calculus* **or Wirtinger calculus**

Introduced by Wilhelm Wirtinger in 1927 [2], the *CR-Calculus*, also known as the Wirtinger calculus, provides a way to differentiate nonanalytic functions of complex variables. Specifically, this calculus is applicable to a function *f x*ð Þ given by Eq. (1) if *u a*ð Þ , *b* and *v a*ð Þ , *b* have continuous partial derivatives with respect to *a* and *b*, yielding

$$
\frac{\partial \mathbf{f}}{\partial \mathbf{x}} = \frac{\partial \mathbf{f}}{\partial \mathbf{a}} \frac{\partial \mathbf{a}}{\partial \mathbf{x}} + \frac{\partial \mathbf{f}}{\partial b} \frac{\partial b}{\partial \mathbf{x}} \,. \tag{6}
$$

Since we have

complex variables because they are nonanalytic in their arguments. Consequently, for these functions Taylor series expansions do not exist. Hence, for many decades this problem has been solved redefining the nonlinear functions as separate functions of the real and imaginary parts of their complex arguments so that standard methods can be applied. Although not widely known, it is also possible to construct an extended nonlinear function that includes not only the original complex state variables but also their complex conjugates, and then the Wirtinger calculus can be applied [2]. This property lies on the fact that if a function is analytic in the space spanned by ℜf g*x* and f g*x* in , it is also analytic in the space spanned by *x* and *x*\* in . In complex analysis of one and several complex variables, *Wirtinger operators* are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to *holomorphic* functions, *non-holomorphic* functions, or simply differentiable functions on complex domain. These operators allow the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables [2, 3]. Then, taken into account the

Wirtinger calculus, this chapter shows how the Jacobian matrix patterns emerge in complex plane corresponding to the steady-state models of power flow analysis and

In this chapter the classical Newton-Raphson and Gauss-Newton methods in complex plane aiming the numerical solution of the power flow analysis and power system state estimation are derived, respectively. Moreover, the factorization methods addressed to deal with the Jacobian matrices emerged from these

This chapter is organized as follows. The theoretical foundation which is based on *Wirtinger calculus* is summed up in Section 2. Section 3 describes two algorithms suggested to factorize Jacobian matrix in complex plane regardless if it is exactly determined or overdetermined. In Section 4, the complex-valued static model solution by using Newton-Raphson method is derived, whereas in Section 5, the Gauss-Newton method developed in complex plane is equally presented. Finally, in Section 6 some conclusions are gathered and stated the next issues to be investi-

where *<sup>x</sup>* <sup>¼</sup> *<sup>a</sup>* <sup>þ</sup> *j b* and *u a*ð Þ , *<sup>b</sup>* , *v a*ð Þ , *<sup>b</sup>* are real functions, *<sup>u</sup>*, *<sup>v</sup>*: <sup>2</sup> ! . Functions like Eq. (1) are in general complex but may be real-valued in special cases, e.g.,

requires that the derivatives defined as the limit be independent of the direction in

ð Þ¼ *x*<sup>0</sup> lim Δ*x*!0

This requires that the Cauchy-Riemann equations be satisfied, i.e.,

*f x*ð Þ¼ *u a*ð Þþ , *b jv a*ð Þ , *b* , (1)

. The definition of complex differentiability

<sup>Δ</sup>*<sup>x</sup> :* (2)

*f x*ð Þ� þ Δ*x f x*ð Þ

power system state estimation, respectively.

*Advances in Complex Analysis and Applications*

approaches are included [4].

gated in the near future.

**2. Theoretical foundation**

**2.1 Complex differentiability**

A complex function is defined as

squared error cost function <sup>J</sup> **<sup>e</sup>**<sup>2</sup>

**42**

which Δ*x* approaches 0 in complex plane:

*f* 0

$$\mathfrak{a} = \begin{array}{c} \frac{(\mathfrak{x} + \mathfrak{x}^\*)}{2}, \qquad \mathfrak{d}\mathfrak{a} = \begin{array}{c} \frac{(\mathfrak{d}\mathfrak{x} + \partial\mathfrak{x}^\*)}{2}, \end{array} \tag{7}$$

$$b = j \xrightarrow[2]{} \frac{(\mathfrak{x}^\* - \mathfrak{x})}{2}, \qquad \partial b = j \xrightarrow[2]{} \frac{(\partial \mathfrak{x}^\* - \partial \mathfrak{x})}{}, \tag{8}$$

and by setting *<sup>∂</sup>x*<sup>∗</sup> *<sup>∂</sup><sup>x</sup>* to zero, it follows that

$$\frac{\partial f}{\partial \alpha} = \frac{1}{2} \left( \frac{\partial f}{\partial a} - j \left. \frac{\partial f}{\partial b} \right| . \right) . \tag{9}$$

Note that the Cauchy-Riemann conditions for *f*ð Þ� to be analytic in *x* can be expressed compactly using the gradient as *<sup>∂</sup><sup>f</sup> <sup>∂</sup>x*<sup>∗</sup> ¼ 0, i.e., *f*ð Þ� is a function of only *x*. Similarly, if we take the derivative of *<sup>f</sup>*ð Þ� with respect to *<sup>x</sup>*<sup>∗</sup> , that is,

$$\frac{\partial \mathbf{f}}{\partial \mathbf{x}^\*} = \frac{\partial \mathbf{f}}{\partial a} \frac{\partial a}{\partial \mathbf{x}^\*} + \frac{\partial \mathbf{f}}{\partial b} \frac{\partial b}{\partial \mathbf{x}^\*} \,. \tag{10}$$

By setting *<sup>∂</sup><sup>x</sup> <sup>∂</sup>x*<sup>∗</sup> to zero, we get *Advances in Complex Analysis and Applications*

$$\frac{\partial \mathbf{f}}{\partial \mathbf{x}^\*} = \frac{1}{2} \left( \frac{\partial \mathbf{f}}{\partial a} + j \left. \frac{\partial \mathbf{f}}{\partial b} \right| . \right) . \tag{11}$$

Hereafter, a real-valued or complex-valued function and its argument are provided with a subscript *c* if it is a function in the complex conjugate coordinates, i.e.,

As well in the real domain, there are two classes of methods for the numerical

• Direct methods: Which produce the exact solution assuming the absence of truncation and round-off errors, by performing a finite number of *flops* in a finite known number of steps. These methods are usually recommended when most of the entries in the coefficient matrix are nonzero and the dimension of the system is not too large, for instance, the Gaussian elimination, the LU

• Iterative methods: This class of methods is beyond of this work. Notice the solution provided by these methods is approximated and the accuracy is imposed by the user. The number of *ν* accomplished iterations depends on the given precision or convergence criterion. This class of methods is preferred when the majority of the coefficients are equal to zero and the number of unknowns is very large. The methods of Jacobi and Gauss-Seidel are good

The first studied algorithm is the *three-angle complex rotations (TACR)*, which is derived in polar coordinates [6]. Nonetheless, the key idea behind *QR* decomposition is to eliminate the square roots needed for the computation of the cosine and sine which represent a bottleneck in real-time applications. Consequently, this algorithm is devoid of interest for the solution of large linear systems of equations.

On the other hand, the *fast plane rotation* [7] that is derived in complex plane and rectangular coordinates is a square root- and division-free Givens rotations [8]. In this sense, the fast plane rotation which is also referred as the complex-valued fast

*decomposition* of matrices once the computations are performed incrementally, i.e., as the data arrives sequentially in time. Thus, it allows us to reduce the overall latency and hardware resources drastically. In the forthcoming contribution, the *CV* � *FGR* performance will be compared to the well-known approaches which were successfully applied to the power system state estimation [9–11], once they are accordingly converted from real to complex domain. Further proposals in the updated state of the art will be equally considered, e.g., [12] and [13], to cite a few. The complex fast Givens transformation **M** is computed using **Algorithm 1** such

that the second component of **M***<sup>H</sup> x* is zero and **M***<sup>H</sup>***DM** is a diagonal matrix, as

Givens rotations (*CV* � *FGR*) is a very efficient algorithm, aiming a *QR-*

examples, besides the classical Conjugate gradient method [5].

*m*�**1**

**∈** *<sup>m</sup>*�*<sup>n</sup>*

*<sup>x</sup>*, *<sup>x</sup>*<sup>∗</sup> ð Þ. Moreover, when the *CR-Calculus* is extended to the vector case, it is denoted that the multivariate *CR-Calculus* and the basic rules for the scalar case

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

remain unchanged.

**3. Solution of the problem: A***<sup>m</sup>*�*<sup>n</sup> xn*�**<sup>1</sup>** ¼ *y*

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

**3.1 Three-angle complex rotation algorithm**

**3.2 Complex-valued fast givens rotations**

shown below:

**45**

solution of large linear system of equations in complex plane:

decomposition and QR factorization, to cite a few.

Again, the Cauchy-Riemann conditions for *<sup>f</sup>*ð Þ� to be analytic in *<sup>x</sup>*<sup>∗</sup> can be expressed compactly using the gradient as *<sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>x</sup>* <sup>¼</sup> 0, i.e., *<sup>f</sup>*ð Þ� is a function only of *<sup>x</sup>*<sup>∗</sup> .

In other words, the gradient (respectively conjugate gradient) operator acts as a partial derivative with respect to *x* (respectively to *x*<sup>∗</sup> ), treating *x*<sup>∗</sup> (respectively *x*) as a constant. Formally, we have

$$\frac{\partial f(\mathbf{x}\_{\varepsilon})}{\partial \mathbf{x}} = \frac{\partial f(\mathbf{x}, \mathbf{x}^\*)}{\partial \mathbf{x}}\Big|\_{\mathbf{x}^\* = \text{Const}} = \frac{1}{2} \left( \frac{\partial f}{\partial \mathbf{a}} - j \left. \frac{\partial f}{\partial \mathbf{b}} \right| \right), \tag{12}$$

$$\frac{\partial f(\mathbf{x}\_c)}{\partial \mathbf{x}^\*} = \frac{\partial f(\mathbf{x}, \mathbf{x}^\*)}{\partial \mathbf{x}^\*} \Big|\_{\mathbf{x} = \text{Const}} = \frac{1}{2} \left( \frac{\partial f}{\partial a} + j \left. \frac{\partial f}{\partial b} \right| \right). \tag{13}$$

As an example, let *f x*ð Þ¼ *<sup>c</sup> f x*, *<sup>x</sup>*<sup>∗</sup> ð Þ¼ *<sup>x</sup>*<sup>∗</sup> *<sup>x</sup>* <sup>¼</sup> k k*<sup>x</sup>* <sup>2</sup> <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*<sup>2</sup> be a real function of complex variable which is the squared Euclidean distance to the origin, with *x* ¼ *a* þ *j b*. Then,

$$f(\mathbf{x}\_c) = f(\mathbf{x}, \mathbf{x}^\*) = \mathbf{x}^\* \mathbf{x} = \underbrace{a^2 + b^2}\_{=\mathbf{u}} + j\underbrace{(ab - ab)}\_{=\mathbf{v}} = \mathbf{y} \tag{14}$$

as *v* ¼ 0; clearly the Cauchy-Riemann equations do not hold, and hence *f x*ð Þ¼ *<sup>c</sup> f x*, *<sup>x</sup>*<sup>∗</sup> ð Þ¼ *<sup>x</sup>*<sup>∗</sup> *<sup>x</sup>* is not analytic and thus is non-holomorphic function. To overcome this apparent difficult, by applying the *CR-Calculus* leads to

$$\frac{\partial f(\mathbf{x}\_c)}{\partial \mathbf{x}} = \mathbf{x}^\* ; \; \frac{\partial f(\mathbf{x}\_c)}{\partial \mathbf{x}^\*} \quad = \mathbf{x}, \tag{15}$$

which suggests the geometric interpretation shown in **Figure 1**. Its analysis allows us to infer that the direction of maximum rate of change of the objective function is given by the conjugate gradient defined in Eq. (13). Observe that its positive direction is referred to a maximization problem (dot arrow), whereas the opposite direction concerns to the cost function minimization.

**Figure 1.** *Contour plot of the real function of complex variable*.

*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*

Hereafter, a real-valued or complex-valued function and its argument are provided with a subscript *c* if it is a function in the complex conjugate coordinates, i.e., *<sup>x</sup>*, *<sup>x</sup>*<sup>∗</sup> ð Þ. Moreover, when the *CR-Calculus* is extended to the vector case, it is denoted that the multivariate *CR-Calculus* and the basic rules for the scalar case remain unchanged.

#### **3. Solution of the problem: A***<sup>m</sup>*�*<sup>n</sup> xn*�**<sup>1</sup>** <sup>¼</sup> *ym*�**<sup>1</sup> ∈** *<sup>m</sup>*�*<sup>n</sup>*

As well in the real domain, there are two classes of methods for the numerical solution of large linear system of equations in complex plane:


#### **3.1 Three-angle complex rotation algorithm**

The first studied algorithm is the *three-angle complex rotations (TACR)*, which is derived in polar coordinates [6]. Nonetheless, the key idea behind *QR* decomposition is to eliminate the square roots needed for the computation of the cosine and sine which represent a bottleneck in real-time applications. Consequently, this algorithm is devoid of interest for the solution of large linear systems of equations.

#### **3.2 Complex-valued fast givens rotations**

On the other hand, the *fast plane rotation* [7] that is derived in complex plane and rectangular coordinates is a square root- and division-free Givens rotations [8]. In this sense, the fast plane rotation which is also referred as the complex-valued fast Givens rotations (*CV* � *FGR*) is a very efficient algorithm, aiming a *QRdecomposition* of matrices once the computations are performed incrementally, i.e., as the data arrives sequentially in time. Thus, it allows us to reduce the overall latency and hardware resources drastically. In the forthcoming contribution, the *CV* � *FGR* performance will be compared to the well-known approaches which were successfully applied to the power system state estimation [9–11], once they are accordingly converted from real to complex domain. Further proposals in the updated state of the art will be equally considered, e.g., [12] and [13], to cite a few.

The complex fast Givens transformation **M** is computed using **Algorithm 1** such that the second component of **M***<sup>H</sup> x* is zero and **M***<sup>H</sup>***DM** is a diagonal matrix, as shown below:

*∂f <sup>∂</sup>x*<sup>∗</sup> <sup>¼</sup> <sup>1</sup> 2

expressed compactly using the gradient as *<sup>∂</sup><sup>f</sup>*

*Advances in Complex Analysis and Applications*

*<sup>∂</sup>f x*ð Þ*<sup>c</sup>*

*<sup>∂</sup>f x*ð Þ*<sup>c</sup>*

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup>f x*, *<sup>x</sup>*<sup>∗</sup> ð Þ *∂x*

*<sup>∂</sup>x*<sup>∗</sup> <sup>¼</sup> *<sup>∂</sup>f x*, *<sup>x</sup>*<sup>∗</sup> ð Þ *∂x*<sup>∗</sup>

As an example, let *f x*ð Þ¼ *<sup>c</sup> f x*, *<sup>x</sup>*<sup>∗</sup> ð Þ¼ *<sup>x</sup>*<sup>∗</sup> *<sup>x</sup>* <sup>¼</sup> k k*<sup>x</sup>*

*f x*ð Þ¼ *<sup>c</sup> f x*, *<sup>x</sup>*<sup>∗</sup> ð Þ¼ *<sup>x</sup>*<sup>∗</sup> *<sup>x</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*<sup>2</sup>

overcome this apparent difficult, by applying the *CR-Calculus* leads to

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>∗</sup> ;

*<sup>∂</sup>f x*ð Þ*<sup>c</sup>*

opposite direction concerns to the cost function minimization.

as a constant. Formally, we have

*x* ¼ *a* þ *j b*. Then,

**Figure 1.**

**44**

*Contour plot of the real function of complex variable*.

*∂f ∂a*

Again, the Cauchy-Riemann conditions for *<sup>f</sup>*ð Þ� to be analytic in *<sup>x</sup>*<sup>∗</sup> can be

� � � � *x*<sup>∗</sup> ¼*Const*

> � � � � *x*¼*Const*

of complex variable which is the squared Euclidean distance to the origin, with

as *v* ¼ 0; clearly the Cauchy-Riemann equations do not hold, and hence *f x*ð Þ¼ *<sup>c</sup> f x*, *<sup>x</sup>*<sup>∗</sup> ð Þ¼ *<sup>x</sup>*<sup>∗</sup> *<sup>x</sup>* is not analytic and thus is non-holomorphic function. To

which suggests the geometric interpretation shown in **Figure 1**. Its analysis allows us to infer that the direction of maximum rate of change of the objective function is given by the conjugate gradient defined in Eq. (13). Observe that its positive direction is referred to a maximization problem (dot arrow), whereas the

In other words, the gradient (respectively conjugate gradient) operator acts as a partial derivative with respect to *x* (respectively to *x*<sup>∗</sup> ), treating *x*<sup>∗</sup> (respectively *x*)

<sup>þ</sup> *<sup>j</sup> <sup>∂</sup><sup>f</sup> ∂b*

> ¼ 1 2

¼ 1 2


*<sup>∂</sup>f x*ð Þ*<sup>c</sup>*

*∂f <sup>∂</sup><sup>a</sup>* � *<sup>j</sup> <sup>∂</sup><sup>f</sup> ∂b*

*∂f ∂a*

� �

<sup>þ</sup> *<sup>j</sup> <sup>∂</sup><sup>f</sup> ∂b*

� �

þ *j ab* ð Þ � *ab* |fflfflfflfflfflffl{zfflfflfflfflfflffl} <sup>¼</sup>*<sup>v</sup>*

*<sup>∂</sup>x*<sup>∗</sup> <sup>¼</sup> *<sup>x</sup>*, (15)

*:* (11)

, (12)

*:* (13)

¼ *y* (14)

<sup>2</sup> <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*<sup>2</sup> be a real function

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> 0, i.e., *<sup>f</sup>*ð Þ� is a function only of *<sup>x</sup>*<sup>∗</sup> .

� �

*Advances in Complex Analysis and Applications*

$$\mathbf{M} = \underbrace{\begin{bmatrix} \beta & \mathbf{1} \\ \mathbf{1} & a \end{bmatrix}}\_{type = 1} \text{ or } \underbrace{\begin{bmatrix} \mathbf{1} & a \\ \beta & \mathbf{1} \end{bmatrix}}\_{type = 2} \tag{16}$$

*k* ¼ *n* þ 1;

end if **end for**

**if** *k*< ¼ *n* **then**

end if **end for**

algorithm as follows:

**47**

**for** *j* ¼ 1 : *k* � 1 **do**

**Step 1: Get** *alpha* **and** *beta* **using Algorithm 1:**

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

**Step 2: Update elements based on** *type*

*<sup>f</sup>* , *<sup>d</sup>new*

<sup>¼</sup> <sup>1</sup> *<sup>β</sup> α* 1

<sup>¼</sup> *<sup>β</sup>* <sup>1</sup> 1 *α*

Note that the complex fast Givens QRD does not require any square root operation, and during each incremental QRD-update step, the incoming input data row-vector is stored, i.e., vector **new**. In the sequence, the input data row-vector elements are zero-out (inner for loop) in order to update upper triangular matrix **R**. The **new** vector is overwritten each time till the *QRD*-algorithm has exhausted all

Aiming the solution of any set of exactly determined equations in complex plane, the vector of unknowns is regularly taken into account in the iterative

and the residual vector hereafter termed as "mismatches" vector leads to

where in Eq. (21), *Ys* is a vector of specified quantities, i.e., constant term; *Ye xc* ð Þ¼ **J***<sup>c</sup>* Δ*xc* is a vector of calculated quantities at each iteration. Consequently,

<sup>1</sup> , *x*<sup>∗</sup>

� �*<sup>T</sup>*

ð Þ *<sup>ν</sup>*�<sup>1</sup> Δ*x*ð Þ*<sup>ν</sup>*

� �*<sup>T</sup>*

<sup>2</sup> , … , *x*<sup>∗</sup>

<sup>1</sup> , *M*<sup>∗</sup>

*N*�1

<sup>2</sup> , … , *M*<sup>∗</sup>

*M xc* ð Þ¼ *Ye xc* ð Þ� *Ys* ¼ 0, (21)

*N*�1

*<sup>c</sup>* ¼ 0, (22)

, (19)

*:* (20)

the input data, i.e., the upper triangular matrix **R** is entirely updated.

*xc* <sup>¼</sup> *<sup>x</sup>*1, *<sup>x</sup>*2, … , *xN*�1, *<sup>x</sup>*<sup>∗</sup>

*<sup>M</sup> xc* ð Þ¼ *<sup>M</sup>*1, *<sup>M</sup>*2, … , *MN*�1, *<sup>M</sup>*<sup>∗</sup>

Nonetheless, here the goal is to calculate *xc* that satisfies

the linearization of Eq. (21) from one step to the sequel leads to

*M x*ð Þ *<sup>ν</sup>*�<sup>1</sup> *<sup>c</sup>* � � <sup>þ</sup> **<sup>J</sup>**

" #*<sup>H</sup> R j* ð Þ , *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>n</sup>*

" #*<sup>H</sup> R j* ð Þ , *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> : *<sup>n</sup>*

*<sup>g</sup>* ]=**fast.givens** (**R**ð Þ *j*, *j* , **new**ð Þ 1, *j* , *d <sup>f</sup>* , *dg*);

*new*ð Þ 1, *j* þ 1 : *n*

*new*ð Þ 1, *j* þ 1 : *n*

" #

" #

[*α*, *<sup>β</sup>*, **<sup>R</sup>**ð Þ *<sup>j</sup>*, *<sup>j</sup>* , *type*, *dnew*

**else if** *J* <*n* and *type*=2 **then** *R j* ð Þ , *j* þ 1 : *n new*ð Þ 1, *j* þ 1 : *n*

*R k*ð Þ¼ , 1 : *n new*ð Þ 1, 1 : *n* ; *d k*ð Þ¼ *dnew*;

**4. Complex-valued Newton-Raphson method**

**if** *j*<*n* and *type*=1 **then** *R j* ð Þ , *j* þ 1 : *n new*ð Þ 1, *j* þ 1 : *n*

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

" #

" #

end if

$$\mathbf{M}^H \underline{\mathbf{x}} = \underbrace{\begin{bmatrix} r \\ 0 \end{bmatrix}}\_{=\mathbf{D}} \otimes \mathbf{M}^H \underbrace{\begin{bmatrix} d\_f & \mathbf{0} \\ \mathbf{0} & d\_\mathbf{g} \end{bmatrix}}\_{=\mathbf{D}} \mathbf{M} = \underbrace{\begin{bmatrix} d\_f^{new} & \mathbf{0} \\ \mathbf{0} & d\_\mathbf{g}^{new} \end{bmatrix}}\_{=\mathbf{D}^{new}}.\tag{17}$$

Notice the superscript *H* denotes the *Hermitian* operation, i.e., complex conjugate transpose.

**Algorithm 1.** Complex fast Givens transform.

$$\begin{array}{l} \{a,\emptyset,\text{r},type,d\_{f}^{new},d\_{g}^{new}\} = \textbf{fast.givens} \ (\text{f, g, }d\_{f},d\_{g});\\ \textbf{if } f = 0 \text{ then} \\ type = 1; a = \beta = 0; r = g;\\ \begin{array}{l} d\_{f}^{new} = d\_{f}; \ d\_{g}^{new} = d\_{f};\\ \textbf{else } f = 0 \text{ then} \\ type = 2; a = \beta = 0; r = f;\\ \begin{array}{l} d\_{f}^{new} = d\_{f}; \ d\_{g}^{new} = d\_{g};\\ \textbf{else } \textbf{if } \|f\|^{2} \leq \|g\|^{2} \text{ then} \\ \textbf{type } = 1; i = f/g; s = d\_{g}/d\_{f};\\ \textbf{e} = -i, \beta = s \ast i;\\ \textbf{y } = s \ast \|i\|^{2}; r = g\*(1+\gamma);\\ \textbf{else } \|d\_{f}^{new} = (1+\gamma)\*d\_{\xi}; \ d\_{\xi}^{new} = (1+\gamma)\*d\_{f};\\ \textbf{else } \\ \textbf{type = 2; i = g/f; s = d\_{f}/d\_{\xi};\\ \textbf{e} = s \ast \|i\|^{2}; r = f\*(1+\gamma);\\ \textbf{f} = \textbf{e}\* \|(\text{i}\|^{2})^{\*}; r = f\*(1+\gamma);\\ \textbf{d}\_{g}^{new} = d\_{g}^{new} = (1+\gamma)\*d\_{f}; \ d\_{\xi}^{new} = (1+\gamma)\*d\_{\xi};\\ \textbf{end if} \end{array} \end{array}$$

The matrices **Q**, **M**, and **D** are connected to the following equation:

$$\mathbf{Q} = \mathbf{M} \left( \mathbf{D}^{\text{new}} \right)^{-1/2} = \mathbf{M} \operatorname{diag} \left( \mathbf{1} / \text{spt} \left( d\_i^{\text{new}} \right) \right). \tag{18}$$

In the sequence the *QR-decomposition* using complex fast Givens transformations is presented as **Algorithm 2**.

**Algorithm 2.** Sequential fast Givens QRD decomposition.

```
[Qα, Qβ, type, R]=fast.givens_QRD (A);
[m,n]=size(A);
R ¼ zeros nð Þ; D ¼ In;
Rð Þ¼ 1, 1 : n Að Þ 1, 1 : n ;
for i ¼ 2 : m do
   new¼ A ið Þ , 1 : n ; dnew ¼ 1;
   if i < ¼ n then
     k ¼ i;
   else
```
*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*

```
k ¼ n þ 1;
  end if
  for j ¼ 1 : k � 1 do
      Step 1: Get alpha and beta using Algorithm 1:
         [α, β, Rð Þ j, j , type, dnew
                             f , dnew
                                  g ]=fast.givens (Rð Þ j, j , newð Þ 1, j , d f , dg);
      Step 2: Update elements based on type
      if j<n and type=1 then
           R j ð Þ , j þ 1 : n
          newð Þ 1, j þ 1 : n
        " # ¼ 1 β
                                 α 1
                                " #H R j ð Þ , j þ 1 : n
                                           newð Þ 1, j þ 1 : n
                                          " #
      else if J <n and type=2 then
           R j ð Þ , j þ 1 : n
          newð Þ 1, j þ 1 : n
        " # ¼ β 1
                                 1 α
                                " #H R j ð Þ , j þ 1 : n
                                           newð Þ 1, j þ 1 : n
                                          " #
      end if
  end for
  if k< ¼ n then
    R kð Þ¼ , 1 : n newð Þ 1, 1 : n ; d kð Þ¼ dnew;
  end if
end for
```
Note that the complex fast Givens QRD does not require any square root operation, and during each incremental QRD-update step, the incoming input data row-vector is stored, i.e., vector **new**. In the sequence, the input data row-vector elements are zero-out (inner for loop) in order to update upper triangular matrix **R**. The **new** vector is overwritten each time till the *QRD*-algorithm has exhausted all the input data, i.e., the upper triangular matrix **R** is entirely updated.

#### **4. Complex-valued Newton-Raphson method**

Aiming the solution of any set of exactly determined equations in complex plane, the vector of unknowns is regularly taken into account in the iterative algorithm as follows:

$$\underline{\mathbf{x}}\_{\star} = \begin{bmatrix} \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_{N-1}, \mathbf{x}\_1^\*, \mathbf{x}\_2^\*, \dots, \mathbf{x}\_{N-1}^\* \end{bmatrix}^T,\tag{19}$$

and the residual vector hereafter termed as "mismatches" vector leads to

$$\underline{\mathbf{M}}(\underline{\mathbf{x}}\_{\varepsilon}) = \begin{bmatrix} \mathbf{M}\_1, \mathbf{M}\_2, \dots, \mathbf{M}\_{N-1}, \mathbf{M}\_1^\*, \mathbf{M}\_2^\*, \dots, \mathbf{M}\_{N-1}^\* \end{bmatrix}^T. \tag{20}$$

Nonetheless, here the goal is to calculate *xc* that satisfies

$$
\underline{\mathbf{M}}(\underline{\mathbf{x}}\_{\circ}) = \underline{Y\_{\underline{\mathbf{c}}}(\underline{\mathbf{x}}\_{\circ})} - \underline{Y\_{\underline{\mathbf{c}}}} = \mathbf{0},\tag{21}
$$

where in Eq. (21), *Ys* is a vector of specified quantities, i.e., constant term; *Ye xc* ð Þ¼ **J***<sup>c</sup>* Δ*xc* is a vector of calculated quantities at each iteration. Consequently, the linearization of Eq. (21) from one step to the sequel leads to

$$
\underline{\mathbf{M}} \Big( \underline{\mathbf{x}}^{(\nu - 1)} \Big) + \mathbf{J}^{(\nu - 1)} \,\,\Delta \underline{\mathbf{x}}^{(\nu)} = \mathbf{0},\tag{22}
$$

**<sup>M</sup>** <sup>¼</sup> *<sup>β</sup>* <sup>1</sup>

& **M***<sup>H</sup>*

**<sup>M</sup>***<sup>H</sup> <sup>x</sup>* <sup>¼</sup> *<sup>r</sup>*

*Advances in Complex Analysis and Applications*

gate transpose.

[*α*, *β*, r, *type*, *dnew*

**else if** *g* ¼ 0 **then**

**else if** k k*<sup>f</sup>* <sup>2</sup> <sup>≤</sup>k k*<sup>g</sup>*

*<sup>f</sup>* <sup>¼</sup> *dg* ; *dnew*

*<sup>f</sup>* <sup>¼</sup> *<sup>d</sup> <sup>f</sup>* ; *<sup>d</sup>new*

*α* ¼ �*i*; *β* ¼ *s* ∗ *i*; *<sup>γ</sup>* <sup>¼</sup> *<sup>s</sup>* <sup>∗</sup> k k*<sup>i</sup>* <sup>2</sup>

*α* ¼ �*i*; *β* ¼ *s* ∗ *i*; *<sup>γ</sup>* <sup>¼</sup> *<sup>s</sup>* <sup>∗</sup> k k*<sup>i</sup>* <sup>2</sup>

is presented as **Algorithm 2**.

[m,n]=size(A); *R* ¼ *zeros n*ð Þ; *D* ¼ *In*; *R*ð Þ¼ 1, 1 : *n A*ð Þ 1, 1 : *n* ; **for** *i* ¼ 2 : *m* **do**

> **if** *i* < ¼ *n* **then** *k* ¼ *i*; else

**46**

**if** *f* ¼ 0 **then**

*dnew*

*dnew*

*dnew*

*dnew*

**end if**

**else**

0

**Algorithm 1.** Complex fast Givens transform.

*<sup>f</sup>* , *dnew*

*<sup>g</sup>* ¼ *d <sup>f</sup>* ;

*<sup>g</sup>* ¼ *dg* ;

<sup>2</sup> **then**

; *r* ¼ *g* ∗ ð Þ 1 þ *γ* ;

; *r* ¼ *f* ∗ ð Þ 1 þ *γ* ;

[*Qα*, *Qβ*, *type*, *R*]=**fast.givens\_QRD** (A);

**new**¼ *A i*ð Þ , 1 : *n* ; *dnew* ¼ 1;

*type*¼ 1; *α* ¼ *β* ¼ 0; *r* ¼ *g*;

*type*¼ 2; *α* ¼ *β* ¼ 0; *r* ¼ *f*;

*type*¼ 1; *i* ¼ *f =g*; *s* ¼ *dg=d <sup>f</sup>* ;

*<sup>f</sup>* <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>γ</sup>* <sup>∗</sup> *dg*; *dnew*

*type*¼ 2; *i* ¼ *g=f*; *s* ¼ *d <sup>f</sup> =dg* ;

*<sup>f</sup>* <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *<sup>γ</sup>* <sup>∗</sup> *<sup>d</sup> <sup>f</sup>* ; *dnew*

" #

1 *α* � �

*or*

*d <sup>f</sup>* 0 0 *dg*

" #


Notice the superscript *H* denotes the *Hermitian* operation, i.e., complex conju-

*<sup>g</sup>* ] = **fast.givens** (f, g, *d <sup>f</sup>* , *dg*);

*<sup>g</sup>* ¼ ð Þ 1 þ *γ* ∗ *d <sup>f</sup>* ;

*<sup>g</sup>* ¼ ð Þ 1 þ *γ* ∗ *dg*;

The matrices **Q**, **M**, and **D** are connected to the following equation:

**Algorithm 2.** Sequential fast Givens QRD decomposition.

**<sup>Q</sup>** <sup>¼</sup> **M Dnew** ð Þ�1*=*<sup>2</sup> <sup>¼</sup> **<sup>M</sup>** *diag* <sup>1</sup>*=sqrt dnew*

In the sequence the *QR-decomposition* using complex fast Givens transformations

1 *α β* 1 � �

(16)

*:* (17)


**M** ¼

*dnew <sup>f</sup>* 0 0 *dnew g*

*i*

� � � � *:* (18)

" #



or

$$
\Delta \underline{\mathbf{x}}\_{\epsilon}^{(\nu)} = - \left( \mathbf{J}^{(\nu - 1)} \right)^{-1} \underline{M} \left( \underline{\mathbf{x}}\_{\epsilon}^{(\nu - 1)} \right), \tag{23}
$$

**4.1 Jacobian matrix factorization**

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

which the dimension is 2*n* � ð Þ 2*n* þ 1 , resulting

Δ*x*ð Þ*<sup>ν</sup> c* � � � �

equations which is summarized thereafter.

Finally, Eq. (23) is solved by performing a back-substitution via

Δ*x*ð Þ*<sup>ν</sup>*

<sup>∞</sup> *and M xc* ð Þð Þ*<sup>ν</sup>* � � �

*x*ð Þ*<sup>ν</sup>*

This approach requires the nodal admittance matrix building, e.g.,

thus the complex nodal power can be expressed as

yielding

systems [14], yielding

is updated as shown below:

**4.2 Nodal equation**

**49**

In order to factorize the Jacobian matrix required in Eq. (23), the recommended

h i � � , (25)

h i � � , (26)

� �*:* (27)

<sup>∞</sup> <sup>≤</sup> *tol e:g:*, 10�<sup>12</sup> � �*:* (28)

*<sup>c</sup>* , (29)

*I* ¼ **Ybus** *V*, (30)

*<sup>S</sup>* <sup>¼</sup> *diag*ð Þ *<sup>V</sup> <sup>I</sup>* <sup>∗</sup> , (31)

� �

procedure is to operate the factorization on the augmented Jacobian matrix,

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

**<sup>J</sup>***<sup>a</sup>* <sup>¼</sup> **<sup>J</sup>** *<sup>M</sup> <sup>x</sup>*ð Þ *<sup>ν</sup>*�<sup>1</sup> *<sup>c</sup>*

<sup>~</sup>**J***<sup>a</sup>* <sup>¼</sup> **<sup>T</sup>***<sup>c</sup> <sup>M</sup>*<sup>~</sup> *<sup>x</sup>*ð Þ *<sup>ν</sup>*�<sup>1</sup> *<sup>c</sup>*

where **Tc** is an upper triangular matrix of dimension (2*<sup>n</sup>* � <sup>2</sup>*n*) and *<sup>M</sup>*<sup>~</sup> *<sup>x</sup>*ð Þ *<sup>ν</sup>*�<sup>1</sup> *<sup>c</sup>*

comprises the corresponding rows in the updated *rhs* vector of dimension (2*n* � 1).

*<sup>c</sup>* <sup>¼</sup> **Tc** *<sup>M</sup>*<sup>~</sup> *<sup>x</sup>*ð Þ *<sup>ν</sup>*�<sup>1</sup> *<sup>c</sup>*

On the other hand, it is recommendable to perform the convergence checking over the infinity norm of two vectors. Firstly, as the former, it occurs over the corrections to be applied to the state variables and simultaneously over the mismatches vector. This latter is included, aiming to be aware against ill-conditioned

> � � �

If Eq. (28) is satisfied, stop and print out the results. Otherwise, the state vector

*<sup>c</sup>* <sup>¼</sup> *<sup>x</sup>*ð Þ *<sup>ν</sup>*�<sup>1</sup> *<sup>c</sup>* <sup>þ</sup> <sup>Δ</sup>*x*ð Þ*<sup>ν</sup>*

and the iteration counter is increased followed by the updating of the mismatch vector and the Jacobian matrix factorization. This latter task can be mandatory or not once the Jacobian matrix may be kept constant throughout the iterative process (approximate, instead of full gain) which is a decision very often adopted after the second iteration aiming to lighten the computational burden. Further details can be found in [14], but in the sequence, a small example is presented forwarded of simulations carried out on large systems. However, as any other application in the industry, the power flow model lies in the solution of a system of linear algebraic

where **J** is the complex-valued Jacobian matrix which the dimension is 2 ð Þ� *N* � 1 2 ð Þ *N* � 1 . It means that at least one complex-valued state variable have to be specified, i.e., is known.

As a further advantage provided by the Wirtinger calculus [2, 3], the Jacobian matrix which emerged in Cartesian coordinates needs lesser algebra task as well as minor implementation effort (encoding) than the former procedure in real domain [14]. Thereby, the Jacobian matrix in expanded form may be represented through four partitions matrix, yielding

**J** ¼ *∂M*<sup>1</sup> *∂x*1 *∂M*<sup>1</sup> *∂x*2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>1</sup> *<sup>∂</sup>xN*�<sup>1</sup> ⋮ *∂M*<sup>1</sup> *∂x*<sup>∗</sup> 1 *∂M*<sup>1</sup> *∂x*<sup>∗</sup> 2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>1</sup> *∂x*<sup>∗</sup> *N*�1 *∂M*<sup>2</sup> *∂x*1 *∂M*<sup>2</sup> *∂x*2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>2</sup> *<sup>∂</sup>xN*�<sup>1</sup> ⋮ *∂M*<sup>2</sup> *∂x*<sup>∗</sup> 1 *∂M*<sup>2</sup> *∂x*<sup>∗</sup> 2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>2</sup> *∂x*<sup>∗</sup> *N*�1 *<sup>∂</sup>MN*�<sup>1</sup> *∂x*1 *<sup>∂</sup>MN*�<sup>1</sup> *∂x*2 <sup>⋯</sup> *<sup>∂</sup>MN*�<sup>1</sup> *<sup>∂</sup>xN*�<sup>1</sup> ⋮ *<sup>∂</sup>MN*�<sup>1</sup> *∂x*<sup>∗</sup> 1 *<sup>∂</sup>MN*�<sup>1</sup> *∂x*<sup>∗</sup> 2 <sup>⋯</sup> *<sup>∂</sup>MN*�<sup>1</sup> *∂x*<sup>∗</sup> *N*�1 ⋯ ⋯ ⋯⋯ ⋯ ⋮ ⋯ ⋯ ⋯ ⋯ *∂M*<sup>∗</sup> 1 *∂x*1 *∂M*<sup>∗</sup> 1 *∂x*2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup> 1 *<sup>∂</sup>xN*�<sup>1</sup> ⋮ *∂M*<sup>∗</sup> 1 *∂x*<sup>∗</sup> 1 *∂M*<sup>∗</sup> 1 *∂x*<sup>∗</sup> 2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup> 1 *∂x*<sup>∗</sup> *N*�1 *∂M*<sup>∗</sup> 2 *∂x*1 *∂M*<sup>∗</sup> 2 *∂x*2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup> 2 *<sup>∂</sup>xN*�<sup>1</sup> ⋮ *∂M*<sup>∗</sup> 2 *∂x*<sup>∗</sup> 1 *∂M*<sup>∗</sup> 2 *∂x*<sup>∗</sup> 2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup> 2 *∂x*<sup>∗</sup> *N*�1 *∂M*<sup>∗</sup> *N*�1 *∂x*1 *∂M*<sup>∗</sup> *N*�1 *∂x*2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup> *N*�1 *∂xN* ⋮ *∂M*<sup>∗</sup> *N*�1 *∂x*<sup>∗</sup> 1 *∂M*<sup>∗</sup> *N*�1 *∂x*<sup>∗</sup> 2 <sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup> *N*�1 *∂x*<sup>∗</sup> *N*�1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 *:* (24)

For instance, **Figure 2** displays the pattern for the IEEE-57 bus system in domain and complex plane. This latter is given by Eq. (24).

**Figure 2.**

*Sparsity structure of (a) real-valued Jacobian matrix; (b) complex-valued Jacobian matrix of the IEEE 57-bus system.*

*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*

#### **4.1 Jacobian matrix factorization**

or

**J** ¼

**Figure 2.**

*system.*

**48**

to be specified, i.e., is known.

four partitions matrix, yielding

*∂M*<sup>1</sup> *∂x*2

*∂M*<sup>2</sup> *∂x*2

*<sup>∂</sup>MN*�<sup>1</sup> *∂x*2

> *∂M*<sup>∗</sup> 1 *∂x*2

> *∂M*<sup>∗</sup> 2 *∂x*2

*∂M*<sup>∗</sup> *N*�1 *∂x*2

*∂M*<sup>1</sup> *∂x*1

*∂M*<sup>2</sup> *∂x*1

*<sup>∂</sup>MN*�<sup>1</sup> *∂x*1

> *∂M*<sup>∗</sup> 1 *∂x*1

> *∂M*<sup>∗</sup> 2 *∂x*1

*∂M*<sup>∗</sup> *N*�1 *∂x*1

Δ*x*ð Þ*<sup>ν</sup>*

*Advances in Complex Analysis and Applications*

*<sup>c</sup>* ¼ � **J**

<sup>⋯</sup> *<sup>∂</sup>M*<sup>1</sup> *<sup>∂</sup>xN*�<sup>1</sup>

<sup>⋯</sup> *<sup>∂</sup>M*<sup>2</sup> *<sup>∂</sup>xN*�<sup>1</sup>

<sup>⋯</sup> *<sup>∂</sup>MN*�<sup>1</sup> *<sup>∂</sup>xN*�<sup>1</sup>

<sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup>

<sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup>

<sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup>

domain and complex plane. This latter is given by Eq. (24).

1 *<sup>∂</sup>xN*�<sup>1</sup>

2 *<sup>∂</sup>xN*�<sup>1</sup>

> *N*�1 *∂xN*

For instance, **Figure 2** displays the pattern for the IEEE-57 bus system in -

*Sparsity structure of (a) real-valued Jacobian matrix; (b) complex-valued Jacobian matrix of the IEEE 57-bus*

where **J** is the complex-valued Jacobian matrix which the dimension is 2 ð Þ� *N* � 1 2 ð Þ *N* � 1 . It means that at least one complex-valued state variable have

As a further advantage provided by the Wirtinger calculus [2, 3], the Jacobian matrix which emerged in Cartesian coordinates needs lesser algebra task as well as minor implementation effort (encoding) than the former procedure in real domain [14]. Thereby, the Jacobian matrix in expanded form may be represented through

⋮

⋮

⋮

⋯ ⋯ ⋯⋯ ⋯ ⋮ ⋯ ⋯ ⋯ ⋯

⋮

⋮

⋮

*∂M*<sup>1</sup> *∂x*<sup>∗</sup> 1

*∂M*<sup>2</sup> *∂x*<sup>∗</sup> 1

*<sup>∂</sup>MN*�<sup>1</sup> *∂x*<sup>∗</sup> 1

> *∂M*<sup>∗</sup> 1 *∂x*<sup>∗</sup> 1

> *∂M*<sup>∗</sup> 2 *∂x*<sup>∗</sup> 1

*∂M*<sup>∗</sup> *N*�1 *∂x*<sup>∗</sup> 1

*∂M*<sup>1</sup> *∂x*<sup>∗</sup> 2

*∂M*<sup>2</sup> *∂x*<sup>∗</sup> 2

*<sup>∂</sup>MN*�<sup>1</sup> *∂x*<sup>∗</sup> 2

> *∂M*<sup>∗</sup> 1 *∂x*<sup>∗</sup> 2

> *∂M*<sup>∗</sup> 2 *∂x*<sup>∗</sup> 2

*∂M*<sup>∗</sup> *N*�1 *∂x*<sup>∗</sup> 2

ð Þ *<sup>ν</sup>*�<sup>1</sup> � ��<sup>1</sup> *<sup>M</sup> <sup>x</sup>*ð Þ *<sup>ν</sup>*�<sup>1</sup> *<sup>c</sup>*

� �

, (23)

<sup>⋯</sup> *<sup>∂</sup>M*<sup>1</sup> *∂x*<sup>∗</sup> *N*�1

*:*

<sup>⋯</sup> *<sup>∂</sup>M*<sup>2</sup> *∂x*<sup>∗</sup> *N*�1

<sup>⋯</sup> *<sup>∂</sup>MN*�<sup>1</sup> *∂x*<sup>∗</sup> *N*�1

<sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup> 1 *∂x*<sup>∗</sup> *N*�1

<sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup> 2 *∂x*<sup>∗</sup> *N*�1

<sup>⋯</sup> *<sup>∂</sup>M*<sup>∗</sup>

*N*�1 *∂x*<sup>∗</sup> *N*�1

(24)

In order to factorize the Jacobian matrix required in Eq. (23), the recommended procedure is to operate the factorization on the augmented Jacobian matrix, yielding

$$\mathbf{J}\_a \quad = \left[ \mathbf{J}\_a \underline{\mathbf{M}} (\underline{\mathbf{x}}\_{\varepsilon}^{(\nu - 1)}) \right],\tag{25}$$

which the dimension is 2*n* � ð Þ 2*n* þ 1 , resulting

$$\tilde{\mathbf{J}}\_{a} \quad = \begin{bmatrix} \mathbf{T}\_{c} \ \tilde{\mathbf{M}} \left( \underline{\mathbf{x}}\_{c}^{(\nu - 1)} \right) \end{bmatrix}, \tag{26}$$

where **Tc** is an upper triangular matrix of dimension (2*<sup>n</sup>* � <sup>2</sup>*n*) and *<sup>M</sup>*<sup>~</sup> *<sup>x</sup>*ð Þ *<sup>ν</sup>*�<sup>1</sup> *<sup>c</sup>* � � comprises the corresponding rows in the updated *rhs* vector of dimension (2*n* � 1). Finally, Eq. (23) is solved by performing a back-substitution via

$$
\Delta \underline{\mathbf{x}}\_{\epsilon}^{(\nu)} = \mathbf{T}\_{\mathbf{c}} \underline{\tilde{M}} \left( \underline{\mathbf{x}}\_{\epsilon}^{(\nu - 1)} \right). \tag{27}
$$

On the other hand, it is recommendable to perform the convergence checking over the infinity norm of two vectors. Firstly, as the former, it occurs over the corrections to be applied to the state variables and simultaneously over the mismatches vector. This latter is included, aiming to be aware against ill-conditioned systems [14], yielding

$$\left\|\left|\Delta\underline{\mathbf{x}}\_{\varepsilon}^{(\nu)}\right|\right\|\_{\infty} \operatorname{and} \left\|\mathbf{M}(\underline{\mathbf{x}}\_{\varepsilon})^{(\nu)}\right\|\_{\infty} \leq \operatorname{tol}\left(\mathrm{e.g.}, \mathbf{10}^{-12}\right). \tag{28}$$

If Eq. (28) is satisfied, stop and print out the results. Otherwise, the state vector is updated as shown below:

$$
\underline{\mathfrak{x}}^{(\nu)} = \underline{\mathfrak{x}}^{(\nu-1)} + \Delta \underline{\mathfrak{x}}^{(\nu)},\tag{29}
$$

and the iteration counter is increased followed by the updating of the mismatch vector and the Jacobian matrix factorization. This latter task can be mandatory or not once the Jacobian matrix may be kept constant throughout the iterative process (approximate, instead of full gain) which is a decision very often adopted after the second iteration aiming to lighten the computational burden. Further details can be found in [14], but in the sequence, a small example is presented forwarded of simulations carried out on large systems. However, as any other application in the industry, the power flow model lies in the solution of a system of linear algebraic equations which is summarized thereafter.

#### **4.2 Nodal equation**

This approach requires the nodal admittance matrix building, e.g.,

$$
\underline{I} = \mathbf{Y\_{bus}} \underline{V},
\tag{30}
$$

thus the complex nodal power can be expressed as

$$\underline{\mathbf{S}} = \operatorname{diag}(\underline{\mathbf{V}}) \, \underline{\mathbf{I}}^\*,\tag{31}$$

or

$$\underline{\mathbf{S}} = \operatorname{diag}(\underline{\mathbf{V}}) \, \mathbf{Y}\_{bus}^\* \, \underline{\mathbf{V}}^\*. \tag{32}$$

Then, the nodal complex power at *bus* � *k*, i.e., *Sk*, is

$$\mathcal{S}\_k = V\_k \mathcal{y}\_{kk}^\* \mathcal{V}\_k^\* + V\_k \sum\_{m=0 \atop m \neq k}^N \mathcal{Y}\_{km}^\* \mathcal{V}\_m^\*,\tag{33}$$

where *N* is the number of network nodes. Therefore, the unknowns to be determined are the voltages at each node or bus into the system and the general power flow equations that model any type of branch in an electrical network, i.e., transmission lines and phase- and phase-shifting-transformers can be written, yielding

$$S\_{km} = V\_k \left(\frac{\mathcal{Y}\_{km}^\*}{t\_{km} t\_{km}^\*} - j \, b\_{km}^{sh}\right) \, V\_k^\* - V\_k \, \frac{\mathcal{Y}\_{km}^\*}{t\_{km}} \, V\_m^\*,\tag{34}$$

$$\mathcal{S}\_{mk} = V\_m \left( y\_{km}^\* - j \left. b\_{km}^{sh} \right| \right) V\_m^\* - V\_m \frac{y\_{km}^\*}{t\_{km}^\*} \left. V\_k^\* \right. \tag{35}$$

and their complex conjugate counterpart are

$$\mathcal{S}\_{km}^\* = V\_k^\* \left( \frac{\mathcal{Y}\_{km}}{t\_{km}^\* t\_{km}} + j \; b\_{km}^{sh} \right) \; V\_k - V\_k^\* \; \frac{\mathcal{Y}\_{km}}{t\_{km}^\*} \; V\_m,\tag{36}$$

$$\mathcal{S}\_{mk}^{\*} = V\_{m}^{\*} \left( y\_{km} + j \, b\_{km}^{sh} \right) \, V\_{m} - V\_{m}^{\*} \, \frac{\mathcal{Y}\_{km}}{t\_{km}} \, \mathcal{V}\_{k}. \tag{37}$$

As in the real domain, the elements of the complex-valued Jacobian matrix remain practically unchanged after the second iteration, which suggest that we may keep them constant thereafter. Moreover, the computation of some entries can be avoided because they are complex conjugates of other entries; it turns out that these

**Branch Series Shunt**

**Bus Specified quantities in pu**

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

1-3 0.0150 0.0400 2-3 0.3000 1.6000

*i* ! *j* **R X Charging Y/2**

1-2 0.0012 0.0021 39.2 0.196

**Type** *Pg V Pload Qload* PV-2 1.0000 1.0000 0.2160 0.0918 PQ-3 2.700 1.620

**pu pu MVAr pu**

elements are PV-nodes.

**Table 1.**

**51**

**Figure 3.** *Small 3-bus system.*

*Branch and bus data.*

In Eqs. (34)–(37), *tkm* <sup>¼</sup> *akm <sup>e</sup>*�*jφkm* is the general off-nominal tap transformer model which is composed by an ideal transformer with complex turns ratio *tkm* : 1 in series with its admittance or impedance. Thus, if the corresponding branch is referred to.


#### **4.3 Small example**

The power flow model in complex plane as detailed in [14] is applied to a small example system in which the diagram is shown in **Figure 3**, while the corresponding branch parameters and bus data, both in *pu* (*Vbase* ¼ 230 *kV*; *Sbase* ¼ 100 *MVA*), are presented in **Table 1**.

The nodal admittance matrix, **Y***bus*, leads to

$$\mathbf{Y}\_{bus} = \begin{bmatrix} +213.3474 - j \ \ $0.8922 & -205.1282 + j \ \$ 58.9744 & -8.2192 + j \ 21.9178 \\ -205.1282 + j \ \ $58.9744 & +205.2414 - j \ \$ 59.3821 & -0.1132 + j \ \ $.6037 \\ -8.2192 + j \ \text{21.9178} & -0.1132 + j \ \$ .6037 & +8.3324 - j \ \text{22.3256} \end{bmatrix}. \tag{38}$$

The whole set of intermediary results throughout the iterative process is presented in the sequence.

*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*

**Figure 3.** *Small 3-bus system.*

or

referred to.

2. Pure-shifter: *bsh*

3. Phase-shifter: *bsh*

**4.3 Small example**

presented in **Table 1**.

presented in the sequence.

2 6 4

**Y***bus* ¼

**50**

*<sup>S</sup>* <sup>¼</sup> *diag*ð Þ *<sup>V</sup>* **<sup>Y</sup>**<sup>∗</sup>

*kk V* <sup>∗</sup>

lines and phase- and phase-shifting-transformers can be written, yielding

� �

*km* � *j bsh km*

� �

� �

*<sup>m</sup> ykm* <sup>þ</sup> *j bsh*

� �

� *j bsh km*

<sup>þ</sup> *j bsh km*

*km*

In Eqs. (34)–(37), *tkm* <sup>¼</sup> *akm <sup>e</sup>*�*jφkm* is the general off-nominal tap transformer model which is composed by an ideal transformer with complex turns ratio *tkm* : 1 in series with its admittance or impedance. Thus, if the corresponding branch is

*y* ∗ *km tkmt* <sup>∗</sup> *km*

*ykm t* ∗ *kmtkm*

*km* ¼ 0 and *akm* ¼ 1.

*<sup>k</sup>* þ *Vk*

where *N* is the number of network nodes. Therefore, the unknowns to be determined are the voltages at each node or bus into the system and the general power flow equations that model any type of branch in an electrical network, i.e., transmission

X *<sup>m</sup>*¼<sup>0</sup> *<sup>m</sup>*6¼*<sup>k</sup>*

> *V* <sup>∗</sup> *<sup>k</sup>* � *Vk*

*Vk* � *<sup>V</sup>* <sup>∗</sup> *k ykm t* ∗ *km*

*Vm* � *<sup>V</sup>* <sup>∗</sup> *m ykm tkm*

*km* ¼ 0 and *φkm* ¼ 0.

The power flow model in complex plane as detailed in [14] is applied to a small example system in which the diagram is shown in **Figure 3**, while the corresponding branch parameters and bus data, both in *pu* (*Vbase* ¼ 230 *kV*; *Sbase* ¼ 100 *MVA*), are

> þ213*:*3474 � *j* 380*:*8922 �205*:*1282 þ *j* 358*:*9744 �8*:*2192 þ *j* 21*:*9178 �205*:*1282 þ *j* 358*:*9744 þ205*:*2414 � *j* 359*:*3821 �0*:*1132 þ *j* 0*:*6037 �8*:*2192 þ *j* 21*:*9178 �0*:*1132 þ *j* 0*:*6037 þ8*:*3324 � *j* 22*:*3256

The whole set of intermediary results throughout the iterative process is

*V* <sup>∗</sup> *<sup>m</sup>* � *Vm* *y* ∗ *km tkm V* <sup>∗</sup>

*y* ∗ *km t* ∗ *km V* <sup>∗</sup>

*y* ∗ *km V* <sup>∗</sup>

*N*

Then, the nodal complex power at *bus* � *k*, i.e., *Sk*, is

*Sk* <sup>¼</sup> *Vk <sup>y</sup>* <sup>∗</sup>

*Skm* ¼ *Vk*

*Advances in Complex Analysis and Applications*

*Smk* <sup>¼</sup> *Vm <sup>y</sup>* <sup>∗</sup>

and their complex conjugate counterpart are

*S* ∗ *km* <sup>¼</sup> *<sup>V</sup>* <sup>∗</sup> *k*

> *S* ∗ *mk* <sup>¼</sup> *<sup>V</sup>* <sup>∗</sup>

1. Off-nominal tap transformer: *bsh*

*km* ¼ 0. 4. *π*�transmission line: *akm* ¼ 1 and *φkm* ¼ 0.

The nodal admittance matrix, **Y***bus*, leads to

*bus V* <sup>∗</sup> *:* (32)

*<sup>m</sup>*, (33)

*<sup>m</sup>*, (34)

*<sup>k</sup> :* (35)

*Vm*, (36)

*Vk:* (37)

3 7 5*:*

(38)


#### **Table 1.** *Branch and bus data.*

As in the real domain, the elements of the complex-valued Jacobian matrix remain practically unchanged after the second iteration, which suggest that we may keep them constant thereafter. Moreover, the computation of some entries can be avoided because they are complex conjugates of other entries; it turns out that these elements are PV-nodes.


Interestingly, the numerical values of the state variables corrections, state variables, and mismatches vectors calculated in the complex plane are displayed in the **Tables 2–4**, respectively.

#### **4.4 Performance in larger systems**

The algorithm described earlier was encoded in MATLAB by using sparsity technique and column approximate minimum degree (*colamd*) ordering scheme. The numerical tests were executed by using an Intel® Core™ i5-4200 CPU at 1.60 Hz 2.30 GHz, 6GB of RAM, and 64-bit operating system. A flat start condition is assigned to the state variables in all simulations.

Thereby, **Table 5** presents the performance on larger systems of the Newton-Raphson method in complex plane which is highlighted in bold. The corresponding performance is compared with those of the former Newton-Raphson method developed in polar and rectangular coordinates, both in real domain. In all simulations the convergence criterion of 1 � <sup>10</sup>�<sup>12</sup> is assumed.


#### **Table 2.**

*Unknown correction vector.*


The results presented in the aforementioned table allows us to infer that the Newton-Raphson method in complex plane has very good performance. Except for the SIN-1916 bus system, the time consuming required to achieve the solution is

**Algorithms Number of**

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

IEEE-14 <sup>1</sup>*:RV* � *NRM*ð Þ *<sup>p</sup>* <sup>5</sup> 0.0184 0.1647

IEEE-30 <sup>1</sup>*:RV* � *NRM*ð Þ *<sup>p</sup>* <sup>5</sup> 0.0206 0.1791

IEEE-57 <sup>1</sup>*:RV* � *NRM*ð Þ *<sup>p</sup>* <sup>6</sup> 0.0132 0.1653

IEEE-118 <sup>1</sup>*:RV* � *NRM*ð Þ *<sup>p</sup>* <sup>5</sup> 0.0162 0.1575

SIN-1916 <sup>1</sup>*:RV* � *NRM*ð Þ *<sup>p</sup>* <sup>7</sup> 0.4642 3.4732

**iterations**

<sup>26</sup> � <sup>26</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> 5 0***:***0098 0***:***<sup>0767</sup>**

<sup>58</sup> � <sup>58</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> 6 0***:***0091 0***:***<sup>0645</sup>**

<sup>112</sup> � <sup>112</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> 6 0***:***0110 0***:***<sup>0810</sup>**

<sup>234</sup> � <sup>234</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> 5 0***:***0121 0***:***<sup>1056</sup>**

<sup>3830</sup> � <sup>3830</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> 7 0***:***7699 4***:***<sup>5561</sup>**

<sup>2</sup>*:RV* � *NRM*ð Þ*<sup>r</sup>* <sup>5</sup> 0.0150 0.0942

<sup>2</sup>*:RV* � *NRM*ð Þ*<sup>r</sup>* <sup>6</sup> 0.0105 0.1121

<sup>2</sup>*:RV* � *NRM*ð Þ*<sup>r</sup>* <sup>6</sup> 0.0137 0.1303

<sup>2</sup>*:RV* � *NRM*ð Þ*<sup>r</sup>* <sup>6</sup> 0.0196 0.1840

<sup>2</sup>*:RV* � *NRM*ð Þ*<sup>r</sup>* <sup>8</sup> 0.3095 2.6430

**Time/iteration (s)**

**Total time (s)**

In the next section, the Gauss-Newton method is presented. The goal is to solve

As shown in [3], the complex-valued WLS state estimator minimizes an objec-

<sup>2</sup> *zc* � *hc xc* ð Þ ð Þ *<sup>H</sup>* <sup>Ω</sup>�<sup>1</sup>

where *zc* <sup>=</sup> *<sup>z</sup>*, *<sup>z</sup>* <sup>∗</sup> ð Þ is a vector of specified complex values of dimension (2*<sup>m</sup>* � 1), *xc* <sup>=</sup> *<sup>x</sup>*, *<sup>x</sup>*<sup>∗</sup> ð Þ is a vector of complex-valued state variables of dimension (2*<sup>n</sup>* � 1), *hc*(*xc*) is a vector of nonlinear functions of dimension (2*m* � 1) that maps *zc* to *xc*, *ω<sup>c</sup>* is a vector of random errors in complex plane which dimension is (2*m* � 1), and

*<sup>c</sup> zc* � *hc xc* ð Þ ð Þ , (39)

overdetermined systems of equations, i.e., the former nonlinear least-squares

**5. Complex-valued weighted-least-squares method (CV-WLS)**

lower than the remainder approaches.

*Performance in larger systems—squared matrices.*

*(p)—polar coordinates; (r)—rectangular coordinates; tol.,* <sup>1</sup> � <sup>10</sup>�<sup>12</sup>*.*

**CV-Jacobian matrix dimension 2**ð Þ� *N* � **1 2**ð Þ *N* � **1**

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

argmin *xc*

<sup>J</sup> *xc* ð Þ¼ <sup>1</sup>

method in complex plane.

**Table 5.**

tive function defined as

**53**

#### **Table 3.**

*Unknown vector.*


#### **Table 4.** *Mismatch vector.*


*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*

#### **Table 5.**

Interestingly, the numerical values of the state variables corrections, state variables, and mismatches vectors calculated in the complex plane are displayed in the

The algorithm described earlier was encoded in MATLAB by using sparsity technique and column approximate minimum degree (*colamd*) ordering scheme. The numerical tests were executed by using an Intel® Core™ i5-4200 CPU at 1.60 Hz 2.30 GHz, 6GB of RAM, and 64-bit operating system. A flat start condition

Thereby, **Table 5** presents the performance on larger systems of the Newton-Raphson method in complex plane which is highlighted in bold. The corresponding performance is compared with those of the former Newton-Raphson method developed in polar and rectangular coordinates, both in real domain. In all simula-

**Δ***x* **Δ***x*ð Þ *<sup>ν</sup>*¼**<sup>0</sup> Δ***x*ð Þ *<sup>ν</sup>*¼**<sup>1</sup> Δ***x*ð Þ *<sup>ν</sup>*¼**<sup>2</sup> Δ***x*ð Þ *<sup>ν</sup>*¼**<sup>3</sup>** <sup>Δ</sup>*x*<sup>2</sup> <sup>0</sup>*:*<sup>0023</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>90</sup>*:*<sup>00</sup> <sup>0</sup>*:*<sup>0003</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>90</sup>*:*<sup>39</sup> <sup>0</sup>*:*<sup>0127</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>90</sup>*:*<sup>08</sup> <sup>0</sup>*:*<sup>1708</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>89</sup>*:*<sup>92</sup> <sup>Δ</sup>*x*<sup>3</sup> <sup>0</sup>*:*<sup>1278</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>138</sup>*:*<sup>75</sup> <sup>0</sup>*:*<sup>0185</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>174</sup>*:*<sup>56</sup> <sup>0</sup>*:*<sup>4267</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>179</sup>*:*<sup>44</sup> <sup>0</sup>*:*<sup>2326</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>179</sup>*:*<sup>18</sup>

<sup>2</sup> <sup>0</sup>*:*<sup>0023</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>90</sup>*:*<sup>00</sup> <sup>0</sup>*:*<sup>0003</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>90</sup>*:*<sup>37</sup> <sup>0</sup>*:*<sup>0127</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>90</sup>*:*<sup>07</sup> <sup>0</sup>*:*<sup>1708</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>89</sup>*:*<sup>91</sup>

<sup>3</sup> <sup>0</sup>*:*<sup>1278</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>138</sup>*:*<sup>75</sup> <sup>0</sup>*:*<sup>0203</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>178</sup>*:*<sup>81</sup> <sup>0</sup>*:*<sup>4795</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>173</sup>*:*<sup>22</sup> <sup>0</sup>*:*<sup>2615</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>173</sup>*:*<sup>47</sup>

*<sup>M</sup> <sup>M</sup>*ð Þ *<sup>ν</sup>*¼**<sup>0</sup>** *<sup>M</sup>*ð Þ *<sup>ν</sup>*¼**<sup>1</sup>** *<sup>M</sup>*ð Þ *<sup>ν</sup>*¼**<sup>2</sup>** *<sup>M</sup>*ð Þ *<sup>ν</sup>*¼**<sup>3</sup>** � **<sup>10</sup>**�**<sup>3</sup>** *M*<sup>2</sup> �1*:*5680 þ *j* 0*:*0000 þ0*:*2178 þ *j* 0*:*0000 þ0*:*0093 þ *j* 0*:*0000 þ0*:*1229 þ *j* 0*:*0000 *M*<sup>3</sup> þ2*:*7000 þ *j* 1*:*4240 þ0*:*1363 þ *j* 0*:*3648 þ0*:*0021 þ *j* 0*:*0087 þ0*:*0011 þ *j* 0*:*0047

*x x*ð Þ *<sup>ν</sup>*¼**<sup>0</sup>** *x*ð Þ *<sup>ν</sup>*¼**<sup>1</sup>** *x*ð Þ *<sup>ν</sup>*¼**<sup>2</sup>** *x*ð Þ *<sup>ν</sup>*¼**<sup>3</sup>** *<sup>x</sup>*<sup>2</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup> <sup>j</sup>* <sup>0</sup>*:*<sup>0</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>132</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>115</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>115</sup> *<sup>x</sup>*<sup>3</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup> <sup>j</sup>* <sup>0</sup>*:*<sup>0</sup> <sup>0</sup>*:*<sup>907</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>5</sup>*:*<sup>326</sup> <sup>0</sup>*:*<sup>889</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>5</sup>*:*<sup>548</sup> <sup>0</sup>*:*<sup>889</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>5</sup>*:*<sup>552</sup>

<sup>2</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>180</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>0</sup>*:*<sup>132</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>0</sup>*:*<sup>115</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>0</sup>*:*<sup>115</sup>

<sup>3</sup> <sup>1</sup>*:*<sup>000</sup> � *<sup>e</sup>*�*<sup>j</sup>* <sup>180</sup> <sup>0</sup>*:*<sup>907</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>5</sup>*:*<sup>326</sup> <sup>0</sup>*:*<sup>887</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>5</sup>*:*<sup>421</sup> <sup>0</sup>*:*<sup>887</sup> � *<sup>e</sup>*þ*<sup>j</sup>* <sup>5</sup>*:*<sup>420</sup>

<sup>2</sup> þ0*:*0000 þ *j* 0*:*0000 þ0*:*0000 � *j* 0*:*0000 þ0*:*0000 � *j* 0*:*0000 þ0*:*0000 � *j* 0*:*0000

<sup>3</sup> þ2*:*7000 � *j* 1*:*4240 þ0*:*1363 � *j* 0*:*3648 0*:*0021 � *j* 0*:*0087 0*:*0011 � *j* 0*:*0047

<sup>a</sup> <sup>0</sup>*:*<sup>127809</sup> <sup>0</sup>*:*<sup>020316</sup> <sup>4</sup>*:*<sup>795255</sup> � <sup>10</sup>�<sup>4</sup> <sup>2</sup>*:*<sup>614490</sup> � <sup>10</sup>�<sup>7</sup>

**Tables 2–4**, respectively.

Δ*x*<sup>∗</sup>

Δ*x*<sup>∗</sup>

*a*

**Table 2.**

*x*∗

*x*∗

**Table 3.** *Unknown vector.*

*M*<sup>∗</sup>

*M*<sup>∗</sup>

**Table 4.** *Mismatch vector.*

**52**

*Convergence criteria:* k k *<sup>M</sup>* <sup>∞</sup> <sup>&</sup>lt; *tol:* <sup>≈</sup>10�<sup>4</sup>*.*

k k Δ*x* <sup>∞</sup>

*Unknown correction vector.*

*Convergence criteria:* k k <sup>Δ</sup>*<sup>X</sup>* <sup>∞</sup> <sup>&</sup>lt;*tol:* <sup>≈</sup>10�<sup>4</sup>*.*

**4.4 Performance in larger systems**

*Advances in Complex Analysis and Applications*

is assigned to the state variables in all simulations.

tions the convergence criterion of 1 � <sup>10</sup>�<sup>12</sup> is assumed.

*Performance in larger systems—squared matrices.*

The results presented in the aforementioned table allows us to infer that the Newton-Raphson method in complex plane has very good performance. Except for the SIN-1916 bus system, the time consuming required to achieve the solution is lower than the remainder approaches.

In the next section, the Gauss-Newton method is presented. The goal is to solve overdetermined systems of equations, i.e., the former nonlinear least-squares method in complex plane.

#### **5. Complex-valued weighted-least-squares method (CV-WLS)**

As shown in [3], the complex-valued WLS state estimator minimizes an objective function defined as

$$\underset{\underline{\mathfrak{X}}\mathcal{C}}{\operatorname{argmin}} \mathcal{J}\left(\underline{\mathfrak{x}}\_{\mathsf{c}}\right) = \frac{1}{2} \left(\underline{\mathfrak{z}}\_{\mathsf{c}} - \underline{h}\_{\mathsf{c}}(\underline{\mathfrak{x}}\_{\mathsf{c}})\right)^{H} \boldsymbol{\Omega}\_{\mathsf{c}}^{-1} \left(\underline{\mathfrak{z}}\_{\mathsf{c}} - \underline{h}\_{\mathsf{c}}(\underline{\mathfrak{x}}\_{\mathsf{c}})\right),\tag{39}$$

where *zc* <sup>=</sup> *<sup>z</sup>*, *<sup>z</sup>* <sup>∗</sup> ð Þ is a vector of specified complex values of dimension (2*<sup>m</sup>* � 1), *xc* <sup>=</sup> *<sup>x</sup>*, *<sup>x</sup>*<sup>∗</sup> ð Þ is a vector of complex-valued state variables of dimension (2*<sup>n</sup>* � 1), *hc*(*xc*) is a vector of nonlinear functions of dimension (2*m* � 1) that maps *zc* to *xc*, *ω<sup>c</sup>* is a vector of random errors in complex plane which dimension is (2*m* � 1), and

<sup>Ω</sup>*<sup>c</sup>* <sup>¼</sup> *<sup>E</sup> <sup>ω</sup><sup>c</sup> <sup>ω</sup><sup>H</sup> c* � � is a Hermitian positive-definite covariance matrix of *<sup>ω</sup><sup>c</sup>* which dimension is (2*<sup>m</sup>* � <sup>2</sup>*m*). The superscriptð Þ� *<sup>H</sup>* stands for Hermitian operator, i.e., the transpose complex conjugate operation. Thus, the necessary condition of optimality is given by

$$\frac{\partial \mathcal{J}\left(\underline{\mathbf{x}}\_{\epsilon}\right)}{\partial \underline{\mathbf{x}}\_{\epsilon}} = -\mathbf{H}(\underline{\mathbf{x}}\_{\epsilon})^H \, \underline{\mathbf{Q}}\_{\epsilon}^{-1} \left(\underline{\mathbf{z}}\_{\epsilon} - \underline{h}\_{\epsilon}(\underline{\mathbf{x}}\_{\epsilon})\right) = \mathbf{0}.\tag{40}$$

J *xc* ð Þ∈ )

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

and the property stated in Eq. (48), Eq. (46) becomes

where **I***<sup>n</sup>* is the (*n* � *n*)-identity matrix.

*∂h ∂x*^ � �*<sup>H</sup>*

*<sup>H</sup>* Ω�<sup>1</sup>

*∂***h** *∂***x**^ � �*<sup>H</sup>*

nonzero elements; it is about 45% sparser.

*<sup>H</sup>* Ω�<sup>1</sup> *<sup>c</sup>* **J d <sup>h</sup>** þ **J d h** *<sup>H</sup>* <sup>Ω</sup>�<sup>1</sup> *<sup>c</sup>* **Jh**

defined as

expressed as

where **G***x*<sup>∗</sup> *<sup>x</sup>*<sup>∗</sup> ¼ **G***<sup>x</sup> <sup>x</sup>*

**<sup>G</sup>***<sup>x</sup> <sup>x</sup>* <sup>¼</sup> <sup>1</sup> 2

Similarly, we get

**55**

**Gx**<sup>∗</sup> **<sup>x</sup>** <sup>¼</sup> <sup>1</sup> 2

> ¼ 1 <sup>2</sup> **Jh**

¼ 1 <sup>2</sup> **Jh**

follows from Eq. (47) that

**<sup>H</sup>** *xc* ð Þ¼ **Jh <sup>J</sup>**

**J d h** � � <sup>∗</sup> **Jh** ð Þ <sup>∗</sup>

*∂h*<sup>∗</sup> *<sup>c</sup> xc* ð Þ

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup>hc xc* ð Þ

Therefore, taking into account the chain rule differentiation mentioned earlier

**d h**

!

where **S** is a swap operator that permutes blocks of *m* rows or blocks of *n* columns depending upon whether **S** pre-multiples or post-multiples a matrix, respectively. Moreover, this operator is an isomorphism from to the dual space <sup>∗</sup> , which obeys the properties **<sup>S</sup>**�<sup>1</sup> <sup>¼</sup> **<sup>S</sup>***<sup>T</sup>* <sup>¼</sup> **<sup>S</sup>**. It shows that **<sup>S</sup>** is symmetric and is equal to its own inverse, that is, **<sup>S</sup>**<sup>2</sup> <sup>¼</sup> *<sup>I</sup>*. For instance, as shown in [2], this matrix is

**S** ¼

**G** *x*^*<sup>c</sup>* ð Þ¼

� � <sup>∗</sup> and **<sup>G</sup>***x*<sup>∗</sup> *<sup>x</sup>* <sup>¼</sup> **<sup>G</sup>***<sup>x</sup> <sup>x</sup>*<sup>∗</sup>

*∂h ∂x*^ � �

> **d h** *<sup>H</sup>* <sup>Ω</sup>�<sup>1</sup> *<sup>c</sup>* **J d h**

*∂***h** *∂***x**^ <sup>∗</sup> � �

� �<sup>∗</sup> h i*:*

þ

The investigation of the sparsity structure of the Jacobian matrix in complex plane given by Eq. (49), e.g., **Figure 4**, reveals that the Jacobian matrices in the -domain are sparser than its counterpart in the -domain [15]. **Figure 4** is

referred to the Brazilian equivalent 730-bus system which the Jacobian matrix in the -domain has 10,396 nonzero elements, while in the -domain, it has 18,403

" # ! � � <sup>∗</sup>

� �<sup>∗</sup> h i,

þ

Ω�<sup>1</sup> *c*

*<sup>c</sup>* **Jh** þ **J**

Ω�<sup>1</sup> *c*

Now, the complex-valued gain matrix **G** *x*^*<sup>c</sup>* ð Þ in expanded form can be

<sup>Δ</sup> 0 **I***<sup>n</sup>* **I***<sup>n</sup>* 0

**G***<sup>x</sup> <sup>x</sup>* **G***x*<sup>∗</sup> *<sup>x</sup>* **G***<sup>x</sup> <sup>x</sup>*<sup>∗</sup> **G***x*<sup>∗</sup> *<sup>x</sup>*<sup>∗</sup> !

� � <sup>∗</sup>

*∂h ∂x*^ <sup>∗</sup> � �*<sup>H</sup>*

> *∂***h** *∂***x**^ <sup>∗</sup> � �*<sup>H</sup>*

" # ! � � <sup>∗</sup>

Ω�<sup>1</sup> *c*

> Ω�<sup>1</sup> *c*

*∂***h** *∂***x**^

*∂h ∂x*^ <sup>∗</sup>

*∂x*<sup>∗</sup> � �<sup>∗</sup>

> <sup>¼</sup> **<sup>J</sup>***<sup>c</sup> xc* ð Þ **J** ∗ *<sup>c</sup> xc* ð Þ **S**

!

� �, (50)

, (51)

, all of dimension (*n* � *n*). Then, it

,

,

(52)

(53)

¼ **J d** *h* � � <sup>∗</sup>

*:* (48)

, (49)

By applying a first-order Taylor series expansion of *hc xc* ð Þ about *<sup>x</sup>*ð Þ*<sup>ν</sup> <sup>c</sup>* , we get

$$
\underline{h}\_{\epsilon}(\underline{\mathbf{x}}\_{\epsilon}) = \underline{h}\_{\epsilon}\left(\underline{\mathbf{x}}\_{\epsilon}^{(\nu)}\right) + \mathbf{H}\left(\underline{\mathbf{x}}\_{\epsilon}^{(\nu)}\right)\left(\underline{\mathbf{x}}\_{\epsilon} - \underline{\mathbf{x}}\_{\epsilon}^{(\nu)}\right). \tag{41}
$$

By replacing Eq. (41) into Eq. (40), we obtain

$$\mathbf{H}\left(\underline{\mathbf{x}}\_{\varepsilon}^{(\nu)}\right)^{H}\boldsymbol{\mathfrak{Q}}\_{\varepsilon}^{-1}\left[\underline{\mathbf{z}}\_{\varepsilon}-\underline{\mathbf{h}}\_{\varepsilon}\left(\underline{\mathbf{x}}\_{\varepsilon}^{(\nu)}\right)-\mathbf{H}\left(\underline{\mathbf{x}}\_{\varepsilon}^{(\nu)}\right)\left(\boldsymbol{x}\_{\varepsilon}-\underline{\mathbf{x}}\_{\varepsilon}^{(\nu)}\right)\right]=\mathbf{0},\tag{42}$$

yielding the updated estimated state vector expressed as

$$\begin{split} \underline{\mathbf{x}}\_{\epsilon}^{(\nu+1)} &= \underline{\mathbf{x}}\_{\epsilon}^{(\nu)} + \mathbf{G} \left( \underline{\mathbf{x}}\_{\epsilon}^{(\nu)} \right)^{-1} \mathbf{H} \left( \underline{\mathbf{x}}\_{\epsilon}^{(\nu)} \right)^{H} \boldsymbol{\Omega}\_{\epsilon}^{-1} \, \Delta \underline{\mathbf{x}}\_{\epsilon}^{(\nu)}, \\ &= \underline{\mathbf{x}}\_{\epsilon}^{(\nu)} + \Delta \underline{\mathbf{x}}\_{\epsilon}^{(\nu)}, \end{split} \tag{43}$$

where

$$
\Delta \underline{\mathbf{x}}\_{\epsilon}^{(\nu)} = \mathbf{G} \left( \underline{\mathbf{x}}\_{\epsilon}^{(\nu)} \right)^{-1} \mathbf{H} \left( \underline{\mathbf{x}}\_{\epsilon}^{(\nu)} \right)^{H} \boldsymbol{\Omega}\_{\epsilon}^{-1} \, \Delta \underline{\mathbf{z}}\_{\epsilon}^{(\nu)}.\tag{44}$$

Notice **G** *x*ð Þ*<sup>ν</sup> c* � � <sup>¼</sup> **<sup>H</sup>** *<sup>x</sup>*ð Þ*<sup>ν</sup> c* � �*<sup>H</sup>* Ω�<sup>1</sup> *<sup>c</sup>* **<sup>H</sup>** *<sup>x</sup>*ð Þ*<sup>ν</sup> c* � � and <sup>Δ</sup>*<sup>z</sup>* ð Þ*ν <sup>c</sup>* <sup>¼</sup> *zc* � *hc <sup>x</sup>*ð Þ*<sup>ν</sup> c* � �. Thus, the iterations are stopped when

$$||\Delta \underline{\mathbf{x}}\_{\epsilon}^{(\nu)}||\_{\circ} \le tol, e.g., \mathbf{10}^{-3}, \tag{45}$$

where k k� <sup>∞</sup> is the infinity norm and *ν* is the iteration counter.

Note that in Eq. (40), **H** *xc* ð Þ is the Jacobian matrix of dimension (2*m* � 2*n*) defined in the complex domain, yielding

$$\mathbf{H}(\underline{\mathbf{x}}\_{\text{c}}) \stackrel{\Delta}{=} \frac{\partial \underline{h}\_{\text{c}}(\underline{\mathbf{x}}\_{\text{c}})}{\partial \underline{\mathbf{x}}\_{\text{c}}} \stackrel{\Delta}{=} \begin{pmatrix} \frac{\partial \underline{h}\_{\text{c}}(\underline{\mathbf{x}}\_{\text{c}})}{\partial \underline{\mathbf{x}}} & \frac{\partial \underline{h}\_{\text{c}}(\underline{\mathbf{x}}\_{\text{c}})}{\partial \underline{\mathbf{x}}^{\*}} \\\\ \frac{\partial \underline{h}\_{\text{c}}^{\*}(\underline{\mathbf{x}}\_{\text{c}})}{\partial \underline{\mathbf{x}}} & \frac{\partial \underline{h}\_{\text{c}}^{\*}(\underline{\mathbf{x}}\_{\text{c}})}{\partial \underline{\mathbf{x}}^{\*}} \end{pmatrix}. \tag{46}$$

Let **Jh** <sup>¼</sup> *<sup>∂</sup>hc <sup>x</sup>*ð Þ*<sup>c</sup> <sup>∂</sup><sup>x</sup>* and **J d <sup>h</sup>** <sup>¼</sup> *<sup>∂</sup>hc <sup>x</sup>*ð Þ*<sup>c</sup> <sup>∂</sup>x*<sup>∗</sup> be Jacobian submatrices of dimension (*m* � *n*). They are obtained through the Wirtinger partial derivatives with respect to the complex and the complex conjugate state variables using the chain differentiation rule. Now, let us define the Jacobian matrix as

$$\mathbf{J}\_c(\underline{\mathbf{x}}\_c) = \begin{pmatrix} \mathbf{J}\_{\mathbf{h}} \ \mathbf{J}\_{\mathbf{h}}^{\mathbf{d}} \end{pmatrix}. \tag{47}$$

In the important special case given by Eq. (39) where J *xc* ð Þ is a real-valued function of complex variables, the following property holds

*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*

$$\mathcal{I}(\underline{\mathbf{x}}\_{c}) \in \mathbb{R} \Rightarrow \frac{\partial \underline{h}^{\*} (\underline{\mathbf{x}}\_{c})}{\partial \underline{\mathbf{x}}} = \left(\frac{\partial \underline{h}\_{c} (\underline{\mathbf{x}}\_{c})}{\partial \underline{\mathbf{x}}^{\*}}\right)^{\*} = \left(\mathbf{J}\_{h}^{\mathbf{d}}\right)^{\*}.\tag{48}$$

Therefore, taking into account the chain rule differentiation mentioned earlier and the property stated in Eq. (48), Eq. (46) becomes

$$\mathbf{H}(\underline{\mathbf{x}}\_{\varepsilon}) = \begin{pmatrix} \mathbf{J}\_{\mathbf{h}} & \mathbf{J}\_{\mathbf{h}}^{\mathbf{d}} \\ \left(\mathbf{J}\_{\mathbf{h}}^{\mathbf{d}}\right)^{\*} & \left(\mathbf{J}\_{\mathbf{h}}\right)^{\*} \end{pmatrix} = \begin{pmatrix} \mathbf{J}\_{\varepsilon}(\underline{\mathbf{x}}\_{\varepsilon}) \\ \mathbf{J}\_{\varepsilon}^{\*}\left(\underline{\mathbf{x}}\_{\varepsilon}\right)\mathbf{S} \end{pmatrix},\tag{49}$$

where **S** is a swap operator that permutes blocks of *m* rows or blocks of *n* columns depending upon whether **S** pre-multiples or post-multiples a matrix, respectively. Moreover, this operator is an isomorphism from to the dual space <sup>∗</sup> , which obeys the properties **<sup>S</sup>**�<sup>1</sup> <sup>¼</sup> **<sup>S</sup>***<sup>T</sup>* <sup>¼</sup> **<sup>S</sup>**. It shows that **<sup>S</sup>** is symmetric and is equal to its own inverse, that is, **<sup>S</sup>**<sup>2</sup> <sup>¼</sup> *<sup>I</sup>*. For instance, as shown in [2], this matrix is defined as

$$\mathbf{S} \triangleq \begin{bmatrix} \mathbf{0} & \mathbf{I}\_n \\ \mathbf{I}\_n & \mathbf{0} \end{bmatrix},\tag{50}$$

where **I***<sup>n</sup>* is the (*n* � *n*)-identity matrix.

Now, the complex-valued gain matrix **G** *x*^*<sup>c</sup>* ð Þ in expanded form can be expressed as

$$\mathbf{G}(\underline{\hat{x}}\_{\tau}) = \begin{pmatrix} \mathbf{G}\_{\underline{x}\underline{x}} & \mathbf{G}\_{\underline{x}^\*\underline{x}} \\ \mathbf{G}\_{\underline{x}\underline{x}^\*} & \mathbf{G}\_{\underline{x}^\*\underline{x}^\*} \end{pmatrix},\tag{51}$$

where **G***x*<sup>∗</sup> *<sup>x</sup>*<sup>∗</sup> ¼ **G***<sup>x</sup> <sup>x</sup>* � � <sup>∗</sup> and **<sup>G</sup>***x*<sup>∗</sup> *<sup>x</sup>* <sup>¼</sup> **<sup>G</sup>***<sup>x</sup> <sup>x</sup>*<sup>∗</sup> � � <sup>∗</sup> , all of dimension (*n* � *n*). Then, it follows from Eq. (47) that

$$\begin{split} \mathbf{G}\_{\underline{\mathbf{x}},\underline{\mathbf{x}}} &= \frac{1}{2} \left[ \left( \frac{\partial \underline{\mathbf{l}}}{\partial \underline{\dot{\mathbf{x}}}} \right)^{H} \boldsymbol{\Omega}\_{\boldsymbol{\varepsilon}}^{-1} \left( \frac{\partial \underline{\mathbf{l}}}{\partial \underline{\dot{\mathbf{x}}}} \right) + \left( \left( \frac{\partial \underline{\mathbf{l}}}{\partial \underline{\dot{\mathbf{x}}}^{\*}} \right)^{H} \boldsymbol{\Omega}\_{\boldsymbol{\varepsilon}}^{-1} \left( \frac{\partial \underline{\mathbf{l}}}{\partial \underline{\dot{\mathbf{x}}}^{\*}} \right) \right)^{\*} \right], \\ &= \frac{1}{2} \left[ \mathbf{J}\_{\mathbf{h}}^{-H} \boldsymbol{\Omega}\_{\boldsymbol{\varepsilon}}^{-1} \mathbf{J}\_{\mathbf{h}} + \left( \mathbf{J}\_{\mathbf{h}}^{\mathbf{d}^{H}} \boldsymbol{\Omega}\_{\boldsymbol{\varepsilon}}^{-1} \mathbf{J}\_{\mathbf{h}}^{\mathbf{d}} \right)^{\*} \right], \end{split} \tag{52}$$

Similarly, we get

$$\begin{split} \mathbf{G}\_{\underline{\mathbf{x}}^{\*}\,\underline{\mathbf{x}}} &= \frac{1}{2} \left[ \left( \frac{\partial \underline{\mathbf{h}}}{\partial \underline{\hat{\mathbf{x}}}} \right)^{H} \boldsymbol{\Omega}\_{\boldsymbol{c}}^{-1} \left( \frac{\partial \underline{\mathbf{h}}}{\partial \underline{\hat{\mathbf{x}}}^{\*}} \right) + \left( \left( \frac{\partial \underline{\mathbf{h}}}{\partial \underline{\hat{\mathbf{x}}}^{\*}} \right)^{H} \boldsymbol{\Omega}\_{\boldsymbol{c}}^{-1} \left( \frac{\partial \underline{\mathbf{h}}}{\partial \underline{\hat{\mathbf{x}}}} \right) \right)^{\*} \right], \\ &= \frac{1}{2} \left[ \mathbf{J}\_{\mathbf{h}}{}^{H} \boldsymbol{\Omega}\_{\boldsymbol{c}}^{-1} \mathbf{J}\_{\mathbf{h}}^{\mathrm{d}} + \left( \mathbf{J}\_{\mathbf{h}}^{\mathrm{d}H} \boldsymbol{\Omega}\_{\boldsymbol{c}}^{-1} \mathbf{J}\_{\mathbf{h}} \right)^{\*} \right]. \end{split} \tag{53}$$

The investigation of the sparsity structure of the Jacobian matrix in complex plane given by Eq. (49), e.g., **Figure 4**, reveals that the Jacobian matrices in the -domain are sparser than its counterpart in the -domain [15]. **Figure 4** is referred to the Brazilian equivalent 730-bus system which the Jacobian matrix in the -domain has 10,396 nonzero elements, while in the -domain, it has 18,403 nonzero elements; it is about 45% sparser.

<sup>Ω</sup>*<sup>c</sup>* <sup>¼</sup> *<sup>E</sup> <sup>ω</sup><sup>c</sup> <sup>ω</sup><sup>H</sup>*

*c*

**H** *x*ð Þ*<sup>ν</sup> c* � �*<sup>H</sup>*

where

Notice **G** *x*ð Þ*<sup>ν</sup>*

Let **Jh** <sup>¼</sup> *<sup>∂</sup>hc <sup>x</sup>*ð Þ*<sup>c</sup>*

**54**

*c* � �

iterations are stopped when

*<sup>∂</sup>*<sup>J</sup> *xc* ð Þ *∂xc*

*Advances in Complex Analysis and Applications*

*hc xc* ð Þ¼ *hc <sup>x</sup>*ð Þ*<sup>ν</sup>*

*<sup>c</sup> zc* � *hc <sup>x</sup>*ð Þ*<sup>ν</sup>*

yielding the updated estimated state vector expressed as

By replacing Eq. (41) into Eq. (40), we obtain

*<sup>x</sup>*ð Þ *<sup>ν</sup>*þ<sup>1</sup> *<sup>c</sup>* <sup>¼</sup> *<sup>x</sup>*ð Þ*<sup>ν</sup>*

Δ*x*ð Þ*<sup>ν</sup>*

<sup>¼</sup> **<sup>H</sup>** *<sup>x</sup>*ð Þ*<sup>ν</sup> c* � �*<sup>H</sup>*

defined in the complex domain, yielding

*<sup>∂</sup><sup>x</sup>* and **J**

**d <sup>h</sup>** <sup>¼</sup> *<sup>∂</sup>hc <sup>x</sup>*ð Þ*<sup>c</sup>*

rule. Now, let us define the Jacobian matrix as

<sup>¼</sup> *<sup>x</sup>*ð Þ*<sup>ν</sup>*

Ω�<sup>1</sup>

� � is a Hermitian positive-definite covariance matrix of *<sup>ω</sup><sup>c</sup>* which dimension is (2*<sup>m</sup>* � <sup>2</sup>*m*). The superscriptð Þ� *<sup>H</sup>* stands for Hermitian operator, i.e., the transpose complex conjugate operation. Thus, the necessary condition of optimality is given by

> <sup>þ</sup> **<sup>H</sup>** *<sup>x</sup>*ð Þ*<sup>ν</sup> c* � �

� **<sup>H</sup>** *<sup>x</sup>*ð Þ*<sup>ν</sup> c* � �

> **H** *x*ð Þ*<sup>ν</sup> c* � �*<sup>H</sup>*

**H x**ð Þ*<sup>ν</sup> c* � �*<sup>H</sup>*

<sup>∞</sup> <sup>≤</sup> *tol*,*e:g:*, 10�<sup>3</sup>

*<sup>∂</sup>hc xc* ð Þ *∂x*

*∂h*<sup>∗</sup> *<sup>c</sup> xc* ð Þ *∂x*

Note that in Eq. (40), **H** *xc* ð Þ is the Jacobian matrix of dimension (2*m* � 2*n*)

0

BBB@

They are obtained through the Wirtinger partial derivatives with respect to the complex and the complex conjugate state variables using the chain differentiation

**J***<sup>c</sup> xc* ð Þ¼ **Jh J**

In the important special case given by Eq. (39) where J *xc* ð Þ is a real-valued

**d h**

and Δ*z* ð Þ*ν*

h i � �

*<sup>c</sup> zc* � *hc xc* ð Þ¼ ð Þ 0*:* (40)

*xc* � *<sup>x</sup>*ð Þ*<sup>ν</sup> c* � �

*xc* � *<sup>x</sup>*ð Þ*<sup>ν</sup> c*

> Ω�<sup>1</sup> *<sup>c</sup>* Δ*z*ð Þ*<sup>ν</sup> <sup>c</sup>* ,

Ω�<sup>1</sup> *<sup>c</sup>* Δ*z*ð Þ*<sup>ν</sup>*

*<sup>c</sup>* <sup>¼</sup> *zc* � *hc <sup>x</sup>*ð Þ*<sup>ν</sup>*

*<sup>∂</sup>hc xc* ð Þ *∂x*<sup>∗</sup>

1

� �*:* (47)

*∂h*<sup>∗</sup> *<sup>c</sup> xc* ð Þ *∂x*<sup>∗</sup>

*<sup>∂</sup>x*<sup>∗</sup> be Jacobian submatrices of dimension (*m* � *n*).

*<sup>c</sup>* , we get

*:* (41)

¼ 0, (42)

*<sup>c</sup> :* (44)

. Thus, the

*c* � �

, (45)

CCCA*:* (46)

(43)

¼ �**<sup>H</sup>** *xc* ð Þ*<sup>H</sup>* <sup>Ω</sup>�<sup>1</sup>

By applying a first-order Taylor series expansion of *hc xc* ð Þ about *<sup>x</sup>*ð Þ*<sup>ν</sup>*

*c* � �

*c* � �

*<sup>c</sup>* <sup>þ</sup> **<sup>G</sup> <sup>x</sup>**ð Þ*<sup>ν</sup>* **c** � ��<sup>1</sup>

*<sup>c</sup>* <sup>þ</sup> <sup>Δ</sup>*x*ð Þ*<sup>ν</sup> <sup>c</sup>* ,

*<sup>c</sup>* <sup>¼</sup> **<sup>G</sup>** *<sup>x</sup>*ð Þ*<sup>ν</sup> c* � ��<sup>1</sup>

> Ω�<sup>1</sup> *<sup>c</sup>* **<sup>H</sup>** *<sup>x</sup>*ð Þ*<sup>ν</sup> c* � �

Δ*x*ð Þ*<sup>ν</sup> c* � � � � � � � �

where k k� <sup>∞</sup> is the infinity norm and *ν* is the iteration counter.

*∂xc*

¼ Δ

**<sup>H</sup>** *xc* ð Þ¼<sup>Δ</sup> *<sup>∂</sup>hc xc* ð Þ

function of complex variables, the following property holds

**Figure 4.** *Sparsity structure of (a) real-valued Jacobian matrix; (b) real-valued gain matrix; (c) complex-valued Jacobian matrix; and (d) complex-valued hessian matrix of the Brazilian equivalent 730-bus system.*

Nonetheless, the recommended numerical procedure aiming to solve the complex-valued power system state estimation problem is addressed by solving the weighted form of the right-hand-side (*rhs*) of Eq. (40) instead of Eq. (44), yielding

$$
\tilde{\mathbf{H}}(\hat{\underline{\mathbf{x}}}\_{\varepsilon})\,\Delta\underline{\mathbf{x}}\_{\varepsilon}^{(\nu)} = \Delta\tilde{\underline{z}}\_{\varepsilon},\tag{54}
$$

**<sup>H</sup>**<sup>~</sup> **<sup>a</sup>** *<sup>x</sup>*^*<sup>c</sup>* ð Þ¼ **Tc** <sup>Δ</sup>~~*zc*

Here, **Tc** is an upper triangular matrix of dimension (2*<sup>n</sup>* � <sup>2</sup>*n*), and <sup>Δ</sup>~~*zc* comprises the corresponding rows in the updated *rhs* vector of dimension (2*n* � 1).

The one-line diagram of a 2-bus system is depicted in **Figure 5**. The system is provided with two PMU measurements that meter the nodal voltage magnitudes and phase angles and two real and reactive power flow measurements, which are identified by means of black bullets and red triangles, respectively. In **Table 6**, the

From **Figure 5**, the complex power injections at sending and receiving end,

<sup>1</sup> � *<sup>y</sup>* <sup>∗</sup>

<sup>2</sup> � *<sup>y</sup>* <sup>∗</sup>

<sup>12</sup> *V* <sup>∗</sup> 2 h i, (59)

<sup>12</sup> *V* <sup>∗</sup> 1 h i*:* (60)

<sup>12</sup> � *j bsh* <sup>12</sup> � � *<sup>V</sup>* <sup>∗</sup>

<sup>12</sup> � *j bsh* <sup>12</sup> � � *<sup>V</sup>* <sup>∗</sup>

corresponding power flow measurement, i.e., *S*<sup>12</sup> and *S*21, respectively.

**Branch Series Shunt**

*i* ! *j* **R X Charging Y/2**

1-2 0.0203 0.1318 62.62 0.3131

**pu pu MVAr pu**

Applying the Wirtinger calculus to Eqs. (59) and (60) leads to the Jacobian matrix given by Eq. (47) at each iteration as shown in the sequence. Here, the complex power injection measurement at each end, i.e., *S*<sup>1</sup> and *S*2, is equal to the

Finally, Eq. (55) is solved by performing a back-substitution via

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

transmission line parameters are given in *pu*.

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

*<sup>S</sup>*<sup>1</sup> <sup>¼</sup> *<sup>V</sup>*<sup>1</sup> *<sup>y</sup>* <sup>∗</sup>

*<sup>S</sup>*<sup>2</sup> <sup>¼</sup> *<sup>V</sup>*<sup>2</sup> *<sup>y</sup>* <sup>∗</sup>

respectively, are written yielding

**5.1 Small example**

**Figure 5.** *2-Bus power system.*

**Table 6.** *Branch data.*

**57**

Δ*x*ð Þ*<sup>ν</sup>*

� �*:* (57)

*<sup>c</sup>* <sup>¼</sup> **Tc** <sup>Δ</sup>~~*zc:* (58)

where **<sup>H</sup>**<sup>~</sup> *<sup>x</sup>*^*<sup>c</sup>* ð Þ¼ <sup>Ω</sup>�1*=*<sup>2</sup> *<sup>c</sup>* **<sup>H</sup>** *<sup>x</sup>*^*<sup>c</sup>* ð Þ is of dimension (2*<sup>m</sup>* � <sup>2</sup>*n*) and <sup>Δ</sup>~*zc* <sup>¼</sup> <sup>Ω</sup>�1*=*<sup>2</sup> *<sup>c</sup>* <sup>Δ</sup>*zc* is of dimension (2*m* � 1). Thus, the incremental changes in the state vector are calculated via

$$
\Delta \underline{\mathbf{x}}\_{\cdot}^{(\nu)} = \tilde{\mathbf{H}}(\underline{\hat{\mathbf{x}}}\_{\cdot})^{\dagger} \,\,\Delta \underline{\tilde{\mathbf{z}}}\_{\cdot},\tag{55}
$$

where the † operator is defined as the Moore-Penrose pseudoinverse [3].

Aiming to avoid explicitly store, the *Q*-matrix, we apply the *QR*-transformation to the augmented matrix, **Ha** *x*^*<sup>c</sup>* ð Þ, given by

$$\mathbf{H}\_{\mathbf{a}}(\underline{\hat{x}}\_{\mathcal{c}}) = \begin{bmatrix} \tilde{\mathbf{H}}(\underline{\hat{x}}\_{\mathcal{c}}) \ \Delta \tilde{\mathbf{z}}\_{\mathcal{c}} \end{bmatrix}. \tag{56}$$

By storing the rotations in compact form, the complex-valued Jacobian matrix can be kept constant and only the right-hand-side vector is updated throughout the final iterations. The solution of the state vector increment given by Eq. (55) is found by executing a simple back-substitution of Eq. (56) after performing unitary transformations to the latter matrix, resulting in

*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*

$$
\tilde{\mathbf{H}}\_{\mathbf{a}}(\hat{\underline{\mathbf{x}}}\_{\mathcal{c}}) = \begin{bmatrix} \mathbf{T}\_{\mathbf{c}} & \Delta \tilde{\underline{\mathbf{z}}}\_{\mathcal{c}} \end{bmatrix}. \tag{57}
$$

Here, **Tc** is an upper triangular matrix of dimension (2*<sup>n</sup>* � <sup>2</sup>*n*), and <sup>Δ</sup>~~*zc* comprises the corresponding rows in the updated *rhs* vector of dimension (2*n* � 1). Finally, Eq. (55) is solved by performing a back-substitution via

$$
\Delta \underline{\mathbf{x}}\_{\mathfrak{c}}^{(\boldsymbol{\nu})} = \mathbf{T}\_{\mathfrak{c}} \,\,\Delta \tilde{\underline{\mathbf{z}}}\_{\mathfrak{c}}.\tag{58}
$$

#### **5.1 Small example**

The one-line diagram of a 2-bus system is depicted in **Figure 5**. The system is provided with two PMU measurements that meter the nodal voltage magnitudes and phase angles and two real and reactive power flow measurements, which are identified by means of black bullets and red triangles, respectively. In **Table 6**, the transmission line parameters are given in *pu*.

From **Figure 5**, the complex power injections at sending and receiving end, respectively, are written yielding

$$\mathbf{S}\_1 = V\_1 \left[ \left( \mathbf{y}\_{12}^\* - j \, \mathbf{b}\_{12}^{sh} \right) \, \mathbf{V}\_1^\* - \mathbf{y}\_{12}^\* \, \mathbf{V}\_2^\* \right],\tag{59}$$

$$\mathcal{S}\_2 = V\_2 \left[ \left( \mathcal{y}\_{12}^\* - j \, b\_{12}^{sh} \right) \, \mathcal{V}\_2^\* - \mathcal{y}\_{12}^\* \, \mathcal{V}\_1^\* \right]. \tag{60}$$

Applying the Wirtinger calculus to Eqs. (59) and (60) leads to the Jacobian matrix given by Eq. (47) at each iteration as shown in the sequence. Here, the complex power injection measurement at each end, i.e., *S*<sup>1</sup> and *S*2, is equal to the corresponding power flow measurement, i.e., *S*<sup>12</sup> and *S*21, respectively.

**Figure 5.** *2-Bus power system.*

Nonetheless, the recommended numerical procedure aiming to solve the complex-valued power system state estimation problem is addressed by solving the weighted form of the right-hand-side (*rhs*) of Eq. (40) instead of Eq. (44), yielding

*Sparsity structure of (a) real-valued Jacobian matrix; (b) real-valued gain matrix; (c) complex-valued Jacobian matrix; and (d) complex-valued hessian matrix of the Brazilian equivalent 730-bus system.*

**<sup>H</sup>**<sup>~</sup> *<sup>x</sup>*^*<sup>c</sup>* ð Þ <sup>Δ</sup>*x*ð Þ*<sup>ν</sup>*

Δ*x*ð Þ*<sup>ν</sup>*

to the augmented matrix, **Ha** *x*^*<sup>c</sup>* ð Þ, given by

*Advances in Complex Analysis and Applications*

formations to the latter matrix, resulting in

calculated via

**56**

**Figure 4.**

where **<sup>H</sup>**<sup>~</sup> *<sup>x</sup>*^*<sup>c</sup>* ð Þ¼ <sup>Ω</sup>�1*=*<sup>2</sup> *<sup>c</sup>* **<sup>H</sup>** *<sup>x</sup>*^*<sup>c</sup>* ð Þ is of dimension (2*<sup>m</sup>* � <sup>2</sup>*n*) and <sup>Δ</sup>~*zc* <sup>¼</sup> <sup>Ω</sup>�1*=*<sup>2</sup> *<sup>c</sup>* <sup>Δ</sup>*zc* is of dimension (2*m* � 1). Thus, the incremental changes in the state vector are

where the † operator is defined as the Moore-Penrose pseudoinverse [3]. Aiming to avoid explicitly store, the *Q*-matrix, we apply the *QR*-transformation

**Ha** *<sup>x</sup>*^*<sup>c</sup>* ð Þ¼ **<sup>H</sup>**<sup>~</sup> *<sup>x</sup>*^*<sup>c</sup>* ð Þ <sup>Δ</sup>~*zc*

By storing the rotations in compact form, the complex-valued Jacobian matrix can be kept constant and only the right-hand-side vector is updated throughout the final iterations. The solution of the state vector increment given by Eq. (55) is found by executing a simple back-substitution of Eq. (56) after performing unitary trans-

*<sup>c</sup>* ¼ Δ~*zc*, (54)

*<sup>c</sup>* <sup>¼</sup> **<sup>H</sup>**<sup>~</sup> *<sup>x</sup>*^*<sup>c</sup>* ð Þ† <sup>Δ</sup>~*zc*, (55)

*:* (56)


**Table 6.** *Branch data.*

As observed in the above expressions of **Jc**, the sub-matrix **Jh** is more sparse than the sub-matrix **J d h**, which affects the sparsity structure of the complex-valued Jacobian matrix as shown earlier in **Figure 4**. The state variables at each iteration are provided in **Table 7**, while the estimated measured quantities are given in **Table 8**.


The residual vector and the chi-squared index throughout the iterations are

*r*ð Þ *<sup>ν</sup>*¼*<sup>i</sup> r*ð Þ *<sup>ν</sup>*¼**<sup>0</sup>** *r*ð Þ *<sup>ν</sup>*¼**<sup>1</sup>** *r*ð Þ *<sup>ν</sup>*¼**<sup>2</sup>** *r*ð Þ *<sup>ν</sup>*¼**<sup>3</sup>** *rV*<sup>1</sup> 0*:*0000 þ *j* 0*:*0000 0*:*1991 þ *j* 0*:*0000 �0*:*0097 þ *j* 0*:*0018 0*:*0000 � *j* 0*:*0000 *rV*<sup>2</sup> �0*:*1349 � *j* 0*:*2333 0*:*1634 � *j* 0*:*0000 �0*:*0017 � *j* 0*:*0015 0*:*0006 � *j* 0*:*0001 *rS*<sup>12</sup> 1*:*8827 þ *j* 0*:*7375 0*:*4080 þ *j* 0*:*2499 �0*:*0046 � *j* 0*:*0571 �0*:*0000 � *j* 0*:*0004 *rS*<sup>21</sup> �1*:*7998 � *j* 0*:*1366 �0*:*3984 � *j* 0*:*3790 0*:*0038 þ *j* 0*:*0597 �0*:*0001 þ *j* 0*:*0003 *rS*<sup>1</sup> 1*:*8827 þ *j* 0*:*7375 0*:*4080 þ *j* 0*:*2499 �0*:*0046 � *j* 0*:*0571 �0*:*0000 � *j* 0*:*0004 *rS*<sup>2</sup> �1*:*7998 � *j* 0*:*1366 �0*:*3984 � *j* 0*:*3790 0*:*0038 þ *j* 0*:*0597 �0*:*0001 þ *j* 0*:*0003 <sup>J</sup> ð Þ *<sup>x</sup>*^ ð Þ *<sup>ν</sup>*¼*<sup>i</sup>* <sup>14</sup>*:*<sup>7654</sup> <sup>1</sup>*:*<sup>1288</sup> <sup>0</sup>*:*<sup>013825</sup> <sup>8</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>7</sup>

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

both vector spaces, i.e., real and complex domain.

*Performance in larger systems—overdetermined matrices.*

**CV-Jacobian matrix dimension 2***m* � **2***n*

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

the convergence criterion of 1 � <sup>10</sup>�<sup>3</sup> is assumed.

**6. Conclusions and future developments**

achieve the solution is lower than the remainder approaches.

**5.2 Performance in larger systems**

*(r)—rectangular coordinates; tol.—*<sup>1</sup> � <sup>10</sup>�<sup>3</sup>*.*

**Table 9.** *CV-residual vector.*

**Table 10.**

**59**

presented in **Table 9**. Notice that the estimated values obtained in the -domain are exactly equal to those extracted from the power flow report in the -domain. Furthermore, the Gauss-Newton iterative algorithm converges in three iterations in

**Algorithms Number of**

IEEE-14 <sup>60</sup> � <sup>28</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> 3 0***:***045303 0***:***<sup>108547</sup>** IEEE-30 <sup>124</sup> � <sup>60</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> 3 0***:***067118 0***:***<sup>158415</sup>** IEEE-118 368 � <sup>236</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> 8 0***:***150205 1***:***<sup>064942</sup>** SIN-340 1704 � <sup>680</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> 7 0***:***992872 6***:***<sup>724038</sup>** SIN-1916 9642 � <sup>3832</sup> **<sup>3</sup>***:***CV** � **NRM**ð Þ**<sup>r</sup> <sup>8</sup> <sup>29</sup>***:***856246 242***:***<sup>695989</sup>**

**iterations**

**Time/ iteration (s)** **Total time (s)**

In the sequel, **Table 10** presents the performance on larger systems of the Gauss-Newton method in complex plane which is highlighted in bold. In all simulations

The comparative analysis of the results presented in the aforementioned table allows us to infer that the Newton-Raphson method in complex plane has very good performance. Except for the SIN-1916 bus system, the time consuming required to

The chapter's prime goal is to present the advances aiming to solve nonlinear system of equations in complex plane, regardless if it is exactly or overdetermined. Firstly, it develops the former Newton-Raphson method addressed to solve exactly determined problem. Thereafter, the weighted-least-squares (WLS) is developed

#### **Table 7.**

*Estimated state variables.*


**Table 8.** *CV-estimated quantities.*

*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*


#### **Table 9.**

*CV-residual vector.*


#### **Table 10.**

As observed in the above expressions of **Jc**, the sub-matrix **Jh** is more sparse than

bian matrix as shown earlier in **Figure 4**. The state variables at each iteration are provided in **Table 7**, while the estimated measured quantities are given in **Table 8**.

*x x*ð Þ *<sup>ν</sup>*¼**<sup>0</sup>** *x*ð Þ *<sup>ν</sup>*¼**<sup>1</sup>** *x*ð Þ *<sup>ν</sup>*¼**<sup>2</sup>** *x*ð Þ *<sup>ν</sup>*¼**<sup>3</sup>** *V*<sup>1</sup> 1*:*0000 1*:*0097 1*:*0000 1*:*0001

*V*<sup>2</sup> 1*:*0000 0*:*8973 0*:*8954 0*:*8956

<sup>1</sup> 1*:*0000 1*:*0097 1*:*0000 1*:*0001

<sup>2</sup> 1*:*0000 0*:*8973 0*:*8954 0*:*8956

^*zi* ^*z*ð Þ *<sup>ν</sup>*¼**<sup>0</sup>** ^*z*ð Þ *<sup>ν</sup>*¼**<sup>1</sup>** ^*z*ð Þ *<sup>ν</sup>*¼**<sup>2</sup>** ^*z*ð Þ *<sup>ν</sup>*¼**<sup>3</sup>**

*S*<sup>12</sup> 0.0000 � j 0.3131 +1.4747 + j 0.1745 +1.8873 + j 0.4815 +1.8827 + j 0.4248 *S*<sup>21</sup> 0.0000 � j 0.3131 �1.4014 � j 0.0707 �1.8036 � j 0.5094 �1.7997 � j 0.4500 *S*<sup>1</sup> 0.0000 � j 0.3131 +1.4747 + j 0.1745 +1.8873 + j 0.4815 +1.8827 + j 0.4248 *S*<sup>2</sup> 0.0000 � j 0.3131 �1.4014 � j 0.0707 �1.8036 � j 0.5094 �1.7997 � j 0.4500

**h**, which affects the sparsity structure of the complex-valued Jaco-

*e*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>0</sup> *e*�*<sup>j</sup>* <sup>0</sup>*:*<sup>1015</sup> *e*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>0018</sup> *e*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>0004</sup>

*e*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>0</sup> *e*�*<sup>j</sup>* <sup>14</sup>*:*<sup>9708</sup> *e*�*<sup>j</sup>* <sup>15</sup>*:*<sup>0924</sup> *e*�*<sup>j</sup>* <sup>15</sup>*:*<sup>0904</sup>

*e*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>0</sup> *e*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>1015</sup> *e*�*<sup>j</sup>* <sup>0</sup>*:*<sup>0018</sup> *e*�*<sup>j</sup>* <sup>0</sup>*:*<sup>0004</sup>

*e*þ*<sup>j</sup>* <sup>0</sup>*:*<sup>0</sup> *e*þ*<sup>j</sup>* <sup>14</sup>*:*<sup>9708</sup> *e*þ*<sup>j</sup>* <sup>15</sup>*:*<sup>0924</sup> *e*þ*<sup>j</sup>* <sup>15</sup>*:*<sup>0904</sup>

the sub-matrix **J**

*V* <sup>∗</sup>

*V* <sup>∗</sup>

**Table 7.**

**Table 8.**

**58**

*Estimated state variables.*

*CV-estimated quantities.*

**d**

*Advances in Complex Analysis and Applications*

*Performance in larger systems—overdetermined matrices.*

The residual vector and the chi-squared index throughout the iterations are presented in **Table 9**. Notice that the estimated values obtained in the -domain are exactly equal to those extracted from the power flow report in the -domain. Furthermore, the Gauss-Newton iterative algorithm converges in three iterations in both vector spaces, i.e., real and complex domain.

#### **5.2 Performance in larger systems**

In the sequel, **Table 10** presents the performance on larger systems of the Gauss-Newton method in complex plane which is highlighted in bold. In all simulations the convergence criterion of 1 � <sup>10</sup>�<sup>3</sup> is assumed.

The comparative analysis of the results presented in the aforementioned table allows us to infer that the Newton-Raphson method in complex plane has very good performance. Except for the SIN-1916 bus system, the time consuming required to achieve the solution is lower than the remainder approaches.

#### **6. Conclusions and future developments**

The chapter's prime goal is to present the advances aiming to solve nonlinear system of equations in complex plane, regardless if it is exactly or overdetermined. Firstly, it develops the former Newton-Raphson method addressed to solve exactly determined problem. Thereafter, the weighted-least-squares (WLS) is developed

#### *Advances in Complex Analysis and Applications*

through the Gauss-Newton method. In both cases the applications are carried out on small examples and larger systems. All results demonstrate the advantages of the algorithms developed in complex plane over the former procedure, although the computational burden is still a bottleneck for larger systems. Highlight that there are many enhancements that can be addressed to mitigate this problem. To cite a few, they are shown as follows:

**References**

[1] Steinmetz CP. Complex quantities and their use in electrical engineering. In: Proceedings of the International Electrical Congress. Chicago, IL: AIEE

*DOI: http://dx.doi.org/10.5772/intechopen.92741*

*Solution Methods of Large Complex-Valued Nonlinear System of Equations*

[8] Gentleman WM. Least squares computations by givens transformations

without square roots. Journal of Mathematical Analysis and Applications. 1974;**12**:329-336

[10] Wang JW, Quintana VH. A decoupled orthogonal row processing algorithm for power system state estimation. IEEE Transactions on Power Apparatus and Systems. 1984;**PAS-103**

[11] Vempati N, Slutsker IW, Tinney WF. Enhancement to givens rotations for power system state

Systems. 1991;**6**(2):842-849

[12] Catalyurek UV, Aykanat CT, Kayaaslan E. Hypergraph partitioningbased fill-reducing ordering for symmetric matrices. SIAM Journal on Scientific Computing. 2011;**4**:1996-2023

[13] Kaya O, Kayaaslan E, Uçar B, and Duff I. Fill-in reduction in sparse matrix

[14] Pires R, Mili L, Chagas G. Robust complex-valued Levenberg-Marquardt algorithm as applied to power flow analysis. International Journal of Electrical Power & Energy Systems. 2019;**113**:383-392. DOI: 10.1016/j.

factorizations using hypergraphs [Research Report—8448]. 2014

[15] Pires RC, Mili L, Lemos FAB. Constrained robust estimation of power system state variables and transformer tap positions under erroneous zeroinjections. IEEE Transactions on Power Systems. 2014;**29**(3):1144-1152. ISSN:

ijepes.2019.05.032

0885-8950

estimation. IEEE Transactions on Power

(8):2337-2344

[9] Simões-Costa AJA, Quintana VH. An orthogonal row processing algorithm for power sequential state estimation. IEEE Transactions on Power Apparatus and Systems. 1981;**100**(8):3791-3800

[2] Kreutz-Delgado K. The Complex Gradient Operator and the CR-Calculus.

Computer Engineering—Jacobs School

complex variables. Society for Industrial and Applied Mathematics—SIAM. 2012;

[4] Pires R. Complex-valued steady-state models as applied to power flow analysis and power system state estimation [PhD dissertation]. Itajuba, Brazil: Institute of Electrical Systems and Energy—ISEE; Federal University of Itajuba—UNIFEI,

2018. Available from: https://re positorio.unifei.edu.br/xmlui/handle/

[5] David AHJ. A generalization of the conjugate-gradient method to solve complex systems. IMA Journal of Numerical Analysis. 1986;**6**(4):

[6] Maltsev A, Pestretsov V,

Balsara PT. Complex QR decomposition using fast plane

Maslennikov R, Khoryaev A. Triangular systolic array with reduced latency for QR-decomposition of complex matrices. In: IEEE International Symposium on Circuits and Systems—ISCAS. 2006.

[7] Awasthi A, Guttal R, Al-Dhahir N,

rotations for MIMO applications. IEEE Communications Letters. 2014;**18**(10):

Proceedings; 1893. pp. 33-74

San Diego, USA: Electrical and

of Engineering; University of California; 2009. pp. 1-74. ArXIV e-print, arXIV:0906.4835v1 [math.OC]

[3] Sorber L, Van Barel M, de Lathauwer L. Unconstrained optimization of real functions in

**22**(3):879-898

123456789/55

447-452

pp. 385-388

1743-1746

**61**


### **Nomenclature**


### **Author details**

Robson Pires Institute of Electric Systems and Energy—ISEE, Federal University of Itajubá—UNIFEI, Itajubá, Minas Geriais, Brazil

\*Address all correspondence to: rpires@unifei.edu.br

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Solution Methods of Large Complex-Valued Nonlinear System of Equations DOI: http://dx.doi.org/10.5772/intechopen.92741*

#### **References**

through the Gauss-Newton method. In both cases the applications are carried out on small examples and larger systems. All results demonstrate the advantages of the algorithms developed in complex plane over the former procedure, although the computational burden is still a bottleneck for larger systems. Highlight that there are many enhancements that can be addressed to mitigate this problem. To cite a

1.Ordering schemes based on rows overlapped by columns ordering schemes

2. Suppressing of complex conjugate calculation storing during the Jacobian

*xc* Vector of the state variables in the conjugate coordinate system

**J** Complex-valued Jacobian matrix for the exactly determined system

**H** Complex-valued Jacobian matrix for the overdetermined system of

*x*, *x*<sup>∗</sup> Complex and complex conjugate state variables *t*, *t* <sup>∗</sup> Complex and complex conjugate tap position ℜf g� , f g� Real and imaginary part of a complex variable

few, they are shown as follows:

*Advances in Complex Analysis and Applications*

matrices building.

of equations

*M* Complex-valued mismatch vector

Itajubá—UNIFEI, Itajubá, Minas Geriais, Brazil

provided the original work is properly cited.

\*Address all correspondence to: rpires@unifei.edu.br

ð Þ� *<sup>c</sup>* Quantity in the conjugate coordinate system

Institute of Electric Systems and Energy—ISEE, Federal University of

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

equations

k k� <sup>2</sup> Squared Euclidean norm

k k� <sup>∞</sup> Infinity norm *ν* Iteration counter

**Author details**

Robson Pires

**60**

[12, 13].

**Nomenclature**

[1] Steinmetz CP. Complex quantities and their use in electrical engineering. In: Proceedings of the International Electrical Congress. Chicago, IL: AIEE Proceedings; 1893. pp. 33-74

[2] Kreutz-Delgado K. The Complex Gradient Operator and the CR-Calculus. San Diego, USA: Electrical and Computer Engineering—Jacobs School of Engineering; University of California; 2009. pp. 1-74. ArXIV e-print, arXIV:0906.4835v1 [math.OC]

[3] Sorber L, Van Barel M, de Lathauwer L. Unconstrained optimization of real functions in complex variables. Society for Industrial and Applied Mathematics—SIAM. 2012; **22**(3):879-898

[4] Pires R. Complex-valued steady-state models as applied to power flow analysis and power system state estimation [PhD dissertation]. Itajuba, Brazil: Institute of Electrical Systems and Energy—ISEE; Federal University of Itajuba—UNIFEI, 2018. Available from: https://re positorio.unifei.edu.br/xmlui/handle/ 123456789/55

[5] David AHJ. A generalization of the conjugate-gradient method to solve complex systems. IMA Journal of Numerical Analysis. 1986;**6**(4): 447-452

[6] Maltsev A, Pestretsov V, Maslennikov R, Khoryaev A. Triangular systolic array with reduced latency for QR-decomposition of complex matrices. In: IEEE International Symposium on Circuits and Systems—ISCAS. 2006. pp. 385-388

[7] Awasthi A, Guttal R, Al-Dhahir N, Balsara PT. Complex QR decomposition using fast plane rotations for MIMO applications. IEEE Communications Letters. 2014;**18**(10): 1743-1746

[8] Gentleman WM. Least squares computations by givens transformations without square roots. Journal of Mathematical Analysis and Applications. 1974;**12**:329-336

[9] Simões-Costa AJA, Quintana VH. An orthogonal row processing algorithm for power sequential state estimation. IEEE Transactions on Power Apparatus and Systems. 1981;**100**(8):3791-3800

[10] Wang JW, Quintana VH. A decoupled orthogonal row processing algorithm for power system state estimation. IEEE Transactions on Power Apparatus and Systems. 1984;**PAS-103** (8):2337-2344

[11] Vempati N, Slutsker IW, Tinney WF. Enhancement to givens rotations for power system state estimation. IEEE Transactions on Power Systems. 1991;**6**(2):842-849

[12] Catalyurek UV, Aykanat CT, Kayaaslan E. Hypergraph partitioningbased fill-reducing ordering for symmetric matrices. SIAM Journal on Scientific Computing. 2011;**4**:1996-2023

[13] Kaya O, Kayaaslan E, Uçar B, and Duff I. Fill-in reduction in sparse matrix factorizations using hypergraphs [Research Report—8448]. 2014

[14] Pires R, Mili L, Chagas G. Robust complex-valued Levenberg-Marquardt algorithm as applied to power flow analysis. International Journal of Electrical Power & Energy Systems. 2019;**113**:383-392. DOI: 10.1016/j. ijepes.2019.05.032

[15] Pires RC, Mili L, Lemos FAB. Constrained robust estimation of power system state variables and transformer tap positions under erroneous zeroinjections. IEEE Transactions on Power Systems. 2014;**29**(3):1144-1152. ISSN: 0885-8950

Section 3

Advanced Complex Analysis

and Geometry Studies

**63**

Section 3

## Advanced Complex Analysis and Geometry Studies

**Chapter 5**

**Abstract**

analytic functions

**1. Introduction**

*u* such that

such that

**65**

izations of classical spaces.

for *λ*>0 and *pn* ∈½ Þ 1, ∞ .

Variable Exponent Spaces

*Gerardo A. Chacón and Gerardo R. Chacón*

Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

**Keywords:** variable exponents, Hardy spaces, Bergman spaces, Carleson measures,

In recent years the interest for nonstandard function spaces has risen due to several considerations including their need in some fields of applied mathematics, differential equations, and simply the possibility to study extensions and general-

One of such type of spaces is the variable exponent Lebesgue spaces. A first introduction to them was due to Orlicz in 1931. He considered measurable functions

j j *u x*ð Þ *p x*ð Þ <sup>d</sup>*x*<sup>&</sup>lt; <sup>∞</sup>,

the usual exponent *p* from the classical theory of Lebesgue spaces is replaced by a suitable function *p*ð Þ� . Orlicz was interested in the summability of Fourier series

j j *<sup>λ</sup>an pn* <sup>&</sup>lt; <sup>∞</sup>,

Later generalizations consisted on considering real functions *u* on a domain Ω

*φ*ð Þ *x*, j*u x*ð Þj d*x*< ∞,

ð1 0

> X∞ *n*¼1

which led him to consider spaces of sequences ð Þ *an* so that

ð Ω

of Analytic Functions

#### **Chapter 5**

## Variable Exponent Spaces of Analytic Functions

*Gerardo A. Chacón and Gerardo R. Chacón*

#### **Abstract**

Variable exponent spaces are a generalization of Lebesgue spaces in which the exponent is a measurable function. Most of the research done in this topic has been situated under the context of real functions. In this work, we present two examples of variable exponent spaces of analytic functions: variable exponent Hardy spaces and variable exponent Bergman spaces. We will introduce the spaces together with some basic properties and the main techniques used in the context. We will show that in both cases, the boundedness of the evaluation functionals plays a key role in the theory. We also present a section of possible directions of research in this topic.

**Keywords:** variable exponents, Hardy spaces, Bergman spaces, Carleson measures, analytic functions

#### **1. Introduction**

In recent years the interest for nonstandard function spaces has risen due to several considerations including their need in some fields of applied mathematics, differential equations, and simply the possibility to study extensions and generalizations of classical spaces.

One of such type of spaces is the variable exponent Lebesgue spaces. A first introduction to them was due to Orlicz in 1931. He considered measurable functions *u* such that

$$\int\_0^1 |u(\mathbf{x})|^{p(\mathbf{x})} \, \mathbf{d}\mathbf{x} < \infty,$$

the usual exponent *p* from the classical theory of Lebesgue spaces is replaced by a suitable function *p*ð Þ� . Orlicz was interested in the summability of Fourier series which led him to consider spaces of sequences ð Þ *an* so that

$$\sum\_{n=1}^{\infty} |\lambda a\_n|^{p\_n} < \infty,$$

for *λ*>0 and *pn* ∈½ Þ 1, ∞ .

Later generalizations consisted on considering real functions *u* on a domain Ω such that

$$\int\_{\Omega} \rho(\mathfrak{x}, |u(\mathfrak{x})|) \, \mathrm{d}\mathfrak{x} < \infty,$$

where

$$\varphi: \mathfrak{Q} \times [\mathfrak{O}, \mathfrak{so}) \to [\mathfrak{O}, \mathfrak{so})$$

**2. Variable exponents**

For Ω ⊂ *<sup>d</sup>* we put *p*<sup>þ</sup>

abbreviations *p*<sup>þ</sup> ¼ *p*<sup>þ</sup>

*ρp*ð Þ� ,*<sup>μ</sup>* by

such that

Hölder continuous.

**67**

*Lp*ð Þ� ð Þ <sup>Ω</sup> . Such operator is defined as

defining parameter *p* is allowed to vary.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

and the *Luxemburg-Nakano norm* by

Perhaps the most known function spaces in mathematical analysis are the *Lebesgue spaces Lp*. These are a fundamental basis of measure theory, functional analysis, harmonic analysis, and differential equations, among other areas. A recent generalization is the so-called *variable exponent Lebesgue spaces*. In this case, the

<sup>Ω</sup> ≔ ess sup*x*<sup>∈</sup> <sup>Ω</sup> *p x*ð Þ and *p*�

*ρ<sup>p</sup>*ð Þ� ,*<sup>μ</sup>*ð Þ *φ* ≔

<sup>Ω</sup> and *p*� ¼ *p*�

The basics on variable Lebesgue spaces may be found in the monographs [1, 17, 18].

it a *variable exponent* and define the set of all variable exponents with *p*<sup>þ</sup> < ∞ as Pð Þ Ω . For a complex-valued measurable function *φ* : Ω ! , we define the *modular*

j j *<sup>φ</sup>*ð Þ *<sup>x</sup> p x*ð Þ <sup>d</sup>*μ*ð Þ *<sup>x</sup>*

*C*log log 1ð Þ *<sup>=</sup>*j*<sup>x</sup>* � *<sup>y</sup>*<sup>j</sup> ,

> 2ℓ*C*log log <sup>2</sup><sup>ℓ</sup> ∣*x*�*y*∣ � �

for all *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>Ω</sup> such that <sup>∣</sup>*<sup>x</sup>* � *<sup>y</sup>*∣<sup>&</sup>lt; <sup>ℓ</sup>. We will write *<sup>p</sup>*ð Þ� <sup>∈</sup>Plogð Þ <sup>Ω</sup> when *<sup>p</sup>*ð Þ� is log -

This condition has proven to be very useful in the theory since, among other things, it implies the boundedness of the Hardy-Littlewood maximal operator in

> 1 ∣*B x*ð Þ ,*r* ∣

and represents an important tool in harmonic analysis. We used it in [19] to show

ð

*B x*ð Þ ,*r*

∣*f y*ð Þ∣d*y*

*φ λ* � � <sup>⩽</sup><sup>1</sup> n o*:* (1)

ð

Ω

Given *<sup>p</sup>*ð Þ� <sup>∈</sup>Pð Þ <sup>Ω</sup> , the *variable Lebesgue space Lp*ð Þ� ð Þ <sup>Ω</sup>, *<sup>μ</sup>* is defined as the set of all complex-valued measurable functions *φ* : Ω ! for which the modular is finite, i.e., *<sup>ρ</sup><sup>p</sup>*ð Þ� ,*<sup>μ</sup>*ð Þ *<sup>φ</sup>* <sup>&</sup>lt; <sup>∞</sup>*:* Equipped with the Luxemburg-Nakano norm (1) is a Banach space. Most of the research being done on variable exponent spaces makes use of a regularity condition in the variable exponent in order to have a "fruitful" theory. **Definition 1.** A function *p* : Ω ! is said to be log *-Hölder continuous—*or satisfy the *Dini-Lipschitz condition* on Ω if there exists a positive constant *C*log

<sup>∥</sup>*φ*∥*Lp*ð Þ� ð Þ <sup>Ω</sup>,*<sup>μ</sup>* <sup>≔</sup> inf *<sup>λ</sup>*><sup>0</sup> : *<sup>ρ</sup><sup>p</sup>*ð Þ� ,*<sup>μ</sup>*

∣*p x*ð Þ� *p y*ð Þ∣⩽

for all *x*, *y*∈ Ω such that ∣*x* � *y*∣<1*=*2, from which we obtain

*Mf x*ð Þ ≔ sup

*r*>0

that the Bergman projection is bounded on variable exponent Bergman spaces. Another consequence of the log -Hölder condition is the following inequality

∣*p x*ð Þ� *p y*ð Þ∣ ≤

<sup>Ω</sup> ≔ ess inf *<sup>x</sup>*<sup>∈</sup> <sup>Ω</sup> *p x*ð Þ; we use the

<sup>Ω</sup>. For a measurable function *p* : Ω ! ½ Þ 1, ∞ , we call

is a suitable function. Such spaces are known nowadays as Musielak-Orlicz spaces. We are interested in the case

$$
\rho(\mathfrak{x}, \mathfrak{y}) = \mathfrak{y}^{p(\mathfrak{x})},
$$

which corresponds to variable exponent Lebesgue spaces, with some adequate restrictions for the *exponent p*.

The study of problems in spaces with nonstandard growth has been mainly related to Lebesgue spaces with variable order *p x*ð Þ, to the corresponding Sobolev spaces, and to generalized Orlicz spaces (see surveys [1, 2]). The investigation of structural properties of variable exponent spaces and of operator theory in such spaces is of interest not only due to their intriguing mathematical structure—also worthy of investigation—but because such spaces appear in applications. One of these applications is the mathematical modeling of inhomogeneous materials, like electrorheological fluids. These are special viscous liquids that have the ability to change their mechanical properties significantly when they come in contact with an electric field. This can be explored in technological applications, like photonic crystals, smart inks, heterogeneous polymer composites, etc. [3–6].

Another known application is related to image restoration. Based on variable exponent spaces, the authors of [7] proposed a new model of image restoration combining the strength of the total variation approach and the isotropic diffusion approach. This model uses one value of *p* near the edge of images and another one near smooth regions, which leads to the variable exponent setting of the problem. Other references discussing image processing are [8–13]. This theory has also been applied to differential equations with nonstandard growth (see [14, 15]).

The theory of spaces of analytic functions on domains of the complex plane, or in general of *<sup>n</sup>*, happens in parallel. Starting from the classical Hardy *<sup>H</sup><sup>p</sup>*ð Þ and Bergman *<sup>A</sup><sup>p</sup>*ð Þ spaces, the functional Banach spaces have had a preferential study.

A Banach space X of complex-valued functions on a domain Ω is said to be *a functional Banach space* (cf. [16]) if the following hold:


Classical Hardy and Bergman spaces are examples of functional Banach spaces. In the case of spaces of analytic functions, the evaluation functionals *Kx*ð Þ¼ *f f x*ð Þ (*x*∈ Ω) are an extension of the well-known *reproducing kernels* in Hilbert spaces of analytic functions. The analyticity property of the functions in the space interacts with the Banach space structure by means of the continuity of evaluation functionals raising a fruitful relation between function theory and operator theory.

The setting of nonstandard spaces of analytic functions is a young area of research; a lot is yet to be explored. In this chapter, an introduction to the subject of variable exponent spaces of analytic functions will be given, making emphasis in the special place that evaluation functionals have in the theory.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

#### **2. Variable exponents**

where

in general of *<sup>n</sup>*

**66**

spaces. We are interested in the case

*Advances in Complex Analysis and Applications*

restrictions for the *exponent p*.

*φ* : Ω � ½ Þ! 0, ∞ ½ Þ 0, ∞

is a suitable function. Such spaces are known nowadays as Musielak-Orlicz

*<sup>φ</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> <sup>y</sup>p x*ð Þ,

which corresponds to variable exponent Lebesgue spaces, with some adequate

The study of problems in spaces with nonstandard growth has been mainly related to Lebesgue spaces with variable order *p x*ð Þ, to the corresponding Sobolev spaces, and to generalized Orlicz spaces (see surveys [1, 2]). The investigation of structural properties of variable exponent spaces and of operator theory in such spaces is of interest not only due to their intriguing mathematical structure—also worthy of investigation—but because such spaces appear in applications. One of these applications is the mathematical modeling of inhomogeneous materials, like electrorheological fluids. These are special viscous liquids that have the ability to change their mechanical properties significantly when they come in contact with an electric field. This can be explored in technological applications, like photonic

Another known application is related to image restoration. Based on variable exponent spaces, the authors of [7] proposed a new model of image restoration combining the strength of the total variation approach and the isotropic diffusion approach. This model uses one value of *p* near the edge of images and another one near smooth regions, which leads to the variable exponent setting of the problem. Other references discussing image processing are [8–13]. This theory has also been

The theory of spaces of analytic functions on domains of the complex plane, or

Bergman *<sup>A</sup><sup>p</sup>*ð Þ spaces, the functional Banach spaces have had a preferential study. A Banach space X of complex-valued functions on a domain Ω is said to be *a*

Classical Hardy and Bergman spaces are examples of functional Banach spaces. In the case of spaces of analytic functions, the evaluation functionals *Kx*ð Þ¼ *f f x*ð Þ (*x*∈ Ω) are an extension of the well-known *reproducing kernels* in Hilbert spaces of analytic functions. The analyticity property of the functions in the space interacts with the Banach space structure by means of the continuity of evaluation functionals raising a fruitful relation between function theory and operator theory. The setting of nonstandard spaces of analytic functions is a young area of research; a lot is yet to be explored. In this chapter, an introduction to the subject of variable exponent spaces of analytic functions will be given, making emphasis in the

, happens in parallel. Starting from the classical Hardy *<sup>H</sup><sup>p</sup>*ð Þ and

crystals, smart inks, heterogeneous polymer composites, etc. [3–6].

applied to differential equations with nonstandard growth (see [14, 15]).

• Vectorial operations are precisely the pointwise operations on Ω.

*functional Banach space* (cf. [16]) if the following hold:

• If *f x*ð Þ¼ *g x*ð Þ for every *x*∈ Ω, then *f* ¼ *g*.

• If *f x*ð Þ¼ *f y*ð Þ for every *f* ∈ X, then *x* ¼ *y*.

• Evaluation functionals *x*↦*f x*ð Þ are continuous on X.

special place that evaluation functionals have in the theory.

Perhaps the most known function spaces in mathematical analysis are the *Lebesgue spaces Lp*. These are a fundamental basis of measure theory, functional analysis, harmonic analysis, and differential equations, among other areas. A recent generalization is the so-called *variable exponent Lebesgue spaces*. In this case, the defining parameter *p* is allowed to vary.

The basics on variable Lebesgue spaces may be found in the monographs [1, 17, 18]. For Ω ⊂ *<sup>d</sup>* we put *p*<sup>þ</sup> <sup>Ω</sup> ≔ ess sup*x*<sup>∈</sup> <sup>Ω</sup> *p x*ð Þ and *p*� <sup>Ω</sup> ≔ ess inf *<sup>x</sup>*<sup>∈</sup> <sup>Ω</sup> *p x*ð Þ; we use the abbreviations *p*<sup>þ</sup> ¼ *p*<sup>þ</sup> <sup>Ω</sup> and *p*� ¼ *p*� <sup>Ω</sup>. For a measurable function *p* : Ω ! ½ Þ 1, ∞ , we call it a *variable exponent* and define the set of all variable exponents with *p*<sup>þ</sup> < ∞ as Pð Þ Ω .

For a complex-valued measurable function *φ* : Ω ! , we define the *modular ρp*ð Þ� ,*<sup>μ</sup>* by

$$\rho\_{p(\cdot),\mu}(\rho) := \int\_{\Omega} |\rho(\mathfrak{x})|^{p(\mathfrak{x})} \cdot \mathrm{d}\mu(\mathfrak{x}),$$

and the *Luxemburg-Nakano norm* by

$$\|\|\rho\|\|\_{L^{p(\cdot)}(\Omega,\mu)} := \inf \left\{ \lambda > 0 : \rho\_{p(\cdot),\mu} \left( \frac{\rho}{\lambda} \right) \leqslant 1 \right\}.\tag{1}$$

Given *<sup>p</sup>*ð Þ� <sup>∈</sup>Pð Þ <sup>Ω</sup> , the *variable Lebesgue space Lp*ð Þ� ð Þ <sup>Ω</sup>, *<sup>μ</sup>* is defined as the set of all complex-valued measurable functions *φ* : Ω ! for which the modular is finite, i.e., *<sup>ρ</sup><sup>p</sup>*ð Þ� ,*<sup>μ</sup>*ð Þ *<sup>φ</sup>* <sup>&</sup>lt; <sup>∞</sup>*:* Equipped with the Luxemburg-Nakano norm (1) is a Banach space.

Most of the research being done on variable exponent spaces makes use of a regularity condition in the variable exponent in order to have a "fruitful" theory.

**Definition 1.** A function *p* : Ω ! is said to be log *-Hölder continuous—*or satisfy the *Dini-Lipschitz condition* on Ω if there exists a positive constant *C*log such that

$$|p(\mathbf{x}) - p(\mathbf{y})| \lesssim \frac{C\_{\log}}{\log \left( 1/|\mathbf{x} - \mathbf{y}| \right)},$$

for all *x*, *y*∈ Ω such that ∣*x* � *y*∣<1*=*2, from which we obtain

$$|p(\mathbf{x}) - p(\mathbf{y})| \le \frac{2\ell C\_{\log}}{\log\left(\frac{2\ell}{|\mathbf{x} - \mathbf{y}|}\right)}$$

for all *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>Ω</sup> such that <sup>∣</sup>*<sup>x</sup>* � *<sup>y</sup>*∣<sup>&</sup>lt; <sup>ℓ</sup>. We will write *<sup>p</sup>*ð Þ� <sup>∈</sup>Plogð Þ <sup>Ω</sup> when *<sup>p</sup>*ð Þ� is log - Hölder continuous.

This condition has proven to be very useful in the theory since, among other things, it implies the boundedness of the Hardy-Littlewood maximal operator in *Lp*ð Þ� ð Þ <sup>Ω</sup> . Such operator is defined as

$$M\!f^\circ(\mathbf{x}) \coloneqq \sup\_{r>0} \frac{1}{|B(\mathbf{x},r)|} \int\_{B(\mathbf{x},r)} |f^\circ(\mathbf{y})| \mathbf{d}\mathbf{y}d\mathbf{y}$$

and represents an important tool in harmonic analysis. We used it in [19] to show that the Bergman projection is bounded on variable exponent Bergman spaces.

Another consequence of the log -Hölder condition is the following inequality

$$|B|^{p\_{-}(B)-p\_{+}(B)} \leq \mathcal{C}\_\*$$

<sup>∥</sup>*<sup>f</sup>* <sup>∥</sup>*hp*ð Þ� ð Þ <sup>¼</sup> sup

*every U* <sup>∈</sup> *hp*ð Þ� ð Þ *, there exists u* <sup>∈</sup>*Lp*ð Þ� ð Þ *such that Pu* <sup>¼</sup> *U and moreover,*

*f z*ð Þ¼ <sup>ð</sup>

For each *z*∈ , the functions *Kz* : ! defined as

Consequently, the set of polynomials is also dense in *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ .

*Kz*ð Þ¼ *w*

*as <sup>γ</sup>z*ð Þ¼ *<sup>f</sup> f z*ð Þ*. Then <sup>γ</sup><sup>z</sup> is a bounded operator for every z* <sup>¼</sup> <sup>∣</sup>*z*∣*e<sup>i</sup><sup>θ</sup>* <sup>∈</sup> *and*

<sup>∥</sup>*γz*∥≲ <sup>1</sup>

lowing theorem analogous to the constant exponent context.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

**Theorem 6.** *The following chain of inclusions holds*

and moreover, the inclusions are continuous.

of analytic functions *<sup>f</sup>* : ! such that *<sup>f</sup>* <sup>∈</sup>*h<sup>p</sup>*ð Þ� ð Þ .

as such, *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ is a Banach space. Recall that for all functions *f* ∈ *H*<sup>1</sup>

*on compact subsets of .*

1

**69**

½ �! 0, 2*π* <sup>þ</sup> *as*

Hölder continuity plays a key role.

*hp*<sup>þ</sup>

<sup>∥</sup>*U*∥*hp*ð Þ� ð Þ � <sup>∥</sup>*u*∥*Lp*ð Þ� ð Þ *.*

0 ≤*r* <1

Notice that, since *Pf* is harmonic for *<sup>f</sup>* <sup>∈</sup>*Lp*ð Þ� ð Þ , then Theorem 3 shows that the Poisson transform *<sup>P</sup>* : *Lp*ð Þ� ð Þ! *hp*ð Þ� ð Þ is bounded. Moreover, we have the fol-

**Theorem 5.** *Suppose that p* : ½ �! 0, 2*π* ½ Þ 1, ∞ *is* log *-Hölder continuous. Then for*

ð Þ <sup>⊂</sup>*hp*ð Þ� ð Þ <sup>⊂</sup>*hp*�

**Definition 7.** The variable exponent Hardy space *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ is defined as the space

In an analogous way to the classical setting *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ can be identified with the subspace of functions in *<sup>L</sup><sup>p</sup>*ð Þ� ð Þ whose negative Fourier coefficients are zero, and

> *f*ð Þ*ζ* 1 � *ζz*

> > 1 <sup>1</sup> � *zw* ,

are called *reproducing kernels*. They are bounded on and consequently belong to every space *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ . Moreover, as a consequence of the reproducing formula and the Hahn-Banach theorem, the linear span of f g *Kz* : *<sup>z</sup>*<sup>∈</sup> is dense in *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ .

**Theorem 8.** *For each z*<sup>∈</sup> *consider the linear functional <sup>γ</sup><sup>z</sup>* : *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ! *defined*

ð Þ <sup>1</sup>�j*z*<sup>j</sup> <sup>1</sup>*=p*ð Þ*<sup>θ</sup> :*

*Consequently, the convergence in the H<sup>p</sup>*ð Þ� ð Þ *-norm implies the uniform convergence*

The main tool for proving the previous theorem is following the version of a Forelli-Rudin inequality, adapted to the case of variable exponents. Here, the Log-

**Lemma 9.** *Suppose p* : ½ �! 0, 2*<sup>π</sup>* ½ Þ 1, <sup>∞</sup> *and q* : ½ �! 0, 2*<sup>π</sup>* ½ Þ 1, <sup>∞</sup> *are such that* <sup>1</sup>

*<sup>q</sup>*ð Þ*<sup>θ</sup>* <sup>¼</sup> <sup>1</sup> *for every <sup>θ</sup>* <sup>∈</sup>½ � 0, 2*<sup>π</sup> . Let* <sup>1</sup>*=*2<sup>&</sup>lt; *<sup>r</sup>*<<sup>1</sup> *and z* <sup>¼</sup> <sup>∣</sup>*z*∣*ei<sup>θ</sup>. Define the function <sup>φ</sup>* :

*<sup>φ</sup>*ðÞ¼ *<sup>t</sup>* ð Þ <sup>1</sup>�j*z*<sup>j</sup> <sup>1</sup>*=q*ð Þ*<sup>θ</sup>*

∣1 � ∣*z*∣*rei t*ð Þ �*<sup>θ</sup>* ∣

*:*

d*m*ð Þ*ζ :*

<sup>∥</sup> *fr*∥*Lp*ð Þ� ð Þ <sup>&</sup>lt; <sup>∞</sup>*:*

ð Þ ,

ð Þ , we have the reproducing formula:

*<sup>p</sup>*ð Þ*<sup>θ</sup>* þ

that hold for all balls *B* ⊂ . Here ∣*B*∣ stands for the Lebesgue measure of *B*, *p*þð Þ¼ *B* ess sup*<sup>z</sup>* <sup>∈</sup>*Bp z*ð Þ, and *p*�ð Þ¼ *B* ess inf *<sup>z</sup>*∈*Bp z*ð Þ. This will be a useful tool when studying complex function spaces since it allows to pass from an exponent that varies over a ball to a constant exponent at the center of such ball.

One interesting point of the theory of variable exponents is that, in general, the classical approach to Bergman spaces seems to fail in the variable framework. For example, in [19] it is shown that the Bergman projection is bounded from *Ap*ð Þ� ð Þ to *Ap*ð Þ� ð Þ . This is done by using the theory of Békollé-Bonami weights and a theorem that extends Rubio de Francia's extrapolation theory to variable Lebesgue spaces. This method is quite different from the usual ways of showing that the Bergman projection is bounded, which use either Schur's test or Calderón-Zygmund theory.

#### **3. Variable exponent Hardy spaces**

In this section we will give an introduction to the variable exponent Hardy spaces and the results and techniques that are usually found. We will follow the presentation as in [20].

**Definition 2.** For each *z* in the unit disk , the Poisson kernel *P z*ð Þ , *ζ* is defined as

$$P(z,\zeta) = \frac{1-\left|z\right|^2}{\left|z-\zeta\right|^2},$$

and the Poisson transform of a function *<sup>f</sup>* <sup>∈</sup>*L<sup>p</sup>*ð Þ� ð Þ is defined as

$$Pf(z) = \int\_{\mathbb{T}} P(z,\zeta)f(\zeta)m(\zeta),$$

where *m* denotes the normalized Lebesgue measure on . We will use the following result from [21]:

**Theorem 3.** *Suppose that p is* log *-Hölder continuous. For each* 0≤*r*<1*, and <sup>f</sup>* <sup>∈</sup>*L<sup>p</sup>*ð Þ� ð Þ *, the linear operator Pr* : *Lp*ð Þ� ð Þ! *Lp*ð Þ� ð Þ *defined as*

$$P\_{\mathcal{Y}}f(\zeta) = Pf(r\zeta).$$

is uniformly bounded on *Lp*ð Þ� ð Þ , <sup>∥</sup>*Pr*ð Þ*<sup>f</sup>* <sup>∥</sup>*Lp*ð Þ� ð Þ ≲∥*<sup>f</sup>* <sup>∥</sup>*Lp*ð Þ� ð Þ , and for every *<sup>f</sup>* <sup>∈</sup>*L<sup>p</sup>*ð Þ� ð Þ ,

$$\|f - P\_{\mathfrak{F}}f\|\_{L^{p(\cdot)}(\mathbb{T})} \to \mathbf{0}, \quad \text{as } r \to \mathbf{1}^-.$$

We are now ready to define the harmonic Hardy spaces with variable exponents. Given *f* : ! and 0 ≤*r*<1, the *dilations f <sup>r</sup>* : ! of *f* are defined as *fr*ð Þ¼ *ζ f r*ð Þ*ζ* .

**Definition 4.** The *variable exponent harmonic Hardy space hp*ð Þ� ð Þ is defined as the space of harmonic functions *f* : ! such that

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

j j *<sup>B</sup> <sup>p</sup>*�ð Þ� *<sup>B</sup> <sup>p</sup>*þð Þ *<sup>B</sup>* <sup>≤</sup>*C*,

that hold for all balls *B* ⊂ . Here ∣*B*∣ stands for the Lebesgue measure of *B*, *p*þð Þ¼ *B* ess sup*<sup>z</sup>* <sup>∈</sup>*Bp z*ð Þ, and *p*�ð Þ¼ *B* ess inf *<sup>z</sup>*∈*Bp z*ð Þ. This will be a useful tool when studying complex function spaces since it allows to pass from an exponent

In this section we will give an introduction to the variable exponent Hardy spaces and the results and techniques that are usually found. We will follow the

**Definition 2.** For each *z* in the unit disk , the Poisson kernel *P z*ð Þ , *ζ* is

*P z*ð Þ¼ , *<sup>ζ</sup>* <sup>1</sup> � j j *<sup>z</sup>*

**Theorem 3.** *Suppose that p is* log *-Hölder continuous. For each* 0≤*r*<1*, and*

*Prf*ð Þ¼ *ζ Pf r*ð Þ*ζ*

<sup>∥</sup>*<sup>f</sup>* � *Prf* <sup>∥</sup>*Lp*ð Þ� ð Þ ! 0, *as r* ! <sup>1</sup>�*:*

exponents. Given *f* : ! and 0 ≤*r*<1, the *dilations f <sup>r</sup>* : ! of *f* are defined as

**Definition 4.** The *variable exponent harmonic Hardy space hp*ð Þ� ð Þ is defined as

is uniformly bounded on *Lp*ð Þ� ð Þ , <sup>∥</sup>*Pr*ð Þ*<sup>f</sup>* <sup>∥</sup>*Lp*ð Þ� ð Þ ≲∥*<sup>f</sup>* <sup>∥</sup>*Lp*ð Þ� ð Þ , and for every

We are now ready to define the harmonic Hardy spaces with variable

and the Poisson transform of a function *<sup>f</sup>* <sup>∈</sup>*L<sup>p</sup>*ð Þ� ð Þ is defined as

ð 

*Pf z*ð Þ¼

where *m* denotes the normalized Lebesgue measure on .

*<sup>f</sup>* <sup>∈</sup>*L<sup>p</sup>*ð Þ� ð Þ *, the linear operator Pr* : *Lp*ð Þ� ð Þ! *Lp*ð Þ� ð Þ *defined as*

We will use the following result from [21]:

the space of harmonic functions *f* : ! such that

2 j j *<sup>z</sup>* � *<sup>ζ</sup>* <sup>2</sup> ,

*P z*ð Þ , *ζ f*ð Þ*ζ m*ð Þ*ζ* ,

One interesting point of the theory of variable exponents is that, in general, the classical approach to Bergman spaces seems to fail in the variable framework. For example, in [19] it is shown that the Bergman projection is bounded from *Ap*ð Þ� ð Þ to *Ap*ð Þ� ð Þ . This is done by using the theory of Békollé-Bonami weights and a theorem that extends Rubio de Francia's extrapolation theory to variable Lebesgue spaces. This method is quite different from the usual ways of showing that the Bergman projection is bounded, which use either Schur's test or Calderón-

that varies over a ball to a constant exponent at the center of such ball.

Zygmund theory.

presentation as in [20].

defined as

*<sup>f</sup>* <sup>∈</sup>*L<sup>p</sup>*ð Þ� ð Þ ,

*fr*ð Þ¼ *ζ f r*ð Þ*ζ* .

**68**

**3. Variable exponent Hardy spaces**

*Advances in Complex Analysis and Applications*

$$\|f\|\_{h^{p(\cdot)}(\mathbb{D})} = \sup\_{0 \le r < 1} \|f\_r\|\_{L^{p(\cdot)}(\mathbb{T})} < \infty.$$

Notice that, since *Pf* is harmonic for *<sup>f</sup>* <sup>∈</sup>*Lp*ð Þ� ð Þ , then Theorem 3 shows that the Poisson transform *<sup>P</sup>* : *Lp*ð Þ� ð Þ! *hp*ð Þ� ð Þ is bounded. Moreover, we have the following theorem analogous to the constant exponent context.

**Theorem 5.** *Suppose that p* : ½ �! 0, 2*π* ½ Þ 1, ∞ *is* log *-Hölder continuous. Then for every U* <sup>∈</sup> *hp*ð Þ� ð Þ *, there exists u* <sup>∈</sup>*Lp*ð Þ� ð Þ *such that Pu* <sup>¼</sup> *U and moreover,* <sup>∥</sup>*U*∥*hp*ð Þ� ð Þ � <sup>∥</sup>*u*∥*Lp*ð Þ� ð Þ *.*

**Theorem 6.** *The following chain of inclusions holds*

$$h^{p^{+}}(\mathbb{D}) \subset h^{p(\cdot)}(\mathbb{D}) \subset h^{p^{-}}(\mathbb{D}),$$

and moreover, the inclusions are continuous.

**Definition 7.** The variable exponent Hardy space *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ is defined as the space of analytic functions *<sup>f</sup>* : ! such that *<sup>f</sup>* <sup>∈</sup>*h<sup>p</sup>*ð Þ� ð Þ .

In an analogous way to the classical setting *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ can be identified with the subspace of functions in *<sup>L</sup><sup>p</sup>*ð Þ� ð Þ whose negative Fourier coefficients are zero, and as such, *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ is a Banach space.

Recall that for all functions *f* ∈ *H*<sup>1</sup> ð Þ , we have the reproducing formula:

$$f(\mathbf{z}) = \int\_{\mathbb{T}} \frac{f(\boldsymbol{\zeta})}{\mathbf{1} - \overline{\boldsymbol{\zeta}}\boldsymbol{z}} \, \mathrm{d}m(\boldsymbol{\zeta})\, .$$

For each *z*∈ , the functions *Kz* : ! defined as

$$K\_{\underline{x}}(w) = \frac{1}{1 - \overline{z}w},$$

are called *reproducing kernels*. They are bounded on and consequently belong to every space *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ . Moreover, as a consequence of the reproducing formula and the Hahn-Banach theorem, the linear span of f g *Kz* : *<sup>z</sup>*<sup>∈</sup> is dense in *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ . Consequently, the set of polynomials is also dense in *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ .

**Theorem 8.** *For each z*<sup>∈</sup> *consider the linear functional <sup>γ</sup><sup>z</sup>* : *<sup>H</sup><sup>p</sup>*ð Þ� ð Þ! *defined as <sup>γ</sup>z*ð Þ¼ *<sup>f</sup> f z*ð Þ*. Then <sup>γ</sup><sup>z</sup> is a bounded operator for every z* <sup>¼</sup> <sup>∣</sup>*z*∣*e<sup>i</sup><sup>θ</sup>* <sup>∈</sup> *and*

$$\|\gamma\_x\| \lesssim \frac{1}{\left(1 - |z|\right)^{1/p(\theta)}}.$$

*Consequently, the convergence in the H<sup>p</sup>*ð Þ� ð Þ *-norm implies the uniform convergence on compact subsets of .*

The main tool for proving the previous theorem is following the version of a Forelli-Rudin inequality, adapted to the case of variable exponents. Here, the Log-Hölder continuity plays a key role.

**Lemma 9.** *Suppose p* : ½ �! 0, 2*<sup>π</sup>* ½ Þ 1, <sup>∞</sup> *and q* : ½ �! 0, 2*<sup>π</sup>* ½ Þ 1, <sup>∞</sup> *are such that* <sup>1</sup> *<sup>p</sup>*ð Þ*<sup>θ</sup>* þ 1 *<sup>q</sup>*ð Þ*<sup>θ</sup>* <sup>¼</sup> <sup>1</sup> *for every <sup>θ</sup>* <sup>∈</sup>½ � 0, 2*<sup>π</sup> . Let* <sup>1</sup>*=*2<sup>&</sup>lt; *<sup>r</sup>*<<sup>1</sup> *and z* <sup>¼</sup> <sup>∣</sup>*z*∣*ei<sup>θ</sup>. Define the function <sup>φ</sup>* : ½ �! 0, 2*π* <sup>þ</sup> *as*

$$\wp(t) = \frac{(1 - |\mathbf{z}|)^{1/q(\theta)}}{|\mathbf{1} - |\mathbf{z}| \eta e^{i(t-\theta)} |}.$$

*Then if φ*ð Þ*t* >1*, it holds that*

$$
\varrho(t)^{p(t)} \lesssim \varrho(t)^{p(\theta)}.
$$

The additional term that appears in the previous inequality makes the boundedness of the domain a necessary condition to use such technique. One question to address in future investigations is how the situation is for Bergman spaces defined on unbounded domains. In such case, the error term accumulates despite the regu-

One technique developed to study variable exponent Bergman spaces is a combination of mollifiers and dilations. Such concept was developed for studying the density of the set of polynomials in Bergman spaces. In the classical case, this is a consequence of the fact that convergence in *Ap*ð Þ implies convergence on compact

*fr*ð Þ¼ *z f rz* ð Þ

One way of overcoming this problem in the case of variable exponent Lebesgue spaces is the use of a *mollification operator*. But in the case of the unit disk, changes need to be made. That is, when we had the idea to introduce a *mollified dilation*,

*f rw*ð Þ*ηr*ð Þ *z* � *w* d*A w*ð Þ*:*

2 *η*

0, if ∣*z*∣ ⩾1,

<sup>∥</sup> *fr*∥*Ap*ð Þ� ð Þ ≲ ∥*<sup>f</sup>* <sup>∥</sup>*Ap*ð Þ� ð Þ *:*

As a consequence of the previous theorem, we can approximate a function

2*rz* 1 � *r* � �

, if ∣*z*∣ <1,

<sup>2</sup>*<sup>r</sup>* . Hence, the sequence of Taylor

<sup>2</sup> , 1 � �, define the *mollified dilation*

converge in norm to *f* for *r* ! 1�. This is a simple fact that depends on the dilation-invariant nature of Bergman Spaces, something that does not hold in the case of variable exponent Bergman spaces. This is one of the main difficulties for

larity condition on *p*ð Þ� , and consequently a different technique is needed.

subsets of and that for any *<sup>f</sup>* <sup>∈</sup> *Ap*ð Þ , its *radial dilations*

*fr*ð Þ*z* ≔

where *ρ* stands for the complex disk with radius *ρ*,

*<sup>η</sup>*ð Þ*<sup>z</sup>* <sup>≔</sup> exp

8 ><

>:

sup ½ ⩽*r*<1

*Moreover,* <sup>∥</sup> *<sup>f</sup> <sup>r</sup>* � *<sup>f</sup>* <sup>∥</sup>*Ap*ð Þ� ð Þ ! <sup>0</sup> *as r* ! <sup>1</sup>�*.*

*<sup>f</sup>* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ by a function *fr* which is analytic in <sup>1</sup>þ*<sup>r</sup>*

ð 

*<sup>η</sup>r*ð Þ*<sup>z</sup>* <sup>≔</sup> <sup>4</sup>*r*<sup>2</sup>

ð Þ 1 � *r*

1 j j *z* <sup>2</sup> � <sup>1</sup>

Mollified dilations allow us to approximate analytic functions in the variable exponent Bergman spaces by functions that are analytic on a neighborhood of the

**Theorem 14.** *Let p*∈Plogð Þ *and let f* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ *. Then for* <sup>½</sup> <sup>⩽</sup>*r*<1*,*

!

**4.1 Mollified dilations**

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

generalizing the classical theory.

<sup>2</sup>*<sup>r</sup>* ! of *f* as

*fr* : <sup>1</sup>þ*<sup>r</sup>*

and

unit disk.

**71**

*fr* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ , *and*

given a function *<sup>f</sup>* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ and *<sup>r</sup>*<sup>∈</sup> <sup>1</sup>

#### **4. Variable exponent Bergman spaces**

The theory of Bergman spaces was introduced by S. Bergman in [22] and since the 1990s has gained a great deal of attention mainly due to some major breakthroughs at the time. For details on the theory of Bergman spaces, we refer to the books [5, 23].

**Definition 10.** Given a measurable function *p* ∈Pð Þ , we define the *variable exponent Bergman space Ap*ð Þ� ð Þ as the space of all analytic functions on that belong to the variable exponent Lebesgue space *Lp*ð Þ� ð Þ with respect to the area measure d*A* on the unit disk , i.e.,

$$\mathcal{A}^{p(\cdot)}(\mathbb{D}) = \left\{ f \text{ is an analytic function and} \int\_{\mathbb{D}} |f(z)|^{p(x)} \mathrm{d}A(z) < \infty \right\}.$$

One of the first results proven in [19] about this theory was the boundedness of the evaluation functionals that, as we saw before, will allow us to conclude that convergence in *Ap*ð Þ� ð Þ implies uniform convergence on compact subsets of and consequently *Lp*ð Þ� -limits of sequences in *Ap*ð Þ� ð Þ are analytic. Hence *Ap*ð Þ� ð Þ is a closed subspace of *<sup>L</sup><sup>p</sup>*ð Þ� ð Þ (and hence a Banach space). The first approach to such result used the following interpolation result based on Forelli-Rudin estimates:

**Lemma 11.** *Let p*∈Pð Þ *, then for every function f* <sup>∈</sup>*Lp*ð Þ� ð Þ *we have*

$$\|f\|\_{L^{p(\cdot)}(\mathbb{D})} \lesssim \|f\|\_{L^{1}(\mathbb{D})}^{1/p^{+}} \|f\|\_{L^{\infty}(\mathbb{D})}^{1-1/p^{+}} \cdot \|$$

Using different techniques, the following sharp estimate was obtained in [24]: **Theorem 12.** *Let p*ð Þ� <sup>∈</sup>Plogð Þ *. Then for every z*<sup>∈</sup> *we have that*

$$\|\|\boldsymbol{\chi}\_{x}\|\|\_{p(\cdot)} \asymp \frac{1}{\left(1-|x|^{2}\right)^{2/p(x)}}.\tag{2}$$

The proof of this fact relies on the Log-Hölder condition and on the following version of a Jensen-type inequality for variable exponent spaces (see [17]).

**Lemma 13.** *Suppose that p*ð Þ� <sup>∈</sup>Plogð Þ *and let*

$$\frac{1}{q(z,\boldsymbol{y})} \coloneqq \max\left\{ \frac{1}{p(\boldsymbol{z})} - \frac{1}{p(\boldsymbol{y})}, \mathbf{0} \right\}.$$

*Then for* 0<*γ* < 1*,*

$$\left(\int\_{B(\mathbf{x},r)} |f(\mathbf{y})| \mathbf{d} \mathbf{A}(\mathbf{y})\right)^{p(\mathbf{x})} \lesssim \int\_{B(\mathbf{x},r)} |f(\mathbf{y})|^{p(\mathbf{y})} \mathbf{d} \mathbf{A}(\mathbf{y}) + \int\_{B(\mathbf{x},r)} \mathbf{y}^{q(\mathbf{x},\mathbf{y})} \mathbf{d} \mathbf{A}(\mathbf{y}) \lesssim \frac{\varepsilon}{2}$$

for all *<sup>z</sup>*<sup>∈</sup> *B x*ð Þ ,*<sup>r</sup>* provided that <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup>*Lp*ð Þ� ð Þ <sup>⩽</sup>1*:* The constant depends on *<sup>γ</sup>* and *<sup>p</sup>*ð Þ� .

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

The additional term that appears in the previous inequality makes the boundedness of the domain a necessary condition to use such technique. One question to address in future investigations is how the situation is for Bergman spaces defined on unbounded domains. In such case, the error term accumulates despite the regularity condition on *p*ð Þ� , and consequently a different technique is needed.

#### **4.1 Mollified dilations**

*Then if φ*ð Þ*t* >1*, it holds that*

*Advances in Complex Analysis and Applications*

measure d*A* on the unit disk , i.e.,

*Ap*ð Þ� ð Þ¼ *<sup>f</sup>* is an analytic function and

<sup>∥</sup>*<sup>f</sup>* <sup>∥</sup>*Lp*ð Þ� ð Þ <sup>⩽</sup>∥*<sup>f</sup>* <sup>∥</sup>

**Theorem 12.** *Let p*ð Þ� <sup>∈</sup>Plogð Þ *. Then for every z*<sup>∈</sup> *we have that*

∥*γz*∥*<sup>p</sup>*ð Þ� ≍

**Lemma 13.** *Suppose that p*ð Þ� <sup>∈</sup>Plogð Þ *and let*

j*f y*ð Þjd*A y*ð Þ !*p z*ð Þ

*Then for* 0<*γ* < 1*,*

**70**

⨏ *B x*ð Þ *;r* 1

*q z*ð Þ , *<sup>y</sup>* <sup>≔</sup> max

≲ ⨏ *B x*ð Þ *;r*

books [5, 23].

**4. Variable exponent Bergman spaces**

*<sup>φ</sup>*ð Þ*<sup>t</sup> p t*ð Þ≲*φ*ð Þ*<sup>t</sup> <sup>p</sup>*ð Þ*<sup>θ</sup> :*

The theory of Bergman spaces was introduced by S. Bergman in [22] and since the 1990s has gained a great deal of attention mainly due to some major breakthroughs at the time. For details on the theory of Bergman spaces, we refer to the

**Definition 10.** Given a measurable function *p* ∈Pð Þ , we define the *variable exponent Bergman space Ap*ð Þ� ð Þ as the space of all analytic functions on that belong to the variable exponent Lebesgue space *Lp*ð Þ� ð Þ with respect to the area

> ð

1�1*=p*<sup>þ</sup> *<sup>L</sup>*∞ð Þ *:*

<sup>2</sup> � �<sup>2</sup>*=p z*ð Þ *:* (2)

� �

One of the first results proven in [19] about this theory was the boundedness of the evaluation functionals that, as we saw before, will allow us to conclude that convergence in *Ap*ð Þ� ð Þ implies uniform convergence on compact subsets of and consequently *Lp*ð Þ� -limits of sequences in *Ap*ð Þ� ð Þ are analytic. Hence *Ap*ð Þ� ð Þ is a closed subspace of *<sup>L</sup><sup>p</sup>*ð Þ� ð Þ (and hence a Banach space). The first approach to such result used the following interpolation result based on Forelli-Rudin estimates: **Lemma 11.** *Let p*∈Pð Þ *, then for every function f* <sup>∈</sup>*Lp*ð Þ� ð Þ *we have*

> 1*=p*þ *<sup>L</sup>*1ð Þ <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup>

Using different techniques, the following sharp estimate was obtained in [24]:

1 � j j *z*

The proof of this fact relies on the Log-Hölder condition and on the following

1 *p z*ð Þ � <sup>1</sup>

for all *<sup>z</sup>*<sup>∈</sup> *B x*ð Þ ,*<sup>r</sup>* provided that <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup>*Lp*ð Þ� ð Þ <sup>⩽</sup>1*:* The constant depends on *<sup>γ</sup>* and *<sup>p</sup>*ð Þ� .

*p y*ð Þ, 0

*:*

*B x*ð Þ *;r*

*<sup>γ</sup>q z*ð Þ *;<sup>y</sup>* <sup>d</sup>*A y*ð Þ

� �

j j *f y*ð Þ *p y*ð Þd*A y*ð Þþ <sup>⨏</sup>

version of a Jensen-type inequality for variable exponent spaces (see [17]).

1

j j *f z*ð Þ *p z*ð Þd*A z*ð Þ<sup>&</sup>lt; <sup>∞</sup>

*:*

One technique developed to study variable exponent Bergman spaces is a combination of mollifiers and dilations. Such concept was developed for studying the density of the set of polynomials in Bergman spaces. In the classical case, this is a consequence of the fact that convergence in *Ap*ð Þ implies convergence on compact subsets of and that for any *<sup>f</sup>* <sup>∈</sup> *Ap*ð Þ , its *radial dilations*

$$f\_r(\mathbf{z}) = f(\mathbf{z}\mathbf{z})$$

converge in norm to *f* for *r* ! 1�. This is a simple fact that depends on the dilation-invariant nature of Bergman Spaces, something that does not hold in the case of variable exponent Bergman spaces. This is one of the main difficulties for generalizing the classical theory.

One way of overcoming this problem in the case of variable exponent Lebesgue spaces is the use of a *mollification operator*. But in the case of the unit disk, changes need to be made. That is, when we had the idea to introduce a *mollified dilation*, given a function *<sup>f</sup>* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ and *<sup>r</sup>*<sup>∈</sup> <sup>1</sup> <sup>2</sup> , 1 � �, define the *mollified dilation fr* : <sup>1</sup>þ*<sup>r</sup>* <sup>2</sup>*<sup>r</sup>* ! of *f* as

$$f\_r(z) \coloneqq \int\_{\mathbb{D}} f(rw) \eta\_r(z - w) \mathrm{d}A(w).$$

where *ρ* stands for the complex disk with radius *ρ*,

$$\eta\_r(z) \coloneqq \frac{4r^2}{\left(1-r\right)^2} \eta\left(\frac{2rz}{1-r}\right),$$

and

$$\eta(z) \coloneqq \begin{cases} \exp\left(\frac{1}{\left|z\right|^2 - 1}\right), & \text{if} \quad |z| < 1, \\ 0, & \text{if} \quad |z| \geqslant 1, \end{cases}$$

Mollified dilations allow us to approximate analytic functions in the variable exponent Bergman spaces by functions that are analytic on a neighborhood of the unit disk.

**Theorem 14.** *Let p*∈Plogð Þ *and let f* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ *. Then for* <sup>½</sup> <sup>⩽</sup>*r*<1*, fr* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ , *and*

$$\sup\_{\forall \boldsymbol{\varepsilon} \leqslant r < 1} \|f\_r\|\_{A^{p(\cdot)}(\mathbb{D})} \lesssim \|f\|\_{A^{p(\cdot)}(\mathbb{D})}.$$

*Moreover,* <sup>∥</sup> *<sup>f</sup> <sup>r</sup>* � *<sup>f</sup>* <sup>∥</sup>*Ap*ð Þ� ð Þ ! <sup>0</sup> *as r* ! <sup>1</sup>�*.*

As a consequence of the previous theorem, we can approximate a function *<sup>f</sup>* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ by a function *fr* which is analytic in <sup>1</sup>þ*<sup>r</sup>* <sup>2</sup>*<sup>r</sup>* . Hence, the sequence of Taylor polynomials associated with *fr* converges uniformly on to *fr*. Therefore, given *<sup>ε</sup>*>0, there exists 1*=*2<sup>≤</sup> *<sup>r</sup>*<sup>&</sup>lt; 1 such that <sup>∥</sup>*<sup>f</sup>* � *fr*∥*Ap*ð Þ� ð Þ <sup>&</sup>lt;*ε=*2. And there exists a polynomial *<sup>p</sup>* such that <sup>∥</sup>*<sup>p</sup>* � *fr*∥*Ap*ð Þ� ð Þ <sup>&</sup>lt;*ε=*2.

A sequence of points f g *zn* <sup>⊂</sup> is a zero set for the Bergman space *Ap*ð Þ if there

1 � *r* � � � � , *<sup>r</sup>* ! <sup>1</sup>�*:*

> þ 1 2*π* ð<sup>2</sup>*<sup>π</sup>* 0

<sup>¼</sup> *O nr*ð Þ<sup>1</sup>*=<sup>p</sup>* � �, *<sup>r</sup>* ! <sup>1</sup>�*:*

Finding an extension of such result to variable exponents would include making

A related concept that we would like to address in variable exponent Bergman spaces is that of sampling sequences. A sequence of points f g *zk* ⊂ is sampling in

One difference that is expected to occur in the case of variable exponents is that sampling sequences will probably not be invariant under automorphism of the unit disk. This is true for a constant exponent, and it will be interesting to find counterexamples if the exponent varies. Sampling sequences are related to the concept of "frames" in Hilbert spaces and can be thought as sequences that contain great information of the space. It is expected that a sampling sequence is a somehow "big" and "spread out" subset of the unit disk. The notion is clearly related in *Ap* to the

> 1 � j j *zk* <sup>2</sup> � �<sup>2</sup>

*δzk*

j j *f z*ð Þ*<sup>k</sup> <sup>p</sup>* <sup>≤</sup>*C*∥*<sup>f</sup>* <sup>∥</sup>*<sup>p</sup>*

*p:*

This equation holds for every analytic function on . Consequently, if we want to find out whether a similar result about the growth of the zeroes holds in the variable exponent case, then it is necessary to study the radial growth of functions in *Ap*ð Þ� ð Þ . Such results, in the case of a constant exponent again, depend on the

Although the problem of completely describing the zero sets in *Ap*ð Þ is still open, some information is available. For example, if *n r*ð Þ denotes the number of zeroes of a function *<sup>f</sup>* <sup>∈</sup> *Ap*ð Þ having modulus smaller than *<sup>r</sup>*>0, then it is

<sup>1</sup> � *<sup>r</sup>* log <sup>1</sup>

This result depends on studying the radial growth of functions in the Bergman space and on Jensen's formula for the sequence f g *zk* of zeroes of *f* such

> log <sup>∣</sup>*zk*<sup>∣</sup> *r* � �

�∈f g *zn* .

log ∣*f re<sup>i</sup><sup>θ</sup>* � �∣d*θ:*

exists *<sup>f</sup>* <sup>∈</sup> *Ap*ð Þ such that *f z*ð Þ¼ *<sup>n</sup>* 0 for all *<sup>n</sup>* and *f z*ð Þ 6¼ 0 if *<sup>z</sup>*

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

*n r*ð Þ¼ *<sup>O</sup>* <sup>1</sup>

log <sup>∣</sup>*f*ð Þ <sup>0</sup> <sup>∣</sup> <sup>¼</sup> <sup>X</sup>*n r*ð Þ

.

Y*n r*ð Þ

1 ∣*zk*∣

*k*¼1

*c*∥*f* ∥*<sup>p</sup>*

*<sup>p</sup>* <sup>≤</sup> <sup>X</sup><sup>∞</sup> *k*¼1

*k*¼1

Another result involving zero sets is that if *f*ð Þ 0 6¼ 0, then

sense of the right-hand side term that is not even clear in this context.

*Ap*ð Þ if there exists constants *<sup>c</sup>*<sup>&</sup>gt; 0 and *<sup>C</sup>*<sup>&</sup>gt; 0 such that for all *<sup>f</sup>* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ

*<sup>μ</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *k*¼1

1 � j j *zk* <sup>2</sup> � �<sup>2</sup>

known that

that ∣*zk*Þ<*r*.

subharmonicity of j j *<sup>f</sup> <sup>p</sup>*

**5.2 Sampling sequences**

concept of Carleson measures.

**73**

If we define a discrete measure *μ* as

#### **4.2 Carleson measures**

Another classical problem that has a variable exponent equivalent consists on finding a geometrical characterization of Carleson measures in variable exponent Bergman spaces. Given a positive Borel measure *μ* defined on the unit disk , we say that *<sup>μ</sup>* is a *Carleson measure* for the variable exponent Bergman space *<sup>A</sup>p*ð Þ� ð Þ if there exists a constant *C*>0 such that

$$\|f\|\_{L^{p(\cdot)}(\mathbb{D},\mu)} \leqslant C \|f\|\_{A^{p(\cdot)}(\mathbb{D})}.$$

In other words, *<sup>μ</sup>* is a Carleson measure on *Ap*ð Þ� ð Þ if *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ is continuously embedded in *Lp*ð Þ� ð Þ , *<sup>μ</sup>* .

Lennart Carleson gave a geometric characterization of Carleson measures on Hardy spaces *H<sup>p</sup>* and used this result in his proof of the Corona theorem and some interpolation problems (cf. [25]). After that, Carleson measures have gained interest, and a geometric characterization is usually important in spaces of analytic functions.

In the case of the classical Bergman spaces *Ap*ð Þ , Carleson measures are characterized in terms of the measure on pseudo-hyperbolic disks (see [26]). An interesting fact is the independence of the characterization on the exponent *p*. In the case of variable exponent Bergman spaces, the independence of *p*ð Þ� does not hold unless the *p*ð Þ� satisfies the Log-Hölder condition. The counterexample for the general case and the theorem for the restricted case were found in [24].

**Theorem 15.** *Let <sup>μ</sup> be a finite, positive Borel measure on and p*ð Þ� <sup>∈</sup>Plogð Þ *. Then <sup>μ</sup> is a Carleson measure for Ap*ð Þ� ð Þ *if and only if there exists a constant C*<sup>&</sup>gt; <sup>0</sup> *such that for every* 0< *r*<1 *and every a*∈ *, μ*ð Þ *Dr*ð Þ *a* ≤*C*∣*Dr*ð Þ *a* ∣*, where Dr*ð Þ *a denotes pseudohyperbolic disk with center at a*∈ *and radius r:*

$$D\_r(\mathfrak{a}) \coloneqq \left\{ z \in \mathbb{D} \, : \, \left| \frac{\mathfrak{a} - z}{1 - \overline{a}z} \right| < r \right\}.$$

The main tool for the proof of the previous theorem is Eq. (2) that implies the following estimate for the reproducing kernels.

**Theorem 16.** *Let p*ð Þ� <sup>∈</sup>Plogð Þ *. Then for every z*<sup>∈</sup> *we have that*

$$\|k\_x\|\_{\mathcal{A}^{p(\cdot)}(\mathbb{D})} \asymp \frac{1}{\left(1-\left|\boldsymbol{z}\right|^2\right)^{2\left(1-1/p(\boldsymbol{z})\right)}}\ .$$

#### **5. Open problems**

#### **5.1 Zero sets**

An interesting area will be to study some analytic properties of functions belonging to variable exponent Bergman spaces. One starting point will be to study the structure of zero sets in the space.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

polynomials associated with *fr* converges uniformly on to *fr*. Therefore, given *<sup>ε</sup>*>0, there exists 1*=*2<sup>≤</sup> *<sup>r</sup>*<sup>&</sup>lt; 1 such that <sup>∥</sup>*<sup>f</sup>* � *fr*∥*Ap*ð Þ� ð Þ <sup>&</sup>lt;*ε=*2. And there exists a

Another classical problem that has a variable exponent equivalent consists on finding a geometrical characterization of Carleson measures in variable exponent Bergman spaces. Given a positive Borel measure *μ* defined on the unit disk , we say that *<sup>μ</sup>* is a *Carleson measure* for the variable exponent Bergman space *<sup>A</sup>p*ð Þ� ð Þ if

<sup>∥</sup>*<sup>f</sup>* <sup>∥</sup>*Lp*ð Þ� ð Þ ,*<sup>μ</sup>* <sup>⩽</sup>*C*∥*<sup>f</sup>* <sup>∥</sup>*Ap*ð Þ� ð Þ *:*

In other words, *<sup>μ</sup>* is a Carleson measure on *Ap*ð Þ� ð Þ if *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ is continuously

Lennart Carleson gave a geometric characterization of Carleson measures on Hardy spaces *H<sup>p</sup>* and used this result in his proof of the Corona theorem and some interpolation problems (cf. [25]). After that, Carleson measures have gained interest, and a geometric characterization is usually important in spaces of analytic

In the case of the classical Bergman spaces *Ap*ð Þ , Carleson measures are characterized in terms of the measure on pseudo-hyperbolic disks (see [26]). An interesting fact is the independence of the characterization on the exponent *p*. In the case of variable exponent Bergman spaces, the independence of *p*ð Þ� does not hold unless the *p*ð Þ� satisfies the Log-Hölder condition. The counterexample for the general case

**Theorem 15.** *Let <sup>μ</sup> be a finite, positive Borel measure on and p*ð Þ� <sup>∈</sup>Plogð Þ *. Then <sup>μ</sup> is a Carleson measure for Ap*ð Þ� ð Þ *if and only if there exists a constant C*<sup>&</sup>gt; <sup>0</sup> *such that for every* 0< *r*<1 *and every a*∈ *, μ*ð Þ *Dr*ð Þ *a* ≤*C*∣*Dr*ð Þ *a* ∣*, where Dr*ð Þ *a denotes pseudo-*

> � � �

The main tool for the proof of the previous theorem is Eq. (2) that implies the

1 � j j *z*

An interesting area will be to study some analytic properties of functions belonging to variable exponent Bergman spaces. One starting point will be to study

n o

1 � *az*

1

<sup>2</sup> � �2 1ð Þ �1*=p z*ð Þ *:*

� � � <sup>&</sup>lt;*<sup>r</sup>*

*:*

*Dr*ð Þ *<sup>a</sup>* <sup>≔</sup> *<sup>z</sup>*<sup>∈</sup> : *<sup>a</sup>* � *<sup>z</sup>*

**Theorem 16.** *Let p*ð Þ� <sup>∈</sup>Plogð Þ *. Then for every z*<sup>∈</sup> *we have that*

<sup>∥</sup>*kz*∥*Ap*ð Þ� ð Þ <sup>≍</sup>

and the theorem for the restricted case were found in [24].

*hyperbolic disk with center at a*∈ *and radius r:*

following estimate for the reproducing kernels.

**5. Open problems**

the structure of zero sets in the space.

**5.1 Zero sets**

**72**

polynomial *<sup>p</sup>* such that <sup>∥</sup>*<sup>p</sup>* � *fr*∥*Ap*ð Þ� ð Þ <sup>&</sup>lt;*ε=*2.

*Advances in Complex Analysis and Applications*

there exists a constant *C*>0 such that

**4.2 Carleson measures**

embedded in *Lp*ð Þ� ð Þ , *<sup>μ</sup>* .

functions.

A sequence of points f g *zn* <sup>⊂</sup> is a zero set for the Bergman space *Ap*ð Þ if there exists *<sup>f</sup>* <sup>∈</sup> *Ap*ð Þ such that *f z*ð Þ¼ *<sup>n</sup>* 0 for all *<sup>n</sup>* and *f z*ð Þ 6¼ 0 if *<sup>z</sup>* �∈f g *zn* .

Although the problem of completely describing the zero sets in *Ap*ð Þ is still open, some information is available. For example, if *n r*ð Þ denotes the number of zeroes of a function *<sup>f</sup>* <sup>∈</sup> *Ap*ð Þ having modulus smaller than *<sup>r</sup>*>0, then it is known that

$$m(r) = O\left(\frac{1}{1-r}\log\left(\frac{1}{1-r}\right)\right), \qquad r \to 1^-.$$

This result depends on studying the radial growth of functions in the Bergman space and on Jensen's formula for the sequence f g *zk* of zeroes of *f* such that ∣*zk*Þ<*r*.

$$\log|f(0)| = \sum\_{k=1}^{n(r)} \log\left(\frac{|x\_k|}{r}\right) + \frac{1}{2\pi} \int\_0^{2\pi} \log|f'(re^{i\theta})| \,\mathrm{d}\theta \,\mathrm{d}\theta$$

This equation holds for every analytic function on . Consequently, if we want to find out whether a similar result about the growth of the zeroes holds in the variable exponent case, then it is necessary to study the radial growth of functions in *Ap*ð Þ� ð Þ . Such results, in the case of a constant exponent again, depend on the subharmonicity of j j *<sup>f</sup> <sup>p</sup>* .

Another result involving zero sets is that if *f*ð Þ 0 6¼ 0, then

$$\prod\_{k=1}^{n(r)} \frac{1}{|z\_k|} = O\left(n(r)^{1/p}\right), \qquad r \to 1^{-}.$$

Finding an extension of such result to variable exponents would include making sense of the right-hand side term that is not even clear in this context.

#### **5.2 Sampling sequences**

A related concept that we would like to address in variable exponent Bergman spaces is that of sampling sequences. A sequence of points f g *zk* ⊂ is sampling in *Ap*ð Þ if there exists constants *<sup>c</sup>*<sup>&</sup>gt; 0 and *<sup>C</sup>*<sup>&</sup>gt; 0 such that for all *<sup>f</sup>* <sup>∈</sup> *<sup>A</sup><sup>p</sup>*ð Þ

$$c \| f \|\_{p}^{p} \le \sum\_{k=1}^{\infty} \left( 1 - |z\_k|^2 \right)^2 |f(z\_k)|^p \le C \| f \|\_{p}^{p}.$$

One difference that is expected to occur in the case of variable exponents is that sampling sequences will probably not be invariant under automorphism of the unit disk. This is true for a constant exponent, and it will be interesting to find counterexamples if the exponent varies. Sampling sequences are related to the concept of "frames" in Hilbert spaces and can be thought as sequences that contain great information of the space. It is expected that a sampling sequence is a somehow "big" and "spread out" subset of the unit disk. The notion is clearly related in *Ap* to the concept of Carleson measures.

If we define a discrete measure *μ* as

$$\mu = \sum\_{k=1}^{\infty} \left( \mathbf{1} - |\mathbf{z}\_k|^2 \right)^2 \delta\_{\mathbf{z}\_k}$$

where *δzk* is the Dirac measure with a point mass at *zk*, then the sequence f g *zk* is sampling if *μ* is a Carleson measure and satisfies a "reverse" Carleson condition. Then it becomes relevant to find a geometric characterization of the measures that satisfy a reverse Carleson condition (see [27–29]). The characterization depends on the concept of dominant sets and it is independent of *p*. The methods used by Luecking for Bergman spaces rely on a calculation of the norm of evaluation functionals and a submean value property. Different techniques must be developed for the variable exponent counterpart.

on *Ap*ð Þ� ð Þ . One approach to follow is to address the boundedness problems of linear operators acting on *Ap*ð Þ� ð Þ through the use Rubio de Francia extrapolation. Given an open set Ω ⊂ , we denote by F a family of pairs of non-negative,

ess sup*<sup>z</sup>* <sup>∈</sup> <sup>Ω</sup>

ð Ω

*Given p* <sup>∈</sup>Pð Þ <sup>Ω</sup> *, if p*<sup>0</sup> <sup>≤</sup> *<sup>p</sup>*� <sup>≤</sup>*p*<sup>þ</sup> <sup>&</sup>lt; <sup>∞</sup> *and the maximal operator is bounded on*

<sup>∥</sup>*F*∥*Lp*ð Þ� ð Þ <sup>Ω</sup> <sup>≤</sup>*Cp*ð Þ� <sup>∥</sup>*G*∥*Lp*ð Þ� ð Þ <sup>Ω</sup> *:*

This result allows, under certain conditions, to pass from the of studying operators defined on variable exponent spaces, to study operators defined on weighted constant exponent spaces, and therefore it is possible to use known results from the

Another type of spaces we are interested is the class of analytic Besov spaces *Bp*.

These are the spaces of all analytic functions in the unit disk such that

ð *f* 0 ð Þ*<sup>z</sup>* � � � �

the exponents that leave the space invariant under disk automorphisms.

functions in *Bp*. It can be seen (see, e.g., [33]) that *f* ∈*Bp* if and only if

j j *f z*ð Þ� *f w*ð Þ *<sup>p</sup>* j j 1 � *zw*

*<sup>p</sup>* <sup>1</sup> � j j *<sup>z</sup>* <sup>2</sup> � �*<sup>p</sup>*�<sup>2</sup>

One particular property of this space is that they are Möbius invariant in the sense that ∥*f*∘*φ*∥*Bp* ¼ ∥*f* ∥*Bp* for every automorphism of the unit disk *φ*. This is probably not true in general for a variable exponent version due to the problems with the change of variables formula. A first question to address is to characterize

One useful technique to study such spaces is a derivative-free characterization of

A similar result in the case of variable exponents could serve as a starting point

ð *f* 0 ð Þ*<sup>z</sup>* � � � �

For each *p* this is a normed space with norm

∥*f* ∥*Bp* ¼ ∣*f*ð Þ 0 ∣ þ

ð ð 

for investigating other properties of the space.

**75**

weights *w*. That is, the non-negative functions *w* such that

*F x*ð Þ*<sup>p</sup>*0*w x*ð Þd*x*≤*C*<sup>0</sup>

*of functions* F *is such that for all w* ∈ *A*1*,*

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

> ð Ω

ð Þ Ω *, then*

Muckenhoupt weights theory.

**5.4 Analytic Besov spaces**

*L <sup>p</sup>*ð Þ� *<sup>=</sup><sup>p</sup>* ð Þ<sup>0</sup> 0

measurable functions defined on Ω, and by *A*<sup>1</sup> we denote the class of Muckenhoupt

where *M* denotes the Hardy-Littlewood maximal operator. Rubio de Francia extrapolation in the setting of variable exponent spaces can be stated as follows. **Proposition 17** (Thm. 5.24 in [1]). *Suppose that for some p*<sup>0</sup> ≥ 1 *the family of pairs*

*Mw z*ð Þ *w z*ð Þ <sup>&</sup>lt; <sup>∞</sup>

*G x*ð Þ*<sup>p</sup>*0*w x*ð Þd*x*, ð Þ *<sup>F</sup>*, *<sup>G</sup>* <sup>∈</sup> <sup>F</sup>*:*

d*A z*ð Þ< ∞*:*

*:*

*<sup>p</sup>* <sup>1</sup> � j j *<sup>z</sup>* <sup>2</sup> � �*<sup>p</sup>*�<sup>2</sup> <sup>d</sup>*A z*ð Þ � �<sup>1</sup>*=<sup>p</sup>*

<sup>4</sup> d*A z*ð Þd*A w*ð Þ< ∞*:*

#### **5.3 Operators in variable exponent Bergman spaces**

The characterization of Carleson measures obtained in [24] opens the possibility of studying the boundedness and compactness of certain operators acting on variable exponent Bergman spaces.

Among those operators worth studying are multiplication operators. Those are formally defined on a functions space F as

$$\mathcal{M}\_f(\mathbf{g}) = f \mathbf{g}$$

where *g* ∈ F. One natural question is to find the set of symbols *f* that make the operator *Mf* map the space *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ to itself.

Such question has been addressed in the case of weighted Bergman spaces in [23]. The proof relies on a geometric characterization of Carleson measures, a composition with a disk isomorphism, and a theorem of change of variables. Such tool is difficult to use in the setting of variable exponents since any change of variables will also affect the exponent and consequently the spaces are not necessarily invariant under composition with isomorphisms. This makes the situation different from the case of a constant exponent.

Other operators related to multiplication are Toeplitz operators. Those have already being studied in [30] for the case of weighted Bergman spaces with nonradial weights. Given a function *φ*∈ *L*<sup>1</sup> ð Þ , the Toeplitz operator *T<sup>φ</sup>* is defined on the set of polynomials as

$$T\_{\boldsymbol{\varrho}}p(\boldsymbol{z}) = \int\_{\mathbb{D}} p(\boldsymbol{w}) k\_{\boldsymbol{w}}(\boldsymbol{z}) \boldsymbol{\varrho}(\boldsymbol{w}) \mathrm{d}A(\boldsymbol{w}).$$

One natural question is to ask whether this operator can be extended boundedly to the space *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ . This question is addressed in [31] were a characterization of the boundedness and compactness of *T<sup>φ</sup>* is found in terms of Carleson measures and the Berezin transform associated with *Tφ*. This type of results are sometimes referred to as Axler-Zheng theorems, and similar questions looking to establish the relation between the Berezin transform of a finite product of Toeplitz operators with its compactness have been addressed by several authors; a survey of this type of results is given in [32]. It is then reasonable to ask weather an Axler-Zheng type result for products of Toeplitz operators holds in the context of variable exponents.

The composition operators are another type of operators to be studied. Given an analytic self-map *φ* of the unit disk, the composition operator *C<sup>φ</sup>* can be formally defined as

$$\mathsf{C}\_{\mathfrak{q}}(f) = f \circ \mathfrak{q}.$$

A natural question is to find function-theoretic conditions on *φ* that guarantee the boundedness and/or compactness of its associated composition operator acting

where *δzk* is the Dirac measure with a point mass at *zk*, then the sequence f g *zk* is sampling if *μ* is a Carleson measure and satisfies a "reverse" Carleson condition. Then it becomes relevant to find a geometric characterization of the measures that satisfy a reverse Carleson condition (see [27–29]). The characterization depends on the concept of dominant sets and it is independent of *p*. The methods used by Luecking for Bergman spaces rely on a calculation of the norm of evaluation functionals and a submean value property. Different techniques must be developed for

The characterization of Carleson measures obtained in [24] opens the possibility of studying the boundedness and compactness of certain operators acting on vari-

Among those operators worth studying are multiplication operators. Those are

*Mf*ð Þ¼ *g fg*

where *g* ∈ F. One natural question is to find the set of symbols *f* that make the

Such question has been addressed in the case of weighted Bergman spaces in [23]. The proof relies on a geometric characterization of Carleson measures, a composition with a disk isomorphism, and a theorem of change of variables. Such tool is difficult to use in the setting of variable exponents since any change of variables will also affect the exponent and consequently the spaces are not necessarily invariant under composition with isomorphisms. This makes the situation

Other operators related to multiplication are Toeplitz operators. Those have already being studied in [30] for the case of weighted Bergman spaces with non-

*p w*ð Þ*kw*ð Þ*z φ*ð Þ *w* d*A w*ð Þ*:*

One natural question is to ask whether this operator can be extended boundedly to the space *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ . This question is addressed in [31] were a characterization of the boundedness and compactness of *T<sup>φ</sup>* is found in terms of Carleson measures and the Berezin transform associated with *Tφ*. This type of results are sometimes referred to as Axler-Zheng theorems, and similar questions looking to establish the relation between the Berezin transform of a finite product of Toeplitz operators with its compactness have been addressed by several authors; a survey of this type of results is given in [32]. It is then reasonable to ask weather an Axler-Zheng type result for products of Toeplitz operators holds in the context of variable exponents. The composition operators are another type of operators to be studied. Given an analytic self-map *φ* of the unit disk, the composition operator *C<sup>φ</sup>* can be formally

*Cφ*ð Þ¼ *f f*∘*φ:*

A natural question is to find function-theoretic conditions on *φ* that guarantee the boundedness and/or compactness of its associated composition operator acting

ð Þ , the Toeplitz operator *T<sup>φ</sup>* is defined on

the variable exponent counterpart.

*Advances in Complex Analysis and Applications*

able exponent Bergman spaces.

formally defined on a functions space F as

operator *Mf* map the space *<sup>A</sup><sup>p</sup>*ð Þ� ð Þ to itself.

different from the case of a constant exponent.

*Tφp z*ð Þ¼

ð 

radial weights. Given a function *φ*∈ *L*<sup>1</sup>

the set of polynomials as

defined as

**74**

**5.3 Operators in variable exponent Bergman spaces**

on *Ap*ð Þ� ð Þ . One approach to follow is to address the boundedness problems of linear operators acting on *Ap*ð Þ� ð Þ through the use Rubio de Francia extrapolation. Given an open set Ω ⊂ , we denote by F a family of pairs of non-negative, measurable functions defined on Ω, and by *A*<sup>1</sup> we denote the class of Muckenhoupt weights *w*. That is, the non-negative functions *w* such that

$$\text{ess } \sup\_{z \in \Omega} \frac{Mw(z)}{w(z)} < \infty$$

where *M* denotes the Hardy-Littlewood maximal operator. Rubio de Francia extrapolation in the setting of variable exponent spaces can be stated as follows.

**Proposition 17** (Thm. 5.24 in [1]). *Suppose that for some p*<sup>0</sup> ≥ 1 *the family of pairs of functions* F *is such that for all w* ∈ *A*1*,*

$$\int\_{\mathfrak{U}} F(\mathfrak{x})^{p\_0} w(\mathfrak{x}) \mathrm{d}\mathfrak{x} \le \mathcal{C}\_0 \int\_{\mathfrak{U}} G(\mathfrak{x})^{p\_0} w(\mathfrak{x}) \mathrm{d}\mathfrak{x}, \quad (F, G) \in \mathcal{F}.$$

*Given p* <sup>∈</sup>Pð Þ <sup>Ω</sup> *, if p*<sup>0</sup> <sup>≤</sup> *<sup>p</sup>*� <sup>≤</sup>*p*<sup>þ</sup> <sup>&</sup>lt; <sup>∞</sup> *and the maximal operator is bounded on L <sup>p</sup>*ð Þ� *<sup>=</sup><sup>p</sup>* ð Þ<sup>0</sup> 0 ð Þ Ω *, then*

$$\|F\|\_{L^{p(\cdot)}(\mathfrak{\Omega})} \le C\_{p(\cdot)} \|G\|\_{L^{p(\cdot)}(\mathfrak{\Omega})}.$$

This result allows, under certain conditions, to pass from the of studying operators defined on variable exponent spaces, to study operators defined on weighted constant exponent spaces, and therefore it is possible to use known results from the Muckenhoupt weights theory.

#### **5.4 Analytic Besov spaces**

Another type of spaces we are interested is the class of analytic Besov spaces *Bp*. These are the spaces of all analytic functions in the unit disk such that

$$\int\_{\mathbb{D}} \left| \left. f'(z) \right|^p \left( 1 - \left| z \right|^2 \right)^{p-2} \mathrm{d}A \left( z \right) < \infty. \right| $$

For each *p* this is a normed space with norm

$$\|\|f\|\|\_{B\_p} = |f(\mathbf{0})| + \left(\int\_{\mathbb{D}} |f'(z)|^p \left(\mathbf{1} - |z|^2\right)^{p-2} \mathbf{d}A(z)\right)^{1/p}.$$

One particular property of this space is that they are Möbius invariant in the sense that ∥*f*∘*φ*∥*Bp* ¼ ∥*f* ∥*Bp* for every automorphism of the unit disk *φ*. This is probably not true in general for a variable exponent version due to the problems with the change of variables formula. A first question to address is to characterize the exponents that leave the space invariant under disk automorphisms.

One useful technique to study such spaces is a derivative-free characterization of functions in *Bp*. It can be seen (see, e.g., [33]) that *f* ∈*Bp* if and only if

$$\int\_{\mathcal{D}} \int\_{\mathcal{D}} \frac{|f(z) - f(w)|^p}{|1 - z\overline{w}|^4} \mathbf{d}A(z) \mathbf{d}A(w) < \infty.$$

A similar result in the case of variable exponents could serve as a starting point for investigating other properties of the space.

*Advances in Complex Analysis and Applications*

**References**

2013

461-482

817-822

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*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

> restoration, numerical methods and extensions. In: Proceedings of the 1997 IEEE International Conference on Image Processing, Volume III; 1997.

[11] Bollt EM, Chartrand R, Esedoglu S,

compromise between total variation and

[12] Chen Y, Guo W, Zeng Q, Liu Y. A

reconstruction of apparent diffusion coefficient profiles from diffusion weighted images. Inverse Problems and

Schultz P, Vixie KR. Graduated adaptive image denoising, local

isotropic diffusion. Advances in Computational Mathematics. 2009;**31**

nonstandard smoothing in

Imaging. 2008;**2**(2):205-224

Mathematical Analysis and

[13] Wunderli T. On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudo-solutions. Journal of

Applications. 2010;**364**(2):5915-5998

[14] Harjulehto P, Hästoö P, Lê ÚV, Nuortio M. Overview of differential equations with non-standard growth. Nonlinear Analysis: Theory Methods & Applications. 2010;**72**(12):4551-4574

[15] Mingione G. Regularity of minima, an invitation to the dark side of the calculus of variations. Applications of Mathematics. 2006;**51**(4):355-426

[16] Cowen CC Jr, MacCluer BI. Composition Operators on Spaces of Analytic Functions (Studies in Advanced Mathematics). Boca Ratón,

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FL: CRC Press; 1995

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[2] Samko S. On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators. Integral Transforms and Special Functions. 2005;**16**(5–6):

[3] Acerbi E, Mingione G. Regularity results for electrorheological fluids, the stationary case. Comptes rendus de l'Académie des Sciences. 2002;**334**(9):

[4] Acerbi E, Mingione G. Regularity results for stationary electrorheological fluids. Archive for Rational Mechanics and Analysis. 2002;**164**(3):213-259

[5] Růžička M. Electrorheological Fluids, Modeling and Mathematical Theory. Lecture Notes in Mathematics. Berlin:

mathematical and numerical analysis of electrorheological fluids. Applications of Mathematics. 2004;**49**(6):565-609

[7] Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics. 2006;**66**(4):

Springer-Verlag; 2000

1383-1406 (electronic)

[8] Aboulaich R, Boujena S,

El Guarmah E. Sur un modèle nonlinéaire pour le débruitage de l'image. Comptes Rendus de l'Académie des Sciences. 2007;**345**(8):425-429

[9] Aboulaich R, Meskine D, Souissi A. New diffusion models in image

processing. Computers & Mathematics with Applications. 2008;**56**(4):874-882

[10] Blomgren P, Chan T, Mulet P, Wong CK. Total variation image

**77**

[6] Růžička M. Modeling,

### **Author details**

Gerardo A. Chacón<sup>1</sup> \* and Gerardo R. Chacón<sup>2</sup>


\*Address all correspondence to: gerardo.chacon@gallaudet.edu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Variable Exponent Spaces of Analytic Functions DOI: http://dx.doi.org/10.5772/intechopen.92617*

### **References**

[1] Cruz-Uribe D, Fiorenza A. Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Basel: Birkhäuser; 2013

[2] Samko S. On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators. Integral Transforms and Special Functions. 2005;**16**(5–6): 461-482

[3] Acerbi E, Mingione G. Regularity results for electrorheological fluids, the stationary case. Comptes rendus de l'Académie des Sciences. 2002;**334**(9): 817-822

[4] Acerbi E, Mingione G. Regularity results for stationary electrorheological fluids. Archive for Rational Mechanics and Analysis. 2002;**164**(3):213-259

[5] Růžička M. Electrorheological Fluids, Modeling and Mathematical Theory. Lecture Notes in Mathematics. Berlin: Springer-Verlag; 2000

[6] Růžička M. Modeling, mathematical and numerical analysis of electrorheological fluids. Applications of Mathematics. 2004;**49**(6):565-609

[7] Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics. 2006;**66**(4): 1383-1406 (electronic)

[8] Aboulaich R, Boujena S, El Guarmah E. Sur un modèle nonlinéaire pour le débruitage de l'image. Comptes Rendus de l'Académie des Sciences. 2007;**345**(8):425-429

[9] Aboulaich R, Meskine D, Souissi A. New diffusion models in image processing. Computers & Mathematics with Applications. 2008;**56**(4):874-882

[10] Blomgren P, Chan T, Mulet P, Wong CK. Total variation image

restoration, numerical methods and extensions. In: Proceedings of the 1997 IEEE International Conference on Image Processing, Volume III; 1997. pp. 384-387

[11] Bollt EM, Chartrand R, Esedoglu S, Schultz P, Vixie KR. Graduated adaptive image denoising, local compromise between total variation and isotropic diffusion. Advances in Computational Mathematics. 2009;**31** (1-3):61-85

[12] Chen Y, Guo W, Zeng Q, Liu Y. A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images. Inverse Problems and Imaging. 2008;**2**(2):205-224

[13] Wunderli T. On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudo-solutions. Journal of Mathematical Analysis and Applications. 2010;**364**(2):5915-5998

[14] Harjulehto P, Hästoö P, Lê ÚV, Nuortio M. Overview of differential equations with non-standard growth. Nonlinear Analysis: Theory Methods & Applications. 2010;**72**(12):4551-4574

[15] Mingione G. Regularity of minima, an invitation to the dark side of the calculus of variations. Applications of Mathematics. 2006;**51**(4):355-426

[16] Cowen CC Jr, MacCluer BI. Composition Operators on Spaces of Analytic Functions (Studies in Advanced Mathematics). Boca Ratón, FL: CRC Press; 1995

[17] Diening L, Harjulehto P, Hästö P, Růžička M. Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics. Berlin: Springer-Verlag; 2011

**Author details**

**76**

Gerardo A. Chacón<sup>1</sup>

\* and Gerardo R. Chacón<sup>2</sup>

\*Address all correspondence to: gerardo.chacon@gallaudet.edu

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

1 Universidad Antonio Nariño, Bogotá, Colombia

*Advances in Complex Analysis and Applications*

2 Gallaudet University, Washington, D.C., USA

provided the original work is properly cited.

[18] Rafeiro H, Rojas E. Espacios de Lebesgue con exponente variable: Un espacio de Banach de funciones medibles (Spanish). Vol. 2014. Caracas: IVIC—Instituto Venezolano de Investigaciones Científicas. p. XI+136

[19] Chacón GR, Rafeiro H. Variable exponent Bergman spaces. Nonlinear Analysis: Theory Methods & Applications. 2014;**105**:41-49

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[23] Zhao R. Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces. Annales Academiæ Scientiarum Fennicæ. 2004;**29**:139-150

[24] Chacón GR, Rafeiro H, Vallejo JC. Carleson measures for variable exponent Bergman spaces. Complex Analysis and Operator Theory. 2017;**11**: 1623-1638. DOI: 10.1007/s11785-016- 0573-0

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**Chapter 6**

*Francisco Bulnes*

the reinterpretation of vector fields.

integral operators

**1. Introduction**

**79**

**Abstract**

Integral Geometry and

Riemannian Manifold

Cohomology in Field Theory on

The study of the relationships between the integration invariants and the different classes of operators, as well as of functions inside the context of the integral geometry, establishes diverse homologies in the dual space of the functions. This is given in the class of cohomology of the integral operators that give solution to certain class of differential equations in field theory inside a holomorphic context. By this way, using a cohomological theory of appropriate operators that establish equivalences among cycles and cocycles of closed submanifolds, line bundles and contours can be obtained by a cohomology of general integrals, useful in the evaluation and measurement of fields, particles, and physical interactions of diverse nature that occurs in the space-time geometry and phenomena. Some of the results applied through this study are the obtaining of solutions through orbital integrals for the tensor of curvature *Rμν*, of Einstein's equations, and using the imbedding of cycles in a complex Riemannian manifold through the duality: line bundles with cohomological contours and closed submanifolds with cohomological functional. Concrete results also are obtained in the determination of Cauchy type integral for

**Keywords:** complex Riemannian manifold, cocycles, cohomology, cohomology of

Obtaining an integral cohomology of general integral operators that determine complex analytic solutions through classes of cohomology born of the *<sup>∂</sup>*‐cohomology is necessary to use a holomorphic language with the purpose of obtaining the holomorphic forms that involve exact forms. In fact, this methodology is a way of so many perspectives that suggest the use of complex hyperholomorphic functions in approaching functions in complex analysis, although using fibrations on some quaternionic algebra. The holomorphic forms required in this language, are good to express the integral of complex vector fields as integral of line, which have more than enough lines and hyperplanes, respectively, in <sup>n</sup> and n, visualizing these

cycles, geometrical integration, integral curvature, integration invariants,

**AMS Classification:** 32A19 32C26 32C30 23C15 58 K70 14 J17

the Space-Time as Complex

[29] Mengestie T. Carleson type measures for Fock-Sobolev spaces. Complex Analysis and Operator Theory. 2014;**8**:1225-1256. DOI: 10.1007/s11785- 013-0321-7

[30] Chacón GR. Toeplitz operators on weighted Bergman spaces. Journal of Function Spaces and Applications. 2013; **2013**:5. Article ID: 753153. DOI: 10.1155/ 2013/753153

[31] Chacón GR, Rafeiro H. Toeplitz operators on variable exponent Bergman spaces. Mediterranean Journal of Mathematics. 2016;**13**:3525-3536. DOI: 10.1007/s00009-016-0701-0

[32] Cucković Z, Şahutoğlu S, Zheng A. Type theorem on a class of domains in *<sup>n</sup>*. Integral Equations and Operator Theory. 2013;**77**:397

[33] Zhu K. Operator Theory in Function Spaces, Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society; 2007

#### **Chapter 6**

[18] Rafeiro H, Rojas E. Espacios de Lebesgue con exponente variable: Un espacio de Banach de funciones

IVIC—Instituto Venezolano de Investigaciones Científicas. p. XI+136

Analysis: Theory Methods & Applications. 2014;**105**:41-49

[21] Sharapudinov II. Uniform

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0573-0

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medibles (Spanish). Vol. 2014. Caracas:

*Advances in Complex Analysis and Applications*

[28] Luecking D. Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives. American Journal of Mathematics. 1985;

[29] Mengestie T. Carleson type measures for Fock-Sobolev spaces. Complex Analysis and Operator Theory. 2014;**8**:1225-1256. DOI: 10.1007/s11785-

[30] Chacón GR. Toeplitz operators on weighted Bergman spaces. Journal of Function Spaces and Applications. 2013; **2013**:5. Article ID: 753153. DOI: 10.1155/

[31] Chacón GR, Rafeiro H. Toeplitz operators on variable exponent

Bergman spaces. Mediterranean Journal of Mathematics. 2016;**13**:3525-3536. DOI: 10.1007/s00009-016-0701-0

[32] Cucković Z, Şahutoğlu S, Zheng A. Type theorem on a class of domains in *<sup>n</sup>*. Integral Equations and Operator

[33] Zhu K. Operator Theory in Function Spaces, Mathematical Surveys and Monographs. Providence, RI: American

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Theory. 2013;**77**:397

Mathematical Society; 2007

[19] Chacón GR, Rafeiro H. Variable exponent Bergman spaces. Nonlinear

[20] Chacón GA, Chacón GR. Analytic variable exponent Hardy spaces. Advances in Operator Theory. 2019;**4**:

boundedness in *Lp* (*<sup>p</sup>* <sup>¼</sup> *p x*ð Þ) of some families of convolution operators. Mathematical Notes. 1996;**59**:205-212

[22] Bergman S. The Kernel Function and Conformal Mapping. 2nd ed. Providence, R. I.: American Mathematical Society; 1970

[23] Zhao R. Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces. Annales Academiæ Scientiarum Fennicæ. 2004;**29**:139-150

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[25] Garnett JW. Bounded Analytic Functions. Berlin: Springer-Verlag; 2007

[27] Luecking D. Closed-range restriction operators on weighted Bergman spaces. Pacific Journal of Mathematics. 1984;**110**:145-160

[26] Duren P, Schuster A. Mathematical surveys and monographs, 100. In: Bergman Spaces. Providence, RI: American Mathematical Society; 2004

## Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex Riemannian Manifold

*Francisco Bulnes*

### **Abstract**

The study of the relationships between the integration invariants and the different classes of operators, as well as of functions inside the context of the integral geometry, establishes diverse homologies in the dual space of the functions. This is given in the class of cohomology of the integral operators that give solution to certain class of differential equations in field theory inside a holomorphic context. By this way, using a cohomological theory of appropriate operators that establish equivalences among cycles and cocycles of closed submanifolds, line bundles and contours can be obtained by a cohomology of general integrals, useful in the evaluation and measurement of fields, particles, and physical interactions of diverse nature that occurs in the space-time geometry and phenomena. Some of the results applied through this study are the obtaining of solutions through orbital integrals for the tensor of curvature *Rμν*, of Einstein's equations, and using the imbedding of cycles in a complex Riemannian manifold through the duality: line bundles with cohomological contours and closed submanifolds with cohomological functional. Concrete results also are obtained in the determination of Cauchy type integral for the reinterpretation of vector fields.

**Keywords:** complex Riemannian manifold, cocycles, cohomology, cohomology of cycles, geometrical integration, integral curvature, integration invariants, integral operators

**AMS Classification:** 32A19 32C26 32C30 23C15 58 K70 14 J17

#### **1. Introduction**

Obtaining an integral cohomology of general integral operators that determine complex analytic solutions through classes of cohomology born of the *<sup>∂</sup>*‐cohomology is necessary to use a holomorphic language with the purpose of obtaining the holomorphic forms that involve exact forms. In fact, this methodology is a way of so many perspectives that suggest the use of complex hyperholomorphic functions in approaching functions in complex analysis, although using fibrations on some quaternionic algebra. The holomorphic forms required in this language, are good to express the integral of complex vector fields as integral of line, which have more than enough lines and hyperplanes, respectively, in <sup>n</sup> and n, visualizing these

fields like holomorphic sections of complex holomorphic bundles of fibrations X ! M.

But the *<sup>∂</sup>*‐cohomology exists naturally in coverings of Stein X ! M, like holomorphic forms. Then, the integral can be expressed on spaces M*δ*, and Δz, [1, 2], that are given as lines and hyperplanes of <sup>n</sup> and n, and that as such they are integral orbital of the complex manifolds M ¼ G*=*L and Δ ¼ Γ*=*Σ, belonging to a *<sup>∂</sup>*‐cohomology in holomorphic language.

The cohomologies of functionals and functions, respectively, that they can built through the complex cohomology of hyperspaces are generalizable for vector fields in the same sense of the coverings of Stein and therefore of the *<sup>∂</sup>*‐cohomology.

The following question arises, how to establish isomorphisms of cohomological classes for functions, functional, and vector fields inside the holomorphic context possible? How to determine a cohomological theory of integral operators that establish equivalences among these objects and the geometric objects of closed submanifolds, bundles of lines, and Feynman diagrams? How everything can decrease to a single cohomology of general integrals on contours or a cohomology of generalized functionals?

Before giving an answer to the previous questions, we give some preliminary definitions that we will use to fix concepts and outlines of the wanted general theory.

Let M, be a complex Riemannian manifold and be a sheaf of germs of holomorphic sections of a vector holomorphic sheaf.

**Definition 1.1.** We say that a spaceH•ð Þ M, <sup>J</sup> , is an integral cohomology (not in the sense of the set , but yes in the sense of the integrals of partial differential equations) of those *<sup>∂</sup>*‐equations, if this is a class of solutions or general integrals of these equations in M [1, 3].

**Definition 1.2.** An integral as generalized solution of a *<sup>∂</sup>*‐equation is a realization of an irreducible representation of a *<sup>∂</sup>*‐cohomology of complex closed submanifolds [2–4].

If the irreducible representations are unitary, then we have a complex L<sup>2</sup> � cohomology or *<sup>∂</sup>*‐cohomology with coefficients in *<sup>L</sup>*<sup>2</sup> . The corresponding integral operators to their integral cohomology are those of the complex Fourier analysis, which in the complex geometrical context (geometrical analysis) could be integrals constructed through integral transforms as the Hilbert transforms and other [3].

In the case of a real reductive Lie group, the generalized integrals come to be determined by their orbital integrals. Let G, be a real form of G, and P, their parabolic subgroup. The generalized integrals in G, are the integrals on open orbits of the generalized flag manifold G*=*P. For this way, if G <sup>¼</sup> U n, 1 ð Þ, and the generalized flag manifold is then n, the open orbit is the group of positive lines þ, which is an U n, 1 ð Þ�orbit in <sup>n</sup>ð Þ . The integrals are of John type [3, 5, 6]:

$$\rho(\mathbb{P}) = \bigcap\_{\mathbb{L}^+ \subset \mathbb{P}^a(\mathbb{C})} \mathbf{f},\tag{1}$$

(see **Figure 1**). Then, it is possible to assign a vector bundle of lines with a unitary

*Electromagnetic waves in conformal actions of the group* SU 2, 2 ð Þ *on a two-dimensional flat model of the*

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

The concepts of general integral and generalized integral are different because one refers to the whole class of cohomology of solutions of those *<sup>∂</sup>*‐equations about a complex analytic manifold, and the other refers to the classes of cohomology of solutions on cycles or cocycles of the complex Riemannian manifold [1, 2].

Another example in the recovery of a space of functions mainly the space M, is the recovery of real functions in the space n, through values of certain integral

<sup>∧</sup>ð Þ *<sup>ξ</sup>*, p ½ � <sup>p</sup> � ð Þ *<sup>ξ</sup>*, x �<sup>n</sup>

where the integral on p, is understood in terms of its regularization (role that carries out the Hilbert transform). The constant c depends on the dimension parity

To answer the first question, we need a structure of complexes that induce

the corresponding fibers X ! M and X ! Δ, of the double fibration [1].

**Definition 1.3.** A covering of Stein is a set of manifolds of Stein<sup>1</sup> M*<sup>δ</sup>* and Δz, of

Let us consider the complexes given in Ref. [1], and let us consider the structure defined by a covering of Stein given by the set of open Mf g*<sup>δ</sup>* , and f g Δ<sup>z</sup> , in the topology

<sup>1</sup> A Stein manifold is an open orbit of a semi-simple Lie group G, in a generalized flag manifold whose nilpotent radical is opposite to the parabolic subgroup P, of G [12]. A definition of the Stein manifold that

Let be G ¼ G0, a real form of G, and *F*<sup>D</sup> ¼ G*=*H, an open orbit in the flag manifold *F* ¼ G*=*P ¼ G0*=*U*: F*<sup>D</sup> is Stein if H is compact (or equivalently, if *F*<sup>D</sup> is Hermitian symmetric). Likewise, a Stein

*τ*<sup>X</sup> ¼ f g ð Þ z, *ξ* ∈ M � ΞjX⊂ M � Ξð Þ ⇔M*<sup>δ</sup>* ∩ Ξ<sup>z</sup> , (4)

dp

9 = ;

ωð Þ*ξ* (3)

operators. Such is the case of the formula tof xð Þ, recovered on n,

ð ∞

�∞ f

representation, where it can be classified.

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

**Figure 1.**

*space-time [9–11].*

f xð Þ¼ c

isomorphisms in integral cohomology.

manifold is a Hermitian symmetric flag domain.

**81**

ð

8 < :

Γ

of the space n, where it was carrying the tomography [6].

uses the Hermitian structure of a complex holomorphic manifold is:

The general integral in this case is given by the twistor transform [7] on the corresponding homogeneous bundle of lines, that is to say:

$$\mathcal{T}: \mathbf{H}^1(\mathbb{P}^+, \mathbf{O}(-\mathbf{n} - \mathbf{2})) \to \mathbf{H}^1(\mathbb{P}^{+\*}, \mathbf{O}(\mathbf{n} - \mathbf{2})),\tag{2}$$

Using the twistor transform like intertwining operator of induced tempered representations on a *<sup>∂</sup>*‐cohomology, we have representations of SU 2, 2 ð Þ that are orbits of a fundamental unitary g, K ð Þ�module of the electrodynamics [8]

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

#### **Figure 1.**

fields like holomorphic sections of complex holomorphic bundles of fibrations

But the *<sup>∂</sup>*‐cohomology exists naturally in coverings of Stein X ! M, like holomorphic forms. Then, the integral can be expressed on spaces M*δ*, and Δz, [1, 2], that are given as lines and hyperplanes of <sup>n</sup> and n, and that as such they are integral orbital of the complex manifolds M ¼ G*=*L and Δ ¼ Γ*=*Σ, belonging to a

The cohomologies of functionals and functions, respectively, that they can built through the complex cohomology of hyperspaces are generalizable for vector fields in the same sense of the coverings of Stein and therefore of the *<sup>∂</sup>*‐cohomology. The following question arises, how to establish isomorphisms of cohomological classes for functions, functional, and vector fields inside the holomorphic context possible? How to determine a cohomological theory of integral operators that establish equivalences among these objects and the geometric objects of closed submanifolds, bundles of lines, and Feynman diagrams? How everything can decrease to a single cohomology of general integrals on contours or a cohomology of

Before giving an answer to the previous questions, we give some preliminary definitions that we will use to fix concepts and outlines of the wanted general theory. Let M, be a complex Riemannian manifold and be a sheaf of germs of

**Definition 1.1.** We say that a spaceH•ð Þ M, <sup>J</sup> , is an integral cohomology (not in the sense of the set , but yes in the sense of the integrals of partial differential equations) of those *<sup>∂</sup>*‐equations, if this is a class of solutions or general integrals of

**Definition 1.2.** An integral as generalized solution of a *<sup>∂</sup>*‐equation is a realization of an irreducible representation of a *<sup>∂</sup>*‐cohomology of complex closed

If the irreducible representations are unitary, then we have a complex L<sup>2</sup>

operators to their integral cohomology are those of the complex Fourier analysis, which in the complex geometrical context (geometrical analysis) could be integrals constructed through integral transforms as the Hilbert transforms and other [3]. In the case of a real reductive Lie group, the generalized integrals come to be determined by their orbital integrals. Let G, be a real form of G, and P, their parabolic subgroup. The generalized integrals in G, are the integrals on open orbits

of the generalized flag manifold G*=*P. For this way, if G <sup>¼</sup> U n, 1 ð Þ, and the

which is an U n, 1 ð Þ�orbit in <sup>n</sup>ð Þ . The integrals are of John type [3, 5, 6]:

*φ*ð Þ¼ P

corresponding homogeneous bundle of lines, that is to say:

generalized flag manifold is then n, the open orbit is the group of positive lines þ,

The general integral in this case is given by the twistor transform [7] on the

Using the twistor transform like intertwining operator of induced tempered representations on a *<sup>∂</sup>*‐cohomology, we have representations of SU 2, 2 ð Þ that are orbits of a fundamental unitary g, K ð Þ�module of the electrodynamics [8]

ð

<sup>L</sup><sup>þ</sup> <sup>⊂</sup>nð Þ

<sup>T</sup> : H1 <sup>þ</sup> <sup>ð</sup> , Oð Þ �<sup>n</sup> � <sup>2</sup> Þ ! H1 <sup>þ</sup> <sup>∗</sup> ð Þ ,O nð Þ � <sup>2</sup> , (2)

�

. The corresponding integral

f, (1)

X ! M.

*<sup>∂</sup>*‐cohomology in holomorphic language.

*Advances in Complex Analysis and Applications*

holomorphic sections of a vector holomorphic sheaf.

cohomology or *<sup>∂</sup>*‐cohomology with coefficients in *<sup>L</sup>*<sup>2</sup>

generalized functionals?

these equations in M [1, 3].

submanifolds [2–4].

**80**

*Electromagnetic waves in conformal actions of the group* SU 2, 2 ð Þ *on a two-dimensional flat model of the space-time [9–11].*

(see **Figure 1**). Then, it is possible to assign a vector bundle of lines with a unitary representation, where it can be classified.

The concepts of general integral and generalized integral are different because one refers to the whole class of cohomology of solutions of those *<sup>∂</sup>*‐equations about a complex analytic manifold, and the other refers to the classes of cohomology of solutions on cycles or cocycles of the complex Riemannian manifold [1, 2].

Another example in the recovery of a space of functions mainly the space M, is the recovery of real functions in the space n, through values of certain integral operators. Such is the case of the formula tof xð Þ, recovered on n,

$$\mathbf{f}(\mathbf{x}) = \mathbf{c} \left\{ \left\{ \int\_{-\infty}^{\infty} \mathbf{f}^{\wedge}(\xi, \mathbf{p}) [\mathbf{p} - (\xi, \mathbf{x})]^{-\mathbf{n}} d\mathbf{p} \right\} \alpha(\xi) \tag{3}$$

where the integral on p, is understood in terms of its regularization (role that carries out the Hilbert transform). The constant c depends on the dimension parity of the space n, where it was carrying the tomography [6].

To answer the first question, we need a structure of complexes that induce isomorphisms in integral cohomology.

**Definition 1.3.** A covering of Stein is a set of manifolds of Stein<sup>1</sup> M*<sup>δ</sup>* and Δz, of the corresponding fibers X ! M and X ! Δ, of the double fibration [1].

Let us consider the complexes given in Ref. [1], and let us consider the structure defined by a covering of Stein given by the set of open Mf g*<sup>δ</sup>* , and f g Δ<sup>z</sup> , in the topology

$$\tau\_{\mathbf{X}} = \{ (\mathbf{z}, \boldsymbol{\xi}) \in \mathbf{M} \times \boldsymbol{\Xi} | \mathbf{X} \subset \mathbf{M} \times \boldsymbol{\Xi} (\Leftrightarrow \mathbf{M}\_{\delta} \cap \boldsymbol{\Xi}\_{\mathbf{z}}) \},\tag{4}$$

<sup>1</sup> A Stein manifold is an open orbit of a semi-simple Lie group G, in a generalized flag manifold whose nilpotent radical is opposite to the parabolic subgroup P, of G [12]. A definition of the Stein manifold that uses the Hermitian structure of a complex holomorphic manifold is:

Let be G ¼ G0, a real form of G, and *F*<sup>D</sup> ¼ G*=*H, an open orbit in the flag manifold *F* ¼ G*=*P ¼ G0*=*U*: F*<sup>D</sup> is Stein if H is compact (or equivalently, if *F*<sup>D</sup> is Hermitian symmetric). Likewise, a Stein manifold is a Hermitian symmetric flag domain.

Then, a complex in X is the space such that Ω<sup>r</sup> h, for any complex Ω<sup>r</sup> , in a corresponding long succession is given as follows:

$$
\{\mathfrak{a}^\prime\} \in \mathfrak{b}^\prime\}\tag{5}
$$

For the theorem of Buchdall Eastwood [12], we have that the orbits generalized in X, give us a new cohomological class that is related to the previous for an integral

and such that H• <sup>Ξ</sup>, *<sup>τ</sup>*‐<sup>1</sup> ð Þffi <sup>J</sup>ð Þ*<sup>ν</sup> <sup>T</sup>* kerf g *<sup>D</sup>*‐equations G*=*<sup>H</sup> , has more than enough.

In particular, kerf g *<sup>D</sup>*‐equations , takes place the correspondence with the cycles of H•ð Þ M, <sup>J</sup> . In fact, kerf g *<sup>D</sup>*‐equations , is similar to a compact number of components on which G, acts transitively, and these belong together to the cocycles of

But kerf g *<sup>D</sup>*‐equations only exists as integral of those *<sup>∂</sup>*‐equations in M (with M,

<sup>M</sup> <sup>¼</sup> <sup>∪</sup> *<sup>σ</sup>* <sup>∈</sup>ΣV*σ*, and <sup>Ξ</sup> <sup>¼</sup> <sup>∪</sup> *<sup>γ</sup>* <sup>∈</sup>ΣV*<sup>γ</sup>* , then for n‐dimensional planes of a Grassmann manifold, G1,n had that M*<sup>δ</sup>* ¼ ∪ <sup>M</sup>*π=*z, and Δ<sup>z</sup> ¼ ∪ <sup>Δ</sup>z*=π*, which defines cycles in

However, fixed G exists alone a finite number of flag manifolds of certain biholomorphism of this type. These are in bijective correspondence with the conjugated classes of parabolic subalgebras of g, and each flag manifold admits a finite

Then, kerf g *<sup>D</sup>*‐equations , is made of a finite number of G�orbits, all which are closed and (iii) == > (ii). Then since each one of these G�orbits exists like an K�orbit of the space of classes G*=*K, with Nijenhuis null curvature tensor, then each

Its integrals are orbital, and their extensions to M*δ*, and Δz, are generalized integrals (since they are integral of line along the fibers of M*δ*, and Δz, respectively)

<sup>2</sup> **Theorem.** The K� invariance given for the G�structure SGð Þ <sup>M</sup> , of complex holomorphic M, is induced to each closed submanifold given for the flag manifolds of the corresponding vector holomorphic G� bundle. Furthermore, the given integral cohomology on such complex submanifolds is equivalent to the

<sup>3</sup> **Theorem.** Let be M <sup>¼</sup> <sup>G</sup>*=*K, an internal symmetric simply connected Riemannian manifold and of

<sup>T</sup> <sup>∗</sup> ð Þ <sup>M</sup> <sup>⊗</sup> <sup>E</sup> � � <sup>R</sup><sup>M</sup>

� , �

It consists of a finite number of connected components on each a G, that acts transitively. Further, any

The requirement of the transitive action of G, on the orbits is, for example, indispensable to the spatial isotropy hypothesis in the constructions of an integral cohomology to the space–time curvature, since the curvature integrals must be determined on G�invariant orbits and will be calculated for reduction of the

<sup>I</sup> ðÞ¼<sup>j</sup> <sup>0</sup><sup>g</sup> �

<sup>I</sup> ð Þ<sup>j</sup> <sup>∈</sup>End <sup>∧</sup><sup>2</sup>

flag submanifold is an K�orbit of the vector holomorphic G�bundle of the 2n� dimensional irreducible symmetrical Riemannian manifold J Mð Þ.

� . This g establishes a generalized structure of M, which

the theorem II.2 [1],<sup>3</sup> we have that each canonical fibration of a flag manifold will

, the G�orbits are K�orbits in X. Then (i) == > (ii). Now for

, m<sup>þ</sup> ½ �⊂ m, for integrability). Then, ∀z∈ M

ð Þ Vw , ∀w ∈F. Then,

i

<sup>J</sup>ð Þ*<sup>ν</sup>* � �, (7)

<sup>H</sup>•ð Þ! M, <sup>J</sup>ð Þ*<sup>ν</sup>* <sup>H</sup>• <sup>Ξ</sup>, *<sup>τ</sup>*‐<sup>1</sup>

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

give a G�orbit in *Z*, for some internal symmetrical G�space M.

� � exist z<sup>∈</sup> F, such that TzF <sup>¼</sup> *<sup>σ</sup>*TwF∈<sup>P</sup>

operator *T*, defined for

By the theorem II.1 [1]2

<sup>H</sup>• <sup>Ξ</sup>, *<sup>τ</sup>*‐<sup>1</sup> ð Þ <sup>J</sup>ð Þ*<sup>ν</sup>* .

integrable) if RM

and *<sup>σ</sup>* <sup>∈</sup><sup>Σ</sup> ffi <sup>P</sup>

compact type. Then

**83**

<sup>I</sup> ðÞ¼<sup>j</sup> <sup>0</sup> �

i Vz

number of canonical fibrations.

underlies in its composition (in fact mþ,

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

Hn‐<sup>1</sup> U, *<sup>ρ</sup>* ð Þ ‐<sup>1</sup> O Vð Þ , with U <sup>⊂</sup> M. Then (ii) implies (iii).

for which it is continued. Then, (ii) implies (i).♦.

integral cohomology on submanifolds of a maximum complex torus.

<sup>Z</sup> <sup>¼</sup> <sup>R</sup><sup>M</sup>

flag G�manifold is realized as a such orbit for some M.

corresponding holonomy group on K� invariant orbits.

(i.e., to say, all the subcomplexes Ω<sup>r</sup> h, of the complex Ω<sup>r</sup> ). Then, the integral operator cohomology H•ð Þ M, <sup>J</sup> , in a complex manifold M, is that whose complexes conform a holomorphic structure that induces (in the corresponding integral manifold) a generalized according structure of integral submanifolds.

The integral submanifolds represent solutions of those *<sup>∂</sup>*‐equations in cycles of M. The integral submanifolds are the corresponding cocycles of M, under the integral operators of H•ð Þ M, <sup>J</sup> .

For example, if we take the complex manifold *M*, like a manifold of rational curves Ez, of a twistor manifold I [where should understand each manifold I, as the manifold of integral submanifolds (locally)], then its structure comes from a structure projective of their line on Ez, guided according to the vectors in TzM.

These correspond to sections of a normal bundle N Ez, to the curve Ez (infinitesimal deformations to the curve), that is to say, these conform the holomorphic structure that will induce the corresponding structure (that is to say in the corresponding integral manifold). In this case, the generalized structure of integral submanifolds is the Vð Þ <sup>k</sup> ‐conformal integrable structure given by <sup>I</sup>. The integral cohomology in this case is given by the family of rational curves.

The twistor content in this case helps and is necessary to establish the deformation of the integral curves of the vector sheaf of lines O kð Þ. In such case, the integral cohomology is H•ð Þ¼ M, <sup>J</sup> H0ð Þ <sup>I</sup>,O kð Þ .

This example is interesting not only for the fact of the definition of the integral cohomology, which defines, for this way, a class of integrals for M, but also for the fact of satisfaction of the integrability condition for the equation of the tensor of Weyl Wij <sup>¼</sup> 0, where H<sup>0</sup> <sup>I</sup><sup>þ</sup> ð Þ ,O kð Þ (respectively, H0ð Þ <sup>I</sup>,O kð Þ ) are the solutions or integral of W<sup>þ</sup> ¼ 0 (respectively, W� ¼ 0 [13, 14].

#### **2. Duality: line bundles with cohomological contours and closed submanifolds with cohomological functional**

We consider the following result on integral cohomology for integral geometry. **Proposition 2.1.** In the integral cohomology H•ð Þ M, <sup>J</sup> , on complex manifolds, the following statements are equivalent:


$$\text{c. } \mathbf{M}\_{\delta} = \cup\_{\mathbf{M}} \underline{\mathbf{z}}/\underline{\mathbf{z}} \text{, and } \Delta\_{\mathbf{z}} = \cup\_{\Delta} \underline{\mathbf{z}}/\underline{\mathbf{z}} \text{, where } \mathbf{H}^{\star}(\mathbf{M}, \mathfrak{J}) = \mathbf{H}^{\mathbf{n} \cdot \mathbf{1}}(\mathbf{U}, \rho^{\cdot} \mathbf{O}(\mathbf{V})).$$

*Proof.* The integrals on the open G� orbits satisfy the G�invariant integration:

$$\int\_{\mathbf{G}/\mathcal{H}} \mathbf{f} \mathbf{d} \Phi(\boldsymbol{\phi}^{\mathbf{r}}) = \int\_{\mathbf{G}/\mathcal{H}} (\mathbf{f} \circ \boldsymbol{\Phi}^{\mathbf{r}}) \boldsymbol{\phi}^{\mathbf{r}},\tag{6}$$

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

For the theorem of Buchdall Eastwood [12], we have that the orbits generalized in X, give us a new cohomological class that is related to the previous for an integral operator *T*, defined for

$$\mathbf{H}^\bullet(\mathsf{M}, \mathfrak{J}(\nu)) \to \mathbf{H}^\bullet\left(\Xi, \mathfrak{r}^{\cdot 1}\mathfrak{J}(\nu)\right),\tag{7}$$

and such that H• <sup>Ξ</sup>, *<sup>τ</sup>*‐<sup>1</sup> ð Þffi <sup>J</sup>ð Þ*<sup>ν</sup> <sup>T</sup>* kerf g *<sup>D</sup>*‐equations G*=*<sup>H</sup> , has more than enough. By the theorem II.1 [1]2 , the G�orbits are K�orbits in X. Then (i) == > (ii). Now for the theorem II.2 [1],<sup>3</sup> we have that each canonical fibration of a flag manifold will give a G�orbit in *Z*, for some internal symmetrical G�space M.

In particular, kerf g *<sup>D</sup>*‐equations , takes place the correspondence with the cycles of H•ð Þ M, <sup>J</sup> . In fact, kerf g *<sup>D</sup>*‐equations , is similar to a compact number of components on which G, acts transitively, and these belong together to the cocycles of <sup>H</sup>• <sup>Ξ</sup>, *<sup>τ</sup>*‐<sup>1</sup> ð Þ <sup>J</sup>ð Þ*<sup>ν</sup>* .

But kerf g *<sup>D</sup>*‐equations only exists as integral of those *<sup>∂</sup>*‐equations in M (with M, integrable) if RM <sup>I</sup> ðÞ¼<sup>j</sup> <sup>0</sup> � � . This g establishes a generalized structure of M, which underlies in its composition (in fact mþ, , m<sup>þ</sup> ½ �⊂ m, for integrability). Then, ∀z∈ M and *<sup>σ</sup>* <sup>∈</sup><sup>Σ</sup> ffi <sup>P</sup> i Vz � � exist z<sup>∈</sup> F, such that TzF <sup>¼</sup> *<sup>σ</sup>*TwF∈<sup>P</sup> i ð Þ Vw , ∀w ∈F. Then, <sup>M</sup> <sup>¼</sup> <sup>∪</sup> *<sup>σ</sup>* <sup>∈</sup>ΣV*σ*, and <sup>Ξ</sup> <sup>¼</sup> <sup>∪</sup> *<sup>γ</sup>* <sup>∈</sup>ΣV*<sup>γ</sup>* , then for n‐dimensional planes of a Grassmann manifold, G1,n had that M*<sup>δ</sup>* ¼ ∪ <sup>M</sup>*π=*z, and Δ<sup>z</sup> ¼ ∪ <sup>Δ</sup>z*=π*, which defines cycles in Hn‐<sup>1</sup> U, *<sup>ρ</sup>* ð Þ ‐<sup>1</sup> O Vð Þ , with U <sup>⊂</sup> M. Then (ii) implies (iii).

However, fixed G exists alone a finite number of flag manifolds of certain biholomorphism of this type. These are in bijective correspondence with the conjugated classes of parabolic subalgebras of g, and each flag manifold admits a finite number of canonical fibrations.

Then, kerf g *<sup>D</sup>*‐equations , is made of a finite number of G�orbits, all which are closed and (iii) == > (ii). Then since each one of these G�orbits exists like an K�orbit of the space of classes G*=*K, with Nijenhuis null curvature tensor, then each flag submanifold is an K�orbit of the vector holomorphic G�bundle of the 2n� dimensional irreducible symmetrical Riemannian manifold J Mð Þ.

Its integrals are orbital, and their extensions to M*δ*, and Δz, are generalized integrals (since they are integral of line along the fibers of M*δ*, and Δz, respectively) for which it is continued. Then, (ii) implies (i).♦.

$$Z = \{ \mathsf{R}\_{\mathcal{T}}^{\mathsf{M}}(\mathbf{j}) \in \mathrm{End}\left(\wedge^{2}\mathsf{T}\*(\mathsf{M})\otimes\_{\mathbb{R}}\mathbf{E}\right)|\mathsf{R}\_{\mathcal{T}}^{\mathsf{M}}(\mathbf{j}) = \mathbf{0} \},$$

Then, a complex in X is the space such that Ω<sup>r</sup>

corresponding long succession is given as follows:

(i.e., to say, all the subcomplexes Ω<sup>r</sup>

*Advances in Complex Analysis and Applications*

operators of H•ð Þ M, <sup>J</sup> .

h, for any complex Ω<sup>r</sup>

h, of the complex Ω<sup>r</sup>

operator cohomology H•ð Þ M, <sup>J</sup> , in a complex manifold M, is that whose complexes conform a holomorphic structure that induces (in the corresponding integral manifold) a generalized according structure of integral submanifolds.

The integral submanifolds represent solutions of those *<sup>∂</sup>*‐equations in cycles of M. The integral submanifolds are the corresponding cocycles of M, under the integral

For example, if we take the complex manifold *M*, like a manifold of rational curves Ez, of a twistor manifold I [where should understand each manifold I, as the manifold of integral submanifolds (locally)], then its structure comes from a structure projective of their line on Ez, guided according to the vectors in TzM. These correspond to sections of a normal bundle N Ez, to the curve Ez (infinitesimal deformations to the curve), that is to say, these conform the holomorphic structure that will induce the corresponding structure (that is to say in the

corresponding integral manifold). In this case, the generalized structure of integral submanifolds is the Vð Þ <sup>k</sup> ‐conformal integrable structure given by <sup>I</sup>. The integral

deformation of the integral curves of the vector sheaf of lines O kð Þ. In such case, the

This example is interesting not only for the fact of the definition of the integral cohomology, which defines, for this way, a class of integrals for M, but also for the fact of satisfaction of the integrability condition for the equation of the tensor of Weyl Wij <sup>¼</sup> 0, where H<sup>0</sup> <sup>I</sup><sup>þ</sup> ð Þ ,O kð Þ (respectively, H0ð Þ <sup>I</sup>,O kð Þ ) are the solutions or

We consider the following result on integral cohomology for integral geometry. **Proposition 2.1.** In the integral cohomology H•ð Þ M, <sup>J</sup> , on complex manifolds,

a. The open M*δ*, and Δz, are G�orbits opened up in X, and their integrals are

b. Exists an integral operator *<sup>T</sup>*, such that H•ð Þffi M, <sup>J</sup> *<sup>T</sup>*kerf g *<sup>D</sup>*‐equations ,

c. M*<sup>δ</sup>* <sup>¼</sup> <sup>∪</sup> <sup>M</sup>*π=*z, and <sup>Δ</sup><sup>z</sup> <sup>¼</sup> <sup>∪</sup> <sup>Δ</sup>z*=π*, where H•ð Þ¼ M, <sup>J</sup> <sup>H</sup><sup>n</sup>‐<sup>1</sup> U, *<sup>ρ</sup>* ð Þ ‐1O Vð Þ .

fd<sup>Φ</sup> *<sup>ϕ</sup>*<sup>r</sup> ð Þ¼

*Proof.* The integrals on the open G� orbits satisfy the G�invariant integration:

ð

G*=*H

f ∘ Φ‐<sup>1</sup> � �*ϕ*<sup>r</sup>

, (6)

The twistor content in this case helps and is necessary to establish the

**2. Duality: line bundles with cohomological contours and closed**

cohomology in this case is given by the family of rational curves.

integral cohomology is H•ð Þ¼ M, <sup>J</sup> H0ð Þ <sup>I</sup>,O kð Þ .

integral of W<sup>þ</sup> ¼ 0 (respectively, W� ¼ 0 [13, 14].

the following statements are equivalent:

generalized integral for M,

**82**

**submanifolds with cohomological functional**

ð

G*=*H

, in a

). Then, the integral

ð5Þ

<sup>2</sup> **Theorem.** The K� invariance given for the G�structure SGð Þ <sup>M</sup> , of complex holomorphic M, is induced to each closed submanifold given for the flag manifolds of the corresponding vector holomorphic G� bundle. Furthermore, the given integral cohomology on such complex submanifolds is equivalent to the integral cohomology on submanifolds of a maximum complex torus.

<sup>3</sup> **Theorem.** Let be M <sup>¼</sup> <sup>G</sup>*=*K, an internal symmetric simply connected Riemannian manifold and of compact type. Then

It consists of a finite number of connected components on each a G, that acts transitively. Further, any flag G�manifold is realized as a such orbit for some M.

The requirement of the transitive action of G, on the orbits is, for example, indispensable to the spatial isotropy hypothesis in the constructions of an integral cohomology to the space–time curvature, since the curvature integrals must be determined on G�invariant orbits and will be calculated for reduction of the corresponding holonomy group on K� invariant orbits.

**Proposition 2.2.** The nð Þ� � <sup>1</sup> *<sup>∂</sup>*� complex cohomology with coefficients in a complex holomorphic bundle of M, is a cohomology of hyperlines and hyperplanes4 .

Their demonstration is a simple consequence of the digression in part II of Ref. [1], on some basic integral *<sup>∂</sup>*�cohomologies on n�dimensional complex spaces

(see **Figure 2**), of this same philosophical dissertation of integral operators

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

cohomology of contours (cohomological functional).

connecting map in the succession of Mayer-Vietoris:

with rule of correspondence for a complex coordinates system z1, z2, … , zn,

2 � �<sup>n</sup>‐<sup>1</sup>ð

*D*

the evaluation of f ∈ H Dð Þ, in the complex hyperplane *π*ð Þz , of D, with rule of correspondence:

<sup>∧</sup> � � <sup>⊗</sup> <sup>z</sup> <sup>þ</sup> <sup>f</sup>

followed residual *<sup>∂</sup>*‐cohomology and the complex Radon transform on subvarieties of <sup>1</sup> [16].

where in particular the exterior algebra <sup>∧</sup>0,qð Þ <sup>T</sup> <sup>∗</sup> ð Þ <sup>M</sup> <sup>⊗</sup> <sup>V</sup> , is generated for elements *<sup>π</sup>*ð Þ<sup>z</sup> <sup>∧</sup>*∂*R fð Þð ÞÞ <sup>z</sup> . Then

<sup>∧</sup> <sup>∈</sup>L Dð Þ, we have that:

<sup>∧</sup>ð Þ<sup>z</sup> � � <sup>¼</sup> *<sup>∂</sup>* <sup>f</sup>

<sup>∧</sup> *<sup>ζ</sup>*1, *<sup>ζ</sup>*2, … , *<sup>ζ</sup>*<sup>n</sup> ð Þ¼ <sup>i</sup>

*∂* f

example with cohomology space H3*υ*ð Þ M � Sing M, .

We define the following concept.

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

**Proposition 2.3.** The integrals of contour are generalized function in a

is an integral cohomology of the form H•ð Þ M � Sing M, , where <sup>M</sup>, can be

understood as a twistor space corresponding to M. (see **Figure 3**). For example, this class belongs to the Feynman integrals.

manifolds X, and Y, respectively, *<sup>α</sup>*<sup>∈</sup> <sup>H</sup>pð Þ X, S , and *<sup>β</sup>* <sup>∈</sup> Hqð Þ Y, S .

*<sup>∂</sup>* <sup>∗</sup> : Hpþ<sup>q</sup>ð Þ! <sup>X</sup><sup>∩</sup> Y, S <sup>⊗</sup> <sup>T</sup> Hpþqþ<sup>1</sup>

**Definition 2.1.** Cohomological function of a cohomology H• <sup>M</sup> � SingM, <sup>Ω</sup><sup>r</sup> ð Þ, <sup>5</sup>

We consider p and differential q forms of the cohomologies on the complex

We consider their cup product given for *<sup>α</sup>* <sup>∪</sup> *<sup>β</sup>* <sup>∈</sup> <sup>H</sup>pþqð Þ <sup>X</sup>∩Y, S <sup>⊗</sup> <sup>T</sup> , and the

R : H Dð Þ! L Dð Þ,

We consider *<sup>∂</sup>*, complex scalar mapping defined to Dolbeault cohomology. Let D <sup>∗</sup> ¼ Lð Þ H Dð Þ, , be the set of

evf : Lð Þ! H Dð Þ, ,

*π*ð Þz ð Þ¼ f <*π*ð Þz , f >,

<sup>R</sup>*∂*ð Þ<sup>f</sup> <sup>⊗</sup> evf <sup>¼</sup> <sup>&</sup>lt;*π*ð Þ<sup>z</sup> f, *<sup>∂</sup>*ð Þ RF <sup>&</sup>gt;,

<sup>5</sup> Here, <sup>M</sup>, is the product of all the twistor spaces and Sing M, is the union of all the subspaces on which the Zð Þ *<sup>α</sup>*W*<sup>α</sup>* ‐<sup>1</sup>

, are singular factors. Its differential form is integrated over a contour, which can be traditional contour, for

Therefore p <sup>¼</sup> 0. Then R*∂*ð Þ<sup>f</sup> <sup>⊗</sup> evf <sup>∈</sup> A0,qð Þ <sup>V</sup> . Then H0,q <sup>n</sup> ð Þ¼ *<sup>=</sup>*D, V H0,qð Þ D1, V , which is a Dolbeault cohomology. Of this form, it can be established that through the hyperspaces geometry can be determined a certain class of holomorphic complex functions using an appropriate *<sup>∂</sup>*‐ cohomology. Also Gindikin generalizes this idea using the ð Þ <sup>n</sup>‐1‐<sup>q</sup> ‐*∂*‐cohomology determining the Radon transform on hyperplanes defined as linearly concave domains of dimension q, first to the real case and after to the complex case [16]. For example, a good modern research to the respect is the

f zð Þ 1, z2, … , zn *<sup>δ</sup>* <sup>s</sup>‐ *<sup>ζ</sup>*1, *<sup>ζ</sup>*2, … , *<sup>ζ</sup>*<sup>n</sup> ð Þ ½ � ð Þ, zð Þ 1, z2, … , zn •

<dz1, dz2, … , dzn, dz1, dz2, … , dznÞ,

<sup>∧</sup>*∂*ð Þ¼ <sup>z</sup> *<sup>∂</sup>*ð Þ<sup>z</sup> <sup>⊗</sup> R fð Þ <sup>∈</sup> <sup>A</sup>p,q‐<sup>1</sup>

ð Þ V ,

,

ð Þ X∪Y, S ⊗ T , (8)

published in 2007.

f

hyperplanes corresponding to D. Let

Due to that *<sup>∂</sup>* R fð Þ¼ <sup>R</sup>*∂*ð Þ<sup>f</sup> , <sup>∀</sup><sup>f</sup>

∀f ∈ H Dð Þ.

and Z*<sup>α</sup>* ð Þ <sup>A</sup>*<sup>α</sup>* ‐<sup>1</sup>

**85**

Let be M ffi n, and we consider concave linearly domains D, in <sup>n</sup> (or so better in <sup>n</sup> [16]). D, has structure of complex vector space. Let be D1 <sup>¼</sup> *<sup>ξ</sup>*<sup>n</sup>*=*D, a holomorphic convex linear domain conformed for holomorphic hyperplanes *<sup>π</sup>*ið Þ <sup>D</sup> , i <sup>¼</sup> 1, 2, 3, … , in D1. Let be H<sup>1</sup> ð Þ D , the complex holomorphic functions space defined on D1. Let D ! M, be a fiber vector bundle seated in the complex holomorphic manifold M. Let Ap,qð Þ <sup>D</sup> , be the p, q ð Þ‐forms space on M, with values in D (that is to say, the global sections space of the fiber tangent bundle <sup>∧</sup>p,qT <sup>∗</sup> ð Þ <sup>M</sup> <sup>⊗</sup> D). Of this way, the bi-graded algebra is the space

$$\mathcal{A}(\mathbf{D}) = \oplus\_{\mathbf{n}+\mathbf{m}-\mathbf{p}} \mathbf{A}^{\mathbf{n},\mathbf{m}}(\mathbf{D}),$$

Let *<sup>∂</sup>*‐be the scalar operator on the complex manifold M, with values on the vector bundle of global sections <sup>E</sup> <sup>∧</sup>p,q ð Þ <sup>T</sup> <sup>∗</sup> ð Þ <sup>M</sup> <sup>⊗</sup> <sup>D</sup> , that is to say, the differential operator

$$\overline{\mathcal{O}} : \mathcal{E}(\wedge^{\mathbb{P},\mathbb{q}} \mathcal{T} \* (\mathbb{M}) \otimes \mathcal{D} \mid) \to \mathcal{E}\left(\wedge^{\mathbb{P},\mathbb{q}+1} \mathcal{T} \* (\mathbb{M}) \otimes \mathcal{D} \mid\right),$$

The operator *∂*, complies certain anticommutative properties [1]. Now, we consider the Radon transform on the complex holomorphic functions H Dð Þ, and its analogous for the complex holomorphic functionals H Dð Þ ∗ , through the corresponding diagram:

$$\begin{array}{l} \mathrm{D} \xrightarrow{\mathrm{f}} \mathrm{H}(\mathrm{D}) \xrightarrow{\mathrm{R}\_{7}} \mathrm{L}(\mathrm{D})\\ \downarrow \mathrm{F} \text{unctional} \downarrow \mathrm{F} \text{unctional} \downarrow \mathrm{F} \text{unctional} \downarrow,\\ \mathrm{D} \ast \xrightarrow{\wp} \mathrm{H}(\mathrm{D} \ast) \xrightarrow{\mathrm{R}\_{7}\ast} \mathrm{L}(\mathrm{D} \ast) \end{array}$$

and we consider an *<sup>∂</sup>*‐operator in D. The before diagram represents a first cohomological advance on the relation between functionals of the nð Þ� � <sup>1</sup> dimensional *<sup>∂</sup>*‐ cohomology with coefficients in a complex vector bundle <sup>Ω</sup>n(holomorphic complex bundle) and the integration of the cohomology on hyperplanes in D, which an integral geometry is equivalent to consider an adequate Radon transform in D. Likewise, and using the satellites *∂* R, and *∂*R�<sup>1</sup> , of the before diagram and composing the diagram with the correspondences to R, on D, and D ∗ , we can have:

$$
\overline{\partial}\,\mathsf{R}(\mathsf{f}) = \mathsf{R}\overline{\partial}(\mathsf{f}),
$$

The details of the demonstration of this identity can be seen in Ref. [1].

Due to that R, is injective, this is equivalent to have the exact succession of Radon transform images:

$$\mathbf{0} \to \mathsf{R}\_{\mathsf{D}} \mathbf{A}^{0,0}(\mathbf{L}(\mathbf{D})^{\prime}) \to \mathbf{R} \mathbf{A}^{0,1}(\mathsf{H}(\mathbf{D})^{\prime}) \to \mathbf{R} \ast \mathbf{A}^{0,1}(\mathsf{L}(\mathbf{D})^{\prime}) \to \mathbf{0},$$

or in equivalent way, for the exact succession:

$$\mathbf{0} \to \mathbf{R}\_0^{-1} \mathbf{B}^{0,0} (\mathbf{L}(\mathbf{D})^{\ast}) \to \mathbf{R}^{-1} \mathbf{B}^{0,1} (\mathbf{H}(\mathbf{D})^{\ast}) \to \mathbf{R} \ast^{-1} \mathbf{B}^{0,1} (\mathbf{L}(\mathbf{D})^{\prime}) \to \mathbf{0},$$

Here the <sup>00</sup> and <sup>0</sup> denote the projectivizing of spaces for R. Then the Radon transform can be generalized to the *<sup>∂</sup>*‐

cohomology classes on the complex spaces D, respectively D1, in n, as the mapping:

<sup>R</sup>*<sup>∂</sup>* : *<sup>∂</sup>*‐cohomology of dimension n ! *<sup>∂</sup>*‐cohomology of dimension nð Þ ‐<sup>1</sup> ,

whose restriction to a domain D, is the mapping:

$$
\mathsf{R}\_{\mathsf{\eth}}|\_{\mathsf{D}}: \mathsf{H}(\mathsf{D}) \to \mathsf{L}(\mathsf{D}),
$$

which satisfies the diagram of correspondences for functionals of a nð Þ ‐<sup>1</sup> ‐dimensional cohomology with coefficients in a holomorphic vector bundle E ! M.

The natural question arises, what relation there is between the two different corresponding objects to functionals, that is to say, cohomology and functions?

The relation is an integral relation of certain *<sup>∂</sup>*‐cohomology, which comes defined for the Radon transform of the Dolbeault cohomology R*∂*.

The Radon transform can be viewed as the mapping of cohomological spaces:

$$\mathsf{R\_{\mathfrak{F}}} \colon \mathsf{H}^{\mathsf{p},\mathsf{n}}(\mathsf{D},\mathsf{V}) \to \mathsf{H}^{\mathsf{p},\mathsf{n}-1}(\mathsf{D},\mathsf{V}),$$

Therefore, it is enough to demonstrate that R*<sup>∂</sup>* <sup>H</sup>p,n ð Þ ð Þ D, V , is the q*th*‐projection nð Þ ‐<sup>1</sup> ‐dimensional of Hp,nð Þ D, V , in <sup>H</sup>0,qð Þ¼ D, V H0,nð Þ D, V . Remember that the Radon transform in the complex context D, is the continuous and analytic mapping [17]:

<sup>4</sup> We can have a little digression with certain details on the complex Radon transform using submanifolds in the space <sup>1</sup> , to the *<sup>∂</sup>*� cohomology. Let be M, a complex holomorphic manifold (or complex Riemannian manifold [15]). We consider its corresponding reductive homogeneous space determined for the flag manifold *F* ¼ G*=*P, with P, a parabolic subgroup of G. We consider the open orbit given for the Stein manifold *F*<sup>D</sup> ¼ G*=*H (as was defined in the footnote 1) with H, a compact subgroup of the real form G0, of G.

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

Their demonstration is a simple consequence of the digression in part II of Ref. [1], on some basic integral *<sup>∂</sup>*�cohomologies on n�dimensional complex spaces (see **Figure 2**), of this same philosophical dissertation of integral operators published in 2007.

**Proposition 2.3.** The integrals of contour are generalized function in a cohomology of contours (cohomological functional).

We define the following concept.

**Definition 2.1.** Cohomological function of a cohomology H• <sup>M</sup> � SingM, <sup>Ω</sup><sup>r</sup> ð Þ, <sup>5</sup> is an integral cohomology of the form H•ð Þ M � Sing M, , where <sup>M</sup>, can be understood as a twistor space corresponding to M. (see **Figure 3**).

For example, this class belongs to the Feynman integrals.

We consider p and differential q forms of the cohomologies on the complex manifolds X, and Y, respectively, *<sup>α</sup>*<sup>∈</sup> <sup>H</sup>pð Þ X, S , and *<sup>β</sup>* <sup>∈</sup> Hqð Þ Y, S .

We consider their cup product given for *<sup>α</sup>* <sup>∪</sup> *<sup>β</sup>* <sup>∈</sup> <sup>H</sup>pþqð Þ <sup>X</sup>∩Y, S <sup>⊗</sup> <sup>T</sup> , and the connecting map in the succession of Mayer-Vietoris:

$$\partial \ast : \mathbf{H}^{p+q}(\mathbf{X} \cap \mathbf{Y}, \mathbf{S} \otimes \mathbf{T}) \to \mathbf{H}^{p+q+1}(\mathbf{X} \cup \mathbf{Y}, \mathbf{S} \otimes \mathbf{T}),\tag{8}$$

$$
\mathsf{R}: \mathsf{H}(\mathsf{D}) \to \mathsf{L}(\mathsf{D}),
$$

with rule of correspondence for a complex coordinates system z1, z2, … , zn,

$$\begin{split} \mathbf{f}^{\wedge}(\zeta\_{1},\zeta\_{2},\ldots,\zeta\_{n}) &= \left(\frac{\mathbf{i}}{2}\right)^{\bullet \cdot 1} \int\_{D} \mathbf{f}(\mathbf{z}\_{1},\mathbf{z}\_{2},\ldots,\mathbf{z}\_{n}) \delta(\mathbf{s}\cdot[(\zeta\_{1},\zeta\_{2},\ldots,\zeta\_{n}),(\mathbf{z}\_{1},\mathbf{z}\_{2},\ldots,\mathbf{z}\_{n})]) \bullet \\ & \qquad \qquad \qquad \qquad \qquad < \mathrm{d}\mathbf{z}\_{1}, \mathrm{d}\mathbf{z}\_{2}, \ldots,\mathbf{d}\mathbf{z}\_{n}, \mathrm{d}\overline{\mathbf{z}}\_{1}, \operatorname{d}\overline{\mathbf{z}}\_{2}, \ldots,\mathbf{d}\overline{\mathbf{z}}\_{n}), \end{split}$$

∀f ∈ H Dð Þ.

**Proposition 2.2.** The nð Þ� � <sup>1</sup> *<sup>∂</sup>*� complex cohomology with coefficients in a complex holomorphic bundle of M, is a cohomology of hyperlines and hyperplanes4

<sup>4</sup> We can have a little digression with certain details on the complex Radon transform using submanifolds in the space

Let be M ffi n, and we consider concave linearly domains D, in <sup>n</sup> (or so better in <sup>n</sup> [16]). D, has structure of complex vector space. Let be D1 <sup>¼</sup> *<sup>ξ</sup>*<sup>n</sup>*=*D, a holomorphic convex linear domain conformed for holomorphic hyperplanes

values in D (that is to say, the global sections space of the fiber tangent bundle <sup>∧</sup>p,qT <sup>∗</sup> ð Þ <sup>M</sup> <sup>⊗</sup> D).

fiber vector bundle seated in the complex holomorphic manifold M. Let Ap,qð Þ <sup>D</sup> , be the p, q ð Þ‐forms space on M, with

Að Þ¼ <sup>D</sup> <sup>⊕</sup><sup>n</sup>þm¼pAn,mð Þ <sup>D</sup> ,

Let *<sup>∂</sup>*‐be the scalar operator on the complex manifold M, with values on the vector bundle of global sections

The operator *∂*, complies certain anticommutative properties [1]. Now, we consider the Radon transform on the complex holomorphic functions H Dð Þ, and its analogous for the complex holomorphic functionals H Dð Þ ∗ , through the

> L Dð Þ ↓Functional↓Functional↓Functional ,

L Dð Þ ∗ and we consider an *<sup>∂</sup>*‐operator in D. The before diagram represents a first cohomological advance on the relation between functionals of the nð Þ� � <sup>1</sup> dimensional *<sup>∂</sup>*‐ cohomology with coefficients in a complex vector bundle <sup>Ω</sup>n(holomorphic complex bundle) and the integration of the cohomology on hyperplanes in D, which an integral geometry is equivalent to

*<sup>∂</sup>* R fð Þ¼ <sup>R</sup>*∂*ð Þ<sup>f</sup> ,

<sup>0</sup> ! R0A0,0ð Þ! L Dð Þ″ RA0,1ð Þ! H Dð Þ″ <sup>R</sup> <sup>∗</sup> <sup>A</sup>0,1 L Dð Þ<sup>0</sup> ! 0,

R*∂*j<sup>D</sup> :H Dð Þ! L Dð Þ, which satisfies the diagram of correspondences for functionals of a nð Þ ‐<sup>1</sup> ‐dimensional cohomology with coefficients in a

The natural question arises, what relation there is between the two different corresponding objects to functionals, that is

ð Þ D, V ,

The relation is an integral relation of certain *<sup>∂</sup>*‐cohomology, which comes defined for the Radon transform of the

<sup>R</sup>*<sup>∂</sup>* : <sup>H</sup>p,nð Þ! D, V Hp,n‐<sup>1</sup>

Therefore, it is enough to demonstrate that R*<sup>∂</sup>* <sup>H</sup>p,n ð Þ ð Þ D, V , is the q*th*‐projection nð Þ ‐<sup>1</sup> ‐dimensional of Hp,nð Þ D, V , in <sup>H</sup>0,qð Þ¼ D, V H0,nð Þ D, V . Remember that the Radon transform in the complex context D, is the continuous and analytic

Here the <sup>00</sup> and <sup>0</sup> denote the projectivizing of spaces for R. Then the Radon transform can be generalized to the *<sup>∂</sup>*‐

B0,1ð Þ! H Dð Þ″ <sup>R</sup> <sup>∗</sup> �<sup>1</sup>

<sup>B</sup>0,1 L Dð Þ<sup>0</sup> ! 0,

*<sup>∂</sup>* : <sup>E</sup> <sup>∧</sup>p,q <sup>ð</sup> <sup>T</sup> <sup>∗</sup> ð Þ <sup>M</sup> <sup>⊗</sup> <sup>D</sup> Þ!E <sup>∧</sup>p,qþ<sup>1</sup>

*<sup>φ</sup>* H Dð Þ! <sup>∗</sup> <sup>R</sup>*<sup>∂</sup>* <sup>∗</sup>

Due to that R, is injective, this is equivalent to have the exact succession of Radon transform images:

D ! <sup>f</sup> H Dð Þ!<sup>R</sup>*<sup>∂</sup>*

D ∗ !

consider an adequate Radon transform in D. Likewise, and using the satellites *∂* R, and *∂*R�<sup>1</sup>

composing the diagram with the correspondences to R, on D, and D ∗ , we can have:

<sup>0</sup> B0,0ð Þ! L Dð Þ″ <sup>R</sup>‐<sup>1</sup>

cohomology classes on the complex spaces D, respectively D1, in n, as the mapping:

<sup>R</sup>*<sup>∂</sup>* : *<sup>∂</sup>*‐cohomology of dimension n ! *<sup>∂</sup>*‐cohomology of dimension nð Þ ‐<sup>1</sup> ,

The Radon transform can be viewed as the mapping of cohomological spaces:

The details of the demonstration of this identity can be seen in Ref. [1].

or in equivalent way, for the exact succession: <sup>0</sup> ! <sup>R</sup>‐<sup>1</sup>

whose restriction to a domain D, is the mapping:

holomorphic vector bundle E ! M.

to say, cohomology and functions?

Dolbeault cohomology R*∂*.

mapping [17]:

**84**

footnote 1) with H, a compact subgroup of the real form G0, of G.

*Advances in Complex Analysis and Applications*

*<sup>π</sup>*ið Þ <sup>D</sup> , i <sup>¼</sup> 1, 2, 3, … , in D1. Let be H<sup>1</sup>

corresponding diagram:

Of this way, the bi-graded algebra is the space

<sup>E</sup> <sup>∧</sup>p,q ð Þ <sup>T</sup> <sup>∗</sup> ð Þ <sup>M</sup> <sup>⊗</sup> <sup>D</sup> , that is to say, the differential operator

, to the *<sup>∂</sup>*� cohomology. Let be M, a complex holomorphic manifold (or complex Riemannian manifold [15]). We consider its corresponding reductive homogeneous space determined for the flag manifold *F* ¼ G*=*P, with P, a parabolic subgroup of G. We consider the open orbit given for the Stein manifold *F*<sup>D</sup> ¼ G*=*H (as was defined in the

ð Þ D , the complex holomorphic functions space defined on D1. Let D ! M, be a

<sup>T</sup> <sup>∗</sup> ð Þ <sup>M</sup> <sup>⊗</sup> <sup>D</sup> ,

<sup>1</sup>

.

, of the before diagram and

We consider *<sup>∂</sup>*, complex scalar mapping defined to Dolbeault cohomology. Let D <sup>∗</sup> ¼ Lð Þ H Dð Þ, , be the set of hyperplanes corresponding to D. Let

$$\mathsf{ev}\_{\mathsf{f}} : \mathcal{L}(\mathsf{H}(\mathsf{D}), \mathsf{C}) \to \mathsf{C},$$

the evaluation of f ∈ H Dð Þ, in the complex hyperplane *π*ð Þz , of D, with rule of correspondence:

$$
\pi(\mathbf{z})(\mathbf{f}) = <\pi(\mathbf{z}), \mathbf{f} > ,
$$

Due to that *<sup>∂</sup>* R fð Þ¼ <sup>R</sup>*∂*ð Þ<sup>f</sup> , <sup>∀</sup><sup>f</sup> <sup>∧</sup> <sup>∈</sup>L Dð Þ, we have that:

$$
\overline{\partial}\left(\mathbf{f}^{\wedge}(\mathbf{z})\right) = \overline{\partial}(\mathbf{f}^{\wedge}) \otimes \mathbf{z} + \mathbf{f}^{\wedge}\overline{\partial}(\mathbf{z}) = \overline{\partial}(\mathbf{z}) \otimes \mathbb{R}(\mathbf{f}) \in \mathbf{A}^{\mathrm{p},\mathrm{q}\cdot 1}(\mathbf{V}),
$$

where in particular the exterior algebra <sup>∧</sup>0,qð Þ <sup>T</sup> <sup>∗</sup> ð Þ <sup>M</sup> <sup>⊗</sup> <sup>V</sup> , is generated for elements *<sup>π</sup>*ð Þ<sup>z</sup> <sup>∧</sup>*∂*R fð Þð ÞÞ <sup>z</sup> . Then

$$
\mathsf{R}\_{\overline{\partial}}(\mathbf{f}) \otimes \mathsf{ev}\_{\mathbf{f}} = \mathsf{ } \mathsf{s}(\mathbf{z}) \mathbf{f}, \overline{\partial}(\mathbf{RF}) > ,
$$

Therefore p <sup>¼</sup> 0. Then R*∂*ð Þ<sup>f</sup> <sup>⊗</sup> evf <sup>∈</sup> A0,qð Þ <sup>V</sup> . Then H0,q <sup>n</sup> ð Þ¼ *<sup>=</sup>*D, V H0,qð Þ D1, V , which is a Dolbeault cohomology. Of this form, it can be established that through the hyperspaces geometry can be determined a certain class of holomorphic complex functions using an appropriate *<sup>∂</sup>*‐ cohomology. Also Gindikin generalizes this idea using the ð Þ <sup>n</sup>‐1‐<sup>q</sup> ‐*∂*‐cohomology determining the Radon transform on hyperplanes defined as linearly concave domains of dimension q, first to the real case and after to the complex case [16]. For example, a good modern research to the respect is the followed residual *<sup>∂</sup>*‐cohomology and the complex Radon transform on subvarieties of <sup>1</sup> [16].

<sup>5</sup> Here, <sup>M</sup>, is the product of all the twistor spaces and Sing M, is the union of all the subspaces on which the Zð Þ *<sup>α</sup>*W*<sup>α</sup>* ‐<sup>1</sup> , and Z*<sup>α</sup>* ð Þ <sup>A</sup>*<sup>α</sup>* ‐<sup>1</sup> , are singular factors. Its differential form is integrated over a contour, which can be traditional contour, for example with cohomology space H3*υ*ð Þ M � Sing M, .

**Figure 2.** *Convex domains conformed for holomorphic hyperplanes π*ið Þ D *:*

**Figure 3.** *Cohomology of contours isomorphic to* <sup>H</sup>• <sup>M</sup> � Sing M, <sup>Ω</sup><sup>r</sup> ð Þ*:*

We consider for the inner product of *α*, and *β*, the relation is

$$a \bullet \beta = \partial \ast (a \cup \beta),\tag{9}$$

using the description of Dolbeault of the first group, forgetting the bi-graduation ð Þ d, f , (d, f) and reminding only the total grade d þ f. A description of Cěch of this mapping is used for the evaluation of twistor cohomology. In our case, we will only use the duality of Poincaré to know in what moment of the evaluation of an element

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

more general sense the cohomological functional. Likewise, the mapping Eq. (10) is

This contour "cohomologic" is easy to relate it with a traditional in Hdð Þ <sup>Π</sup>‐ℓ, ,

given for iteration of the constant mapping of Mayer-Vietoris (in homology)

For example, for diagram, product can be demonstrated that H8ð Þ <sup>Π</sup>‐ℓ, , and that the image of the generator of this group low two mappings of Mayer-Vietoris as is the usual in the physical contour for the product of diagrams given. This affirms that only exists a cohomological contour for the product climb (as is expected) and

**Definition 2.2.** (Hyperfunction). A hyperfunction on n, is an element of the

*Proof*: Consider a vector holomorphic G‐invariant sheaf and their corresponding bundle of lines associated with those r, 0 ð Þ‐forms on the corresponding topological vector space. Then, the integrals on the fibers of the vector holomorphic sheaf are the integrals of line on the cycles of the sections X, of the vector sheaf, given by

*<sup>γ</sup>*X• *δ*, ∀ *δ*∈ Ω<sup>r</sup> [where Ω<sup>r</sup> is a complex defined in Eq. (5)]. Then the holomorphic structure that constitutes these complexes induces (in the corresponding integral manifold) a conformal generalized structure of integral submanifolds where the arches *γ*, are local parts of integral curves of the fibers of the vector sheaf of lines. In

ð Þ Vz exists locally an integral submanifold S, with z∈S, such

ð Þ Vw , ∀w ∈ S. Then the integral of line can be re-written

ð Þ <sup>Π</sup>‐ℓ, . Indeed, let be T ¼ <sup>n</sup> <sup>þ</sup> iV, the tube

<sup>f</sup> • *<sup>δ</sup>*, <sup>∀</sup> *<sup>δ</sup>*<sup>∈</sup> <sup>Ω</sup><sup>r</sup> , f <sup>∈</sup><sup>T</sup> , (12)

**Proposition 2.4.** The general integrals of line are functional on arches *γ*, in

suggests a method for contours that verifies and observes which belong to

, Ω<sup>d</sup> � �. This can define in a

, <sup>Ω</sup><sup>d</sup> � � ! Hd <sup>Π</sup>‐ℓ<sup>0</sup> ð Þ , , (11)

of Hfþ<sup>d</sup> <sup>Π</sup>‐ℓ<sup>0</sup> ð Þ , , one can need a contour in Hfþ<sup>d</sup> <sup>Π</sup>‐ℓ<sup>0</sup>

Hfþ<sup>d</sup> <sup>Π</sup>‐ℓ<sup>0</sup>

ð Þ <sup>n</sup> � <sup>1</sup> ‐*<sup>∂</sup>* � cohomology Hð Þ <sup>n</sup>‐<sup>1</sup> ð Þ M, <sup>J</sup> , with M <sup>¼</sup> <sup>n</sup>*=*n.

geometry of conformal generalized structure.

i

in this conformal generalized structure as

i

ð

TzS

where T , is the tube domain (in the local structure where the integral submanifold S, exists) T ¼ <sup>n</sup> <sup>þ</sup> iV, where V, is a cone, not necessarily convex (that has applicability on the fibers of the sheaf of lines). The idea is to define the expressionf • *δ*, inside the context of the integral of line in such case that the values of f, on the arch *γ*, are values off, a hyperfunction represented this like a variation of holomorphic functions f zð Þ j*δ* , in a submanifold of Stein M*δ*, such that M*δ*⊃T . Then, the sesquilinear coupling of the hyperfunction corresponding to f, and the function f itself, is an integral of contour, and for Proposition 2.3, a generalized

domain where the cone V, is not necessarily convex. This cone V ¼ ∪ *<sup>γ</sup>* <sup>∈</sup>ΣV*<sup>γ</sup>* , in the conformal generalized structure where the V*γ*, are the convex maximal sub-cones in V. Considers our manifold, complex Riemannian manifold. The idea is that a holomorphic form required in this language is a good expression to write the

ð *γ* X• *δ* ¼

an example of the cohomological functional.

due to that the following mapping exists

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

<sup>f</sup>‐times; one for each field.

cohomologics.

other words, ∀ *γ* ∈P

that Tz*<sup>S</sup>* <sup>¼</sup> *<sup>γ</sup>*, and TwS∈<sup>P</sup>

functional in the cohomology H1

Ð

**87**

This description of the inner product has been used in a new development of the cohomology for twistor diagrams foreseen in Refs. [14, 18]. This new method is almost opposed to the procedure that we want to use in the unification of contour integrals on diagrams, in respect of the Feynman integral, although also proper to the Conway integrals, Cauchy integrals,<sup>6</sup> and some integral transforms as the Hilbert transforms.

We want to assemble a Feynman diagram for applications of the product "cup." The interior edges of a Feynman diagram are taken again as elements of groups H<sup>0</sup> (such extra elements have to be abandoned in a cohomology, for example, <sup>H</sup>• M, *<sup>τ</sup>*‐<sup>1</sup> ð Þ <sup>J</sup>ð Þ*<sup>ν</sup>* , and the interior edges form the fields (assuming that they are elementary states) in several cohomology groups H1 .

Let denote M, for Π, and sing M ¼ ℓ. Likewise, if f, is one of these elements of H1 , this new procedure determines an element of the cohomology H<sup>f</sup> <sup>Π</sup>‐ℓ<sup>0</sup> , Ω<sup>d</sup> , where ℓ<sup>0</sup> , is the union of all the subspaces defined by internal edges, always with the subspaces <sup>1</sup> , on whose elementary states f, are singular.

Then for Proposition 2.1 (b), the following mapping exists

$$\mathbf{H}^{\mathbf{f}}(\Pi \mathbf{\cdot} \boldsymbol{\ell}', \mathbf{\Omega}^{\mathbf{d}}) \to \mathbf{H}^{\mathbf{f}+\mathbf{d}}(\Pi \mathbf{\cdot} \boldsymbol{\ell}', \mathbf{C}),\tag{10}$$

$$\mathbf{f}^{(\mathbf{n})}(\mathbf{a}) = \frac{\mathbf{n}!}{2\pi \mathbf{i}} \oint\_{\mathbb{T}} \frac{\mathbf{f}(\mathbf{z})}{\left(\mathbf{z}\cdot\mathbf{a}\right)^{n+1}} \,\mathrm{d}\mathbf{z}.$$

<sup>6</sup> For example, to this case for holomorphic functions, we have the generalized Cauchy formula:

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

using the description of Dolbeault of the first group, forgetting the bi-graduation ð Þ d, f , (d, f) and reminding only the total grade d þ f. A description of Cěch of this mapping is used for the evaluation of twistor cohomology. In our case, we will only use the duality of Poincaré to know in what moment of the evaluation of an element of Hfþ<sup>d</sup> <sup>Π</sup>‐ℓ<sup>0</sup> ð Þ , , one can need a contour in Hfþ<sup>d</sup> <sup>Π</sup>‐ℓ<sup>0</sup> , Ω<sup>d</sup> � �. This can define in a more general sense the cohomological functional. Likewise, the mapping Eq. (10) is an example of the cohomological functional.

This contour "cohomologic" is easy to relate it with a traditional in Hdð Þ <sup>Π</sup>‐ℓ, , due to that the following mapping exists

$$\mathbf{H}\_{\mathrm{f}+\mathrm{d}}\left(\boldsymbol{\Pi}\cdot\boldsymbol{\ell}^{\prime},\boldsymbol{\Omega}^{\mathrm{d}}\right)\to\mathbf{H}\_{\mathrm{d}}(\boldsymbol{\Pi}\cdot\boldsymbol{\ell}^{\prime},\mathbb{C}),\tag{11}$$

given for iteration of the constant mapping of Mayer-Vietoris (in homology) <sup>f</sup>‐times; one for each field.

For example, for diagram, product can be demonstrated that H8ð Þ <sup>Π</sup>‐ℓ, , and that the image of the generator of this group low two mappings of Mayer-Vietoris as is the usual in the physical contour for the product of diagrams given. This affirms that only exists a cohomological contour for the product climb (as is expected) and suggests a method for contours that verifies and observes which belong to cohomologics.

**Definition 2.2.** (Hyperfunction). A hyperfunction on n, is an element of the ð Þ <sup>n</sup> � <sup>1</sup> ‐*<sup>∂</sup>* � cohomology Hð Þ <sup>n</sup>‐<sup>1</sup> ð Þ M, <sup>J</sup> , with M <sup>¼</sup> <sup>n</sup>*=*n.

**Proposition 2.4.** The general integrals of line are functional on arches *γ*, in geometry of conformal generalized structure.

*Proof*: Consider a vector holomorphic G‐invariant sheaf and their corresponding bundle of lines associated with those r, 0 ð Þ‐forms on the corresponding topological vector space. Then, the integrals on the fibers of the vector holomorphic sheaf are the integrals of line on the cycles of the sections X, of the vector sheaf, given by Ð *<sup>γ</sup>*X• *δ*, ∀ *δ*∈ Ω<sup>r</sup> [where Ω<sup>r</sup> is a complex defined in Eq. (5)]. Then the holomorphic structure that constitutes these complexes induces (in the corresponding integral manifold) a conformal generalized structure of integral submanifolds where the arches *γ*, are local parts of integral curves of the fibers of the vector sheaf of lines. In other words, ∀ *γ* ∈P i ð Þ Vz exists locally an integral submanifold S, with z∈S, such that Tz*<sup>S</sup>* <sup>¼</sup> *<sup>γ</sup>*, and TwS∈<sup>P</sup> i ð Þ Vw , ∀w ∈ S. Then the integral of line can be re-written in this conformal generalized structure as

$$\int\_{\mathcal{I}} \mathbf{X} \bullet \boldsymbol{\delta} = \int\_{\mathbf{T}\_{\mathbf{x}} \mathbf{S}} \mathbf{f} \bullet \boldsymbol{\delta}, \; \forall \quad \boldsymbol{\delta} \in \Omega^{\mathbf{r}}, \mathbf{f} \in \mathcal{T}, \tag{12}$$

where T , is the tube domain (in the local structure where the integral submanifold S, exists) T ¼ <sup>n</sup> <sup>þ</sup> iV, where V, is a cone, not necessarily convex (that has applicability on the fibers of the sheaf of lines). The idea is to define the expressionf • *δ*, inside the context of the integral of line in such case that the values of f, on the arch *γ*, are values off, a hyperfunction represented this like a variation of holomorphic functions f zð Þ j*δ* , in a submanifold of Stein M*δ*, such that M*δ*⊃T .

Then, the sesquilinear coupling of the hyperfunction corresponding to f, and the function f itself, is an integral of contour, and for Proposition 2.3, a generalized functional in the cohomology H1 ð Þ <sup>Π</sup>‐ℓ, . Indeed, let be T ¼ <sup>n</sup> <sup>þ</sup> iV, the tube domain where the cone V, is not necessarily convex. This cone V ¼ ∪ *<sup>γ</sup>* <sup>∈</sup>ΣV*<sup>γ</sup>* , in the conformal generalized structure where the V*γ*, are the convex maximal sub-cones in V. Considers our manifold, complex Riemannian manifold. The idea is that a holomorphic form required in this language is a good expression to write the

We consider for the inner product of *α*, and *β*, the relation is

*Cohomology of contours isomorphic to* <sup>H</sup>• <sup>M</sup> � Sing M, <sup>Ω</sup><sup>r</sup> ð Þ*:*

*Convex domains conformed for holomorphic hyperplanes π*ið Þ D *:*

*Advances in Complex Analysis and Applications*

Hilbert transforms.

H1

**86**

where ℓ<sup>0</sup>

**Figure 3.**

**Figure 2.**

subspaces <sup>1</sup>

This description of the inner product has been used in a new development of the cohomology for twistor diagrams foreseen in Refs. [14, 18]. This new method is almost opposed to the procedure that we want to use in the unification of contour integrals on diagrams, in respect of the Feynman integral, although also proper to the Conway integrals, Cauchy integrals,<sup>6</sup> and some integral transforms as the

We want to assemble a Feynman diagram for applications of the product "cup." The interior edges of a Feynman diagram are taken again as elements of groups H<sup>0</sup>

Let denote M, for Π, and sing M ¼ ℓ. Likewise, if f, is one of these elements of

, is the union of all the subspaces defined by internal edges, always with the

f zð Þ ð Þ <sup>z</sup>‐<sup>a</sup> <sup>n</sup>þ<sup>1</sup> dz*:*

(such extra elements have to be abandoned in a cohomology, for example, <sup>H</sup>• M, *<sup>τ</sup>*‐<sup>1</sup> ð Þ <sup>J</sup>ð Þ*<sup>ν</sup>* , and the interior edges form the fields (assuming that they are

, this new procedure determines an element of the cohomology H<sup>f</sup> <sup>Π</sup>‐ℓ<sup>0</sup>

<sup>6</sup> For example, to this case for holomorphic functions, we have the generalized Cauchy formula:

ð Þ <sup>n</sup> ð Þ¼ <sup>a</sup> <sup>n</sup>! 2*π*i ∮ *γ*

, on whose elementary states f, are singular.

Then for Proposition 2.1 (b), the following mapping exists

Hf <sup>Π</sup>‐ℓ<sup>0</sup>

f

elementary states) in several cohomology groups H1

*<sup>α</sup>* • *<sup>β</sup>* <sup>¼</sup> *<sup>∂</sup>* <sup>∗</sup> ð Þ *<sup>α</sup>* <sup>∪</sup> *<sup>β</sup>* , (9)

.

, <sup>Ω</sup><sup>d</sup> ! <sup>H</sup><sup>f</sup>þ<sup>d</sup> <sup>Π</sup>‐ℓ<sup>0</sup> ð Þ , , (10)

, Ω<sup>d</sup> ,

integral of complex vector fields as an integral of line through more than enough bundles of hyperlines and hyperplanes. As for example, we have more than enough hyperlines and hyperplanes, respectively, in n, and n, visualizing these fields like holomorphic sections of complex holomorphic bundles of fibers X ! M. In Δ, exists q‐dimensional cycles such that V <sup>¼</sup> <sup>∪</sup> *<sup>δ</sup>*∈*<sup>γ</sup>*V*δ*. Let be <sup>T</sup> *<sup>δ</sup>* <sup>¼</sup> <sup>n</sup> <sup>þ</sup> iV*δ*, with covering of Stein T ¼ <sup>∪</sup> *<sup>δ</sup>*∈*<sup>γ</sup>*<sup>T</sup> *<sup>δ</sup>*. Let us consider the vector cohomology Hð Þ <sup>q</sup> ð Þ <sup>T</sup> , <sup>J</sup> , using this covering. Then for proposition 2. 1, incise b), a canonical operator exists (of values frontier for f) defined for

$$\mathbf{H}^{(\mathsf{q})}(\mathcal{T}, \mathfrak{I}) \to \mathbf{H}^{(\mathsf{q})}(\mathbb{C}^{\mathsf{n}}/\mathbb{R}^{\mathsf{n}}, \mathfrak{I}),\tag{13}$$

Now, we consider a closed subset (or relatively closed) F, of a space X, and a sheaf J, on X. In a way more than enough, we choose an open covering Y, of X, with

A relative co-chain of Cěch is a co-chain of Cěch with regard to the covering Y,

<sup>F</sup>ð Þ X, J is the group of relative co-chains. The inherent relative co-chain

dz <sup>¼</sup> <sup>0</sup>‐2*π*i‐2*π*<sup>i</sup> <sup>¼</sup> ‐4*π*i*:*

<sup>F</sup> ð Þ X, <sup>J</sup> give the groups of relative cohomology H<sup>p</sup>

. Then, it

<sup>F</sup>ð Þ X, J .

<sup>F</sup>ð Þ! X*=*F, J 0, (16)

ð Þ! X, J … , (17)

ð Þ X, S , (18)

ð Þ Y, T , (19)

subject to the condition of annulling when we restrict to the subcovering Y0

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

<sup>F</sup>ð Þ! X, <sup>J</sup> <sup>C</sup>pð Þ! X, <sup>J</sup> <sup>C</sup><sup>p</sup>

to a co-opposite operator of the ordinary co-chains and the limit on fine coverings

� <sup>1</sup> *z* � *z*<sup>2</sup>

This is a good example of traditional cohomological functional element of

In this case is not necessary to take the limit since ahead of time one has the relative theorem of Leray, which establish that if Hpð Þ¼ U, <sup>J</sup> 0, p<sup>≥</sup> 1, for each set U, in the covering Y, then this covers enough to calculate the relative cohomology. The exact long succession cohomology of the exact short succession defined in Eq. (16)

where the mappings of the cohomology on X, to the given on X*=*F, are restrictions. Other important result on the relative cohomology is the split theorem, which establishes in shallow terms that the relative cohomology depends only on the immediate neighborhoods of the embedding of F, in X. With more precision, giving

This is the form to induce isomorfisms. In our case, the covering Y, is a covering of Stein where the integral operator cohomology H•ð Þ M, <sup>J</sup> , should exist such as we wish. Why? Because the natural place, where a *<sup>∂</sup>*�cohomology exists, is in a covering of Stein and is because we want to obtain the solutions of *<sup>∂</sup>*�partial differential equations. We apply the relative cohomology to cohomologies of contours because we want

We consider the following general procedure due to Baston [8] for the exhibition of all the cohomological functional on a collection of fields given. This procedure is required for the evaluation of boxes diagram, that is to say, the obtaining of

We consider a complex manifold given for X∪Y, the closed subsets F⊂X, and

<sup>G</sup> <sup>⊂</sup> Y, and elements *<sup>α</sup>* <sup>∈</sup> Hpð Þ <sup>X</sup>‐F, S , and *<sup>β</sup>* <sup>∈</sup> Hqð Þ <sup>Y</sup>‐G, T . Then we can use the connecting mappings in the exact successions of relative cohomology

ð Þ i ¼ 1, 2, 3, 4, … of the field through a local cohomology.

<sup>F</sup> ð Þ! X, S <sup>H</sup><sup>p</sup>þ<sup>1</sup>

<sup>G</sup> ð Þ! Y, T Hqþ<sup>1</sup>

a subcovering Y0

where Cp

<sup>H</sup><sup>f</sup> <sup>Π</sup>‐ℓ<sup>0</sup>

ifH<sup>n</sup>

of the homology of C<sup>∗</sup>

, <sup>Ω</sup><sup>r</sup> ð Þ¼ .

<sup>0</sup> ! H0

<sup>F</sup>ð Þ¼ X, <sup>J</sup> <sup>H</sup><sup>n</sup>

the elementary states *φ*<sup>i</sup>

and

**89**

<sup>F</sup> X<sup>0</sup> ð Þ , J .

∮ C

, of X*=*F.

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

<sup>0</sup> ! <sup>C</sup><sup>p</sup>

g zð Þdz ¼ ∮

C

determines the exact succession of relative cohomology

<sup>F</sup>ð Þ! X, <sup>J</sup> <sup>H</sup><sup>0</sup>ð Þ! X, <sup>J</sup> <sup>C</sup><sup>0</sup>ð Þ! <sup>X</sup>*=*F, <sup>J</sup> <sup>H</sup><sup>1</sup>

an open subset X, such that X*=*X<sup>0</sup> ð Þ ∩ F ¼ ∅, a canonical isomorfism exists

generalized function as solutions of the differential equations [5, 18].

<sup>H</sup><sup>p</sup>ð Þ! X, S Hpð Þ! <sup>X</sup>‐F, S <sup>r</sup> <sup>H</sup><sup>p</sup>þ<sup>1</sup>

Hqð Þ! Y, T Hqð Þ! <sup>Y</sup>‐G, T <sup>r</sup> Hqþ<sup>1</sup>

1‐ 1 z‐z1

had the exact succession of relative co-chains groups:

Then, the integral can be expressed on spaces M*<sup>δ</sup>* and Δz, which are affined to lines and hyperplanes <sup>n</sup> and <sup>n</sup> and that such are orbital integrals of the complex manifolds M <sup>¼</sup> <sup>G</sup>*=*L and <sup>Δ</sup> <sup>¼</sup> <sup>Γ</sup>*=*Σ, belonging to a *<sup>∂</sup>*‐cohomology in holomorphic language.

In particular, if f zð Þ <sup>j</sup>*δ*, dj*<sup>δ</sup>* <sup>∈</sup> <sup>Ω</sup><sup>q</sup> <sup>T</sup> has regular values <sup>∀</sup>z<sup>∈</sup> n, then

$$\rho(\mathbf{x}) = \int\_{\mathcal{I}} \rho(\mathbf{x}|\delta, \mathbf{d}|\delta), \quad \forall \quad \mathbf{x} \in \mathbb{R}^n. \tag{14}$$

Then, in the integral submanifold M*δ*, said integrals take the form

$$\int\_{\mathcal{Y}} \mathbf{X} \bullet \mathbf{G} = \int\_{\mathcal{Y}} \rho(\mathbf{z}|\delta, \mathbf{d}|\delta) \mathbf{f}(\mathbf{z}) = \int\_{\mathcal{Y}} \mathbf{f}(\mathbf{z}|\delta). \tag{15}$$

However, these integrals are integral of contour belonging to a cohomology H1 ð Þ <sup>Π</sup>‐ℓ, of cohomological functional. Then, the integral <sup>Ð</sup> *<sup>γ</sup>* f zð Þ j*δ* is a functional inside the integral cohomology Hð Þ <sup>n</sup>‐<sup>1</sup> <sup>n</sup>*=*<sup>n</sup> ð Þ , <sup>J</sup> (**Figure 4**).

The previous Propositions 2.3 and 2.4 establish that the structure of complexes for the integral operator cohomology does suitable to induce isomorfisms in other object classes of the manifold M, doing arise the question to some procedure that exists inside the relative cohomology on J: can we induce isomorfisms of integral cohomologies?

#### **Figure 4.**

*(A). One state or source of a field. Its contour is well defined by only one Cauchy integral. (B). Two states or sources of a field. This represents the surface of the real part of the function* <sup>g</sup>ð Þ¼ *<sup>z</sup> <sup>z</sup>*<sup>2</sup> *<sup>z</sup>*2þ2*z*þ<sup>2</sup>*. The moduli of these points are less than 2 and thus lie inside one contour. Likewise, the contour integral can be split into two smaller integrals using the Cauchy-Goursat theorem having finally the contour integral [19].*

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

Now, we consider a closed subset (or relatively closed) F, of a space X, and a sheaf J, on X. In a way more than enough, we choose an open covering Y, of X, with a subcovering Y0 , of X*=*F.

A relative co-chain of Cěch is a co-chain of Cěch with regard to the covering Y, subject to the condition of annulling when we restrict to the subcovering Y0 . Then, it had the exact succession of relative co-chains groups:

$$\mathbf{0} \to \mathbf{C}\_{\mathrm{F}}^{\mathrm{p}}(\mathbf{X}, \mathfrak{J}) \to \mathbf{C}^{\mathrm{p}}(\mathbf{X}, \mathfrak{J}) \to \mathbf{C}\_{\mathrm{F}}^{\mathrm{p}}(\mathbf{X}/\mathsf{F}, \mathfrak{J}) \to \mathbf{0},\tag{16}$$

where Cp <sup>F</sup>ð Þ X, J is the group of relative co-chains. The inherent relative co-chain to a co-opposite operator of the ordinary co-chains and the limit on fine coverings of the homology of C<sup>∗</sup> <sup>F</sup> ð Þ X, <sup>J</sup> give the groups of relative cohomology H<sup>p</sup> <sup>F</sup>ð Þ X, J .

$$\oint\_{\mathbf{C}} \mathbf{g}(\mathbf{z}) \mathbf{d}\mathbf{z} = \oint\_{\mathbf{C}} \left( \mathbf{1} \cdot \frac{\mathbf{1}}{\mathbf{z} \cdot \mathbf{z}\_1} - \frac{\mathbf{1}}{z - z\_2} \right) \mathbf{d}\mathbf{z} = \mathbf{0} \cdot 2\pi \mathbf{i} \cdot 2\pi \mathbf{i} = \mathbf{-4} \pi \mathbf{i} \mathbf{i}.$$

This is a good example of traditional cohomological functional element of <sup>H</sup><sup>f</sup> <sup>Π</sup>‐ℓ<sup>0</sup> , <sup>Ω</sup><sup>r</sup> ð Þ¼ .

In this case is not necessary to take the limit since ahead of time one has the relative theorem of Leray, which establish that if Hpð Þ¼ U, <sup>J</sup> 0, p<sup>≥</sup> 1, for each set U, in the covering Y, then this covers enough to calculate the relative cohomology. The exact long succession cohomology of the exact short succession defined in Eq. (16) determines the exact succession of relative cohomology

$$\mathbf{0} \to \mathbf{H}\_{\mathbf{F}}^{0}(\mathbf{X}, \mathfrak{J}) \to \mathbf{H}^{0}(\mathbf{X}, \mathfrak{J}) \to \mathbf{C}^{0}(\mathbf{X}/\mathbf{F}, \mathfrak{J}) \to \mathbf{H}^{1}(\mathbf{X}, \mathfrak{J}) \to \dots,\tag{17}$$

where the mappings of the cohomology on X, to the given on X*=*F, are restrictions.

Other important result on the relative cohomology is the split theorem, which establishes in shallow terms that the relative cohomology depends only on the immediate neighborhoods of the embedding of F, in X. With more precision, giving an open subset X, such that X*=*X<sup>0</sup> ð Þ ∩ F ¼ ∅, a canonical isomorfism exists ifH<sup>n</sup> <sup>F</sup>ð Þ¼ X, <sup>J</sup> <sup>H</sup><sup>n</sup> <sup>F</sup> X<sup>0</sup> ð Þ , J .

This is the form to induce isomorfisms. In our case, the covering Y, is a covering of Stein where the integral operator cohomology H•ð Þ M, <sup>J</sup> , should exist such as we wish. Why? Because the natural place, where a *<sup>∂</sup>*�cohomology exists, is in a covering of Stein and is because we want to obtain the solutions of *<sup>∂</sup>*�partial differential equations.

We apply the relative cohomology to cohomologies of contours because we want generalized function as solutions of the differential equations [5, 18].

We consider the following general procedure due to Baston [8] for the exhibition of all the cohomological functional on a collection of fields given. This procedure is required for the evaluation of boxes diagram, that is to say, the obtaining of the elementary states *φ*<sup>i</sup> ð Þ i ¼ 1, 2, 3, 4, … of the field through a local cohomology.

We consider a complex manifold given for X∪Y, the closed subsets F⊂X, and <sup>G</sup> <sup>⊂</sup> Y, and elements *<sup>α</sup>* <sup>∈</sup> Hpð Þ <sup>X</sup>‐F, S , and *<sup>β</sup>* <sup>∈</sup> Hqð Þ <sup>Y</sup>‐G, T . Then we can use the connecting mappings in the exact successions of relative cohomology

$$\mathbf{H}^{\rm p}(\mathbf{X},\mathbf{S}) \to \mathbf{H}^{\rm p}(\mathbf{X}\cdot\mathbf{F},\mathbf{S}) \xrightarrow{\rm r} \mathbf{H}\_{\rm F}^{\rm p+1}(\mathbf{X},\mathbf{S}) \to \mathbf{H}^{\rm p+1}(\mathbf{X},\mathbf{S}),\tag{18}$$

and

$$\mathbf{H}^{\mathrm{q}}(\mathbf{Y},\mathbf{T}) \to \mathbf{H}^{\mathrm{q}}(\mathbf{Y}\text{-G},\mathbf{T}) \xrightarrow{\mathbf{r}} \mathbf{H}\_{\mathrm{G}}^{\mathrm{q}+1}(\mathbf{Y},\mathbf{T}) \to \mathbf{H}^{\mathrm{q}+1}(\mathbf{Y},\mathbf{T}),\tag{19}$$

integral of complex vector fields as an integral of line through more than enough bundles of hyperlines and hyperplanes. As for example, we have more than enough hyperlines and hyperplanes, respectively, in n, and n, visualizing these fields like holomorphic sections of complex holomorphic bundles of fibers X ! M. In Δ, exists q‐dimensional cycles such that V <sup>¼</sup> <sup>∪</sup> *<sup>δ</sup>*∈*<sup>γ</sup>*V*δ*. Let be <sup>T</sup> *<sup>δ</sup>* <sup>¼</sup> <sup>n</sup> <sup>þ</sup> iV*δ*, with covering of Stein T ¼ <sup>∪</sup> *<sup>δ</sup>*∈*<sup>γ</sup>*<sup>T</sup> *<sup>δ</sup>*. Let us consider the vector cohomology Hð Þ <sup>q</sup> ð Þ <sup>T</sup> , <sup>J</sup> , using this covering. Then for proposition 2. 1, incise b), a canonical operator exists

Then, the integral can be expressed on spaces M*<sup>δ</sup>* and Δz, which are affined to lines and hyperplanes <sup>n</sup> and <sup>n</sup> and that such are orbital integrals of the complex manifolds M <sup>¼</sup> <sup>G</sup>*=*L and <sup>Δ</sup> <sup>¼</sup> <sup>Γ</sup>*=*Σ, belonging to a *<sup>∂</sup>*‐cohomology in holomorphic language.

*φ*ð Þ zj*δ*, dj*δ* f zð Þ¼

However, these integrals are integral of contour belonging to a cohomology

The previous Propositions 2.3 and 2.4 establish that the structure of complexes for the integral operator cohomology does suitable to induce isomorfisms in other object classes of the manifold M, doing arise the question to some procedure that exists inside the relative cohomology on J: can we induce isomorfisms of integral

*(A). One state or source of a field. Its contour is well defined by only one Cauchy integral. (B). Two states or*

*points are less than 2 and thus lie inside one contour. Likewise, the contour integral can be split into two smaller*

*sources of a field. This represents the surface of the real part of the function* <sup>g</sup>ð Þ¼ *<sup>z</sup> <sup>z</sup>*<sup>2</sup>

*integrals using the Cauchy-Goursat theorem having finally the contour integral [19].*

<sup>H</sup>ð Þ <sup>q</sup> ð Þ! <sup>T</sup> , <sup>J</sup> <sup>H</sup>ð Þ <sup>q</sup> <sup>n</sup>*=*<sup>n</sup> ð Þ , <sup>J</sup> , (13)

*<sup>φ</sup>*ð Þ <sup>x</sup>j*δ*, dj*<sup>δ</sup>* , <sup>∀</sup> <sup>x</sup><sup>∈</sup> <sup>n</sup>*:* (14)

f zð Þ j*δ :* (15)

*<sup>γ</sup>* f zð Þ j*δ* is a functional

*<sup>z</sup>*2þ2*z*þ<sup>2</sup>*. The moduli of these*

<sup>T</sup> has regular values <sup>∀</sup>z<sup>∈</sup> n, then

ð

*γ*

(of values frontier for f) defined for

*Advances in Complex Analysis and Applications*

In particular, if f zð Þ <sup>j</sup>*δ*, dj*<sup>δ</sup>* <sup>∈</sup> <sup>Ω</sup><sup>q</sup>

H1

cohomologies?

**Figure 4.**

**88**

*φ*ð Þ¼ x

X• G ¼

ð

*γ*

ð

*γ*

Then, in the integral submanifold M*δ*, said integrals take the form

ð

*γ*

ð Þ <sup>Π</sup>‐ℓ, of cohomological functional. Then, the integral <sup>Ð</sup>

inside the integral cohomology Hð Þ <sup>n</sup>‐<sup>1</sup> <sup>n</sup>*=*<sup>n</sup> ð Þ , <sup>J</sup> (**Figure 4**).

to obtain elements r*α*, and r*β*. Then, the cup product on relative cohomology is defined as:

$$\begin{aligned} \mathsf{T} \mathsf{U}: \mathsf{H}^{\mathsf{p}+\mathsf{q}+1}(\mathsf{X} \cup \mathsf{Y}, \mathsf{S} \otimes \mathsf{T}) & \xrightarrow{\mathsf{r}} \mathsf{H}^{\mathsf{p}+\mathsf{q}+1}(\mathsf{X} \cup \mathsf{Y} \cdot \mathsf{F} \cap \mathsf{G}, \mathsf{S} \otimes \mathsf{T}) \\ & \to \mathsf{H}^{\mathsf{p}+\mathsf{q}+2}\_{\mathsf{F} \cap \mathsf{G}}(\mathsf{X} \cup \mathsf{Y}, \mathsf{S} \otimes \mathsf{T}) \to \mathsf{H}^{\mathsf{p}+\mathsf{q}+2}(\mathsf{X} \cup \mathsf{Y}, \mathsf{S} \otimes \mathsf{T}), \end{aligned} \tag{20}$$

and this demonstrates that

$$a \bullet \beta = \mathbf{r}^1 (\mathbf{r}a \cup \mathbf{r}\beta),\tag{21}$$

Let F ¼ F1 � … � Ff. We denote for Li, a projective line included in Fi, and let L ¼ L1 � … � Lf. For f vector fields, we have an element in the cohomological

where **r** ¼ r1 � … � rf. Each linear continuous functional on these fields is thus

Now well, considering this cohomology of vector fields, is necessary to decide how the interior of a diagram choose some of these functionals. We remember the interior of a diagram as the holomorphic nucleus h<sup>∈</sup> <sup>H</sup>3f,qð Þ <sup>Π</sup>‐ℓ, Oð Þ**<sup>r</sup>** . For example,

<sup>Y</sup>*δ*Z*<sup>δ</sup>* ð Þ<sup>2</sup> <sup>∈</sup> <sup>H</sup>6,qð Þ <sup>Π</sup>‐ℓ0, Oð Þ ‐2‐2‐2‐<sup>2</sup> . Usually q <sup>¼</sup> 0. In these cases h, can be determined for integration without the interior vertices of the twistor diagram, although it is not always easy. If q 6¼ 0, the determination of h in none time is

where such ið Þ *α* ∪ h , is a chosen functional for the interior of the diagram (that is to say h) as required. However, as this was done through *α*, the results are hard to view *α* as a contour. For it, we first note that the embedding of the constant sheaf

We consider the complex cohomology, and also we consider an element

<sup>ð</sup><sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> F; Oð Þ ‐**<sup>r</sup>** Þ ! <sup>H</sup>3f,f

and second, the cohomology groups *<sup>α</sup>*<sup>∈</sup> H5fþ<sup>q</sup>ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> and

normal orientable bundle). Using the Thom isomorphism, we have:

 ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> F; are isomorphic. Now, it is necessary to insist in that *<sup>α</sup>* is in the image of the mapping Eq. (27), which will produce a viewing as contour. Being *α* a contour, we call to ið Þ *α* ∪ h , the functional "associated with" the kernel h, and we remark strongly that this not exists if F⊂ℓ, then H5fþ<sup>q</sup>ð<sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup>Þ ¼ 0, which is hoped. We can refer to this problem as impossible, since necessarily ℓ 6¼ F, for the chosen fields in this cohomology, which are the most general possible. The idea is to wobtain an image of the vector field as an element of a cohomology on homogeneous bundles of lines in each component of the field. We note that our defined fields are generally perfect. In fact, if the vector fields are elemental states, then Fi ¼ Li, and F, is equal to a closed submanifold Λ (of real codimension 4f, with

which is deduced that the viewed contours are given in Hfþ<sup>q</sup>ð Þ Λ � ℓ . If the vector fields are not elemental states along ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> , then ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> , is homotopic to ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup><sup>Λ</sup> , which establishes its generality in homology.

Hfþ<sup>q</sup>ð Þ! <sup>Λ</sup> � <sup>ℓ</sup> ffi H5fþ<sup>q</sup>ð<sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup>ΛÞ ¼ 0, (28)

<sup>ð</sup><sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> F; Þ ! Hf‐<sup>q</sup>

ð Þ U1, Oð Þ �r1 ⊗⋯⊗ H1 Uf ð Þffi , Oð Þ �rf <sup>H</sup>2f

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

an element of the compact relative cohomology group H2f

ð Þ U1, Oð Þ �r1 ⊗⋯⊗ <sup>H</sup><sup>1</sup> Uf ð Þ , Oð Þ �rf . For relative cohomology and projective twistor diagram results [18], the inner product for the line integrals for all these fields is not lost. Then for the Künneth formula to relative cohomology, we have:

<sup>F</sup> ð Þ Π, Oð Þ**r** , (25)

<sup>F</sup> ð Þ <sup>Π</sup>, <sup>Π</sup>‐F, Oð Þ**<sup>r</sup>** . We must

<sup>F</sup> ð Þ <sup>Π</sup>, <sup>Π</sup>‐F, Oð Þ**<sup>r</sup>** , are not in general dual.

<sup>W</sup>*α*Z*<sup>α</sup>* ð Þ<sup>2</sup> <sup>∈</sup> <sup>H</sup>6,qð Þ <sup>Π</sup>‐ℓ, Oð Þ ‐2‐<sup>2</sup> . While in the box

ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> F; Oð Þ ‐**<sup>r</sup>** *:* This is an

ð Þ <sup>Π</sup>, <sup>Π</sup> � F; Oð Þ ‐**<sup>r</sup>** , (26)

ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup> � <sup>ℓ</sup> <sup>∪</sup> F; Oð Þ ‐**<sup>r</sup>** , (27)

group H<sup>1</sup>

H1

<sup>h</sup> <sup>¼</sup> DW∧DZ∧DX∧DY W*α*Z*α*W*β*Y*β*Z*γ*X*<sup>γ</sup>*

*α*∈ H0,f‐<sup>q</sup>

Hf‐<sup>q</sup>

**91**

clear. How to do about it?

induced mapping for the inclusion

i : H3f,f

, in Oð Þ ‐**<sup>r</sup>** , induces a mapping:

H<sup>f</sup>‐<sup>q</sup>

establish that Eq. (25) and the group H2f

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

in the scalar product (spin zero) h <sup>¼</sup> DW∧DZ

ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> *:* Then *<sup>α</sup>* <sup>∪</sup> <sup>h</sup><sup>∈</sup> H3f,f

Due to that in the diagram boxes, the interactive vector fields *φ*, are given as elements of groups H1 , defined on different spaces, we need the vector product in relative cohomology:

$$\times: \mathbf{H}\_{\rm F}^{p+1}(\mathbf{X}, \mathbf{S}) \otimes \mathbf{H}\_{\rm G}^{q+1}(\mathbf{Y}, \mathbf{T}) \to \mathbf{H}\_{\rm F \times G}^{p+q+2}(\mathbf{X} \times \mathbf{Y}, \mathbf{S} \otimes \mathbf{T}),\tag{22}$$

Likewise, for diagram box of four states, we have the cohomology of the left side of Eq. (22) that can be illustrated (**Figure 5**).

Strictly speaking S ⊗ T, could be *π* ∗ XS ⊗ *π* ∗ YT. As before r*α* � r*β*, is in the image of the connecting mapping r, in:

$$\mathbf{H}^{\mathsf{p}+\mathsf{q}+1}(\mathbf{X}\cup\mathbf{Y}\cdot\mathbf{F}\cap\mathbf{G},\mathbf{S}\otimes\mathbf{T}) \stackrel{\mathsf{r}}{\to} \mathbf{H}^{\mathsf{p}+\mathsf{q}+2}\_{\mathsf{F}\times\mathsf{G}}(\mathbf{X}\times\mathbf{Y},\mathbf{S}\otimes\mathbf{T}) \to \mathbf{H}^{\mathsf{p}+\mathsf{q}+2}(\mathbf{X}\times\mathbf{Y},\mathbf{S}\otimes\mathbf{T}),\tag{23}$$

with

$$a \bullet \beta = \mathbf{r}^1 (\mathbf{r}a \times \mathbf{r}\beta) (\in \nu^1 \mathcal{O}(\mathbf{p}, \mathbf{q}, \mathbf{r})),\tag{24}$$

The following technical question arises: how to relate contour cohomology as Hfþ<sup>d</sup> <sup>Π</sup>‐ℓ<sup>0</sup> ð Þ , , with an integral cohomology of vector fields?

Part of the replay to this question is found when are considered the complex components Fi ¼ Pi � Ui, with i ¼ 1, 2, 3, 4, … , f; being Pi, P, P ∗ , and Ui, open subsets of i, belonging to the correct cohomology to the Penrose transform on H1 ð Þ U, Oð Þ �**r** .

The idea is to obtain an image of the vector field as element of a cohomology on homogeneous bundles of lines in each component of the field (that is to say, determine a cohomology for each line integral of each field component). Beforehand this is foreseen that will happen with the Penrose transform, which is an integral transform on the homogeneous bundles of lines.

**Figure 5.** *Feynman boxes diagrams [1].*

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

Let F ¼ F1 � … � Ff. We denote for Li, a projective line included in Fi, and let L ¼ L1 � … � Lf. For f vector fields, we have an element in the cohomological group H<sup>1</sup> ð Þ U1, Oð Þ �r1 ⊗⋯⊗ <sup>H</sup><sup>1</sup> Uf ð Þ , Oð Þ �rf . For relative cohomology and projective twistor diagram results [18], the inner product for the line integrals for all these fields is not lost. Then for the Künneth formula to relative cohomology, we have:

$$\mathbf{H}^{1}(\mathbf{U}\_{1},\mathbf{O}(-\mathbf{r}\_{1})) \otimes \dots \otimes \mathbf{H}^{1}(\mathbf{U}\_{\mathbf{f}},\mathbf{O}(-\mathbf{r}\_{\mathbf{f}})) \cong \mathbf{H}\_{\mathbf{F}}^{2t}(\Pi,\mathbf{O}(\mathbf{r})),\tag{25}$$

where **r** ¼ r1 � … � rf. Each linear continuous functional on these fields is thus an element of the compact relative cohomology group H2f <sup>F</sup> ð Þ <sup>Π</sup>, <sup>Π</sup>‐F, Oð Þ**<sup>r</sup>** . We must establish that Eq. (25) and the group H2f <sup>F</sup> ð Þ <sup>Π</sup>, <sup>Π</sup>‐F, Oð Þ**<sup>r</sup>** , are not in general dual.

Now well, considering this cohomology of vector fields, is necessary to decide how the interior of a diagram choose some of these functionals. We remember the interior of a diagram as the holomorphic nucleus h<sup>∈</sup> <sup>H</sup>3f,qð Þ <sup>Π</sup>‐ℓ, Oð Þ**<sup>r</sup>** . For example, in the scalar product (spin zero) h <sup>¼</sup> DW∧DZ <sup>W</sup>*α*Z*<sup>α</sup>* ð Þ<sup>2</sup> <sup>∈</sup> <sup>H</sup>6,qð Þ <sup>Π</sup>‐ℓ, Oð Þ ‐2‐<sup>2</sup> . While in the box <sup>h</sup> <sup>¼</sup> DW∧DZ∧DX∧DY W*α*Z*α*W*β*Y*β*Z*γ*X*<sup>γ</sup>* <sup>Y</sup>*δ*Z*<sup>δ</sup>* ð Þ<sup>2</sup> <sup>∈</sup> <sup>H</sup>6,qð Þ <sup>Π</sup>‐ℓ0, Oð Þ ‐2‐2‐2‐<sup>2</sup> . Usually q <sup>¼</sup> 0. In these cases h, can be determined for integration without the interior vertices of the twistor dia-

gram, although it is not always easy. If q 6¼ 0, the determination of h in none time is clear. How to do about it?

We consider the complex cohomology, and also we consider an element *α*∈ H0,f‐<sup>q</sup> ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> *:* Then *<sup>α</sup>* <sup>∪</sup> <sup>h</sup><sup>∈</sup> H3f,f ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> F; Oð Þ ‐**<sup>r</sup>** *:* This is an induced mapping for the inclusion

$$\mathbf{i} : \mathbf{H}\_{\mathbb{C}}^{3\mathbf{f},\mathbf{f}}(\Pi - \boldsymbol{\ell}, \Pi \cdot \boldsymbol{\ell} \cup \mathbf{F}; \mathbf{O}(\cdot \mathbf{r})) \to \mathbf{H}\_{\mathbb{C}}^{3\mathbf{f},\mathbf{f}}(\Pi, \Pi - \mathbf{F}; \mathbf{O}(\cdot \mathbf{r})),\tag{26}$$

where such ið Þ *α* ∪ h , is a chosen functional for the interior of the diagram (that is to say h) as required. However, as this was done through *α*, the results are hard to view *α* as a contour. For it, we first note that the embedding of the constant sheaf , in Oð Þ ‐**<sup>r</sup>** , induces a mapping:

$$\mathbf{H}\_{\mathbb{C}}^{\mathbf{f}\cdot\mathbf{q}}(\Pi-\ell,\Pi\text{-}\ell\cup\mathbf{F};\mathbb{R}) \to \mathbf{H}\_{\mathbb{C}}^{\mathbf{f}\cdot\mathbf{q}}(\Pi-\ell,\Pi-\ell\cup\mathbf{F};\mathbb{O}(\mathbf{-r})),\tag{27}$$

and second, the cohomology groups *<sup>α</sup>*<sup>∈</sup> H5fþ<sup>q</sup>ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> and

Hf‐<sup>q</sup> ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> F; are isomorphic. Now, it is necessary to insist in that *<sup>α</sup>* is in the image of the mapping Eq. (27), which will produce a viewing as contour. Being *α* a contour, we call to ið Þ *α* ∪ h , the functional "associated with" the kernel h, and we remark strongly that this not exists if F⊂ℓ, then H5fþ<sup>q</sup>ð<sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup>Þ ¼ 0, which is hoped. We can refer to this problem as impossible, since necessarily ℓ 6¼ F, for the chosen fields in this cohomology, which are the most general possible. The idea is to wobtain an image of the vector field as an element of a cohomology on homogeneous bundles of lines in each component of the field. We note that our defined fields are generally perfect. In fact, if the vector fields are elemental states, then Fi ¼ Li, and F, is equal to a closed submanifold Λ (of real codimension 4f, with normal orientable bundle). Using the Thom isomorphism, we have:

$$\mathbf{H}\_{\mathbf{f}+\mathbf{q}}(\Lambda-\mathcal{E}) \xrightarrow{\simeq} \mathbf{H}\_{\mathbf{5f}+\mathbf{q}}(\Pi-\mathcal{E}, \Pi \cdot \mathcal{E} \cup \Lambda) = \mathbf{0},\tag{28}$$

which is deduced that the viewed contours are given in Hfþ<sup>q</sup>ð Þ Λ � ℓ . If the vector fields are not elemental states along ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> , then ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> , is homotopic to ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup><sup>Λ</sup> , which establishes its generality in homology.

to obtain elements r*α*, and r*β*. Then, the cup product on relative cohomology is

ð Þ <sup>X</sup>∪Y‐<sup>F</sup> <sup>∩</sup> G, S <sup>⊗</sup> <sup>T</sup>

, defined on different spaces, we need the vector product in

<sup>F</sup>�<sup>G</sup> <sup>ð</sup><sup>X</sup> � Y, S <sup>⊗</sup> <sup>T</sup>Þ ! Hpþqþ<sup>2</sup>

ð Þ <sup>X</sup><sup>∪</sup> Y, S <sup>⊗</sup> <sup>T</sup> , (20)

ð Þ r*α* ∪ r*β* , (21)

<sup>F</sup>�<sup>G</sup> ð Þ <sup>X</sup> � Y, S <sup>⊗</sup> <sup>T</sup> , (22)

O p, q, ð Þ**<sup>r</sup>** , (24)

ð Þ X � Y, S ⊗ T ,

(23)

ð Þ! <sup>X</sup>∪Y, S <sup>⊗</sup> <sup>T</sup> <sup>r</sup> <sup>H</sup>pþqþ<sup>1</sup>

<sup>F</sup> <sup>∩</sup> <sup>G</sup> ð Þ! <sup>X</sup>∪Y, S <sup>⊗</sup> <sup>T</sup> <sup>H</sup>pþqþ<sup>2</sup>

*α* • *β* ¼ r

‐1

Due to that in the diagram boxes, the interactive vector fields *φ*, are given as

<sup>G</sup> ð Þ! Y, T <sup>H</sup><sup>p</sup>þqþ<sup>2</sup>

Likewise, for diagram box of four states, we have the cohomology of the left side

Strictly speaking S ⊗ T, could be *π* ∗ XS ⊗ *π* ∗ YT. As before r*α* � r*β*, is in the

ð Þ <sup>r</sup>*<sup>α</sup>* � <sup>r</sup>*<sup>β</sup>* <sup>∈</sup>*ν*‐<sup>1</sup>

The following technical question arises: how to relate contour cohomology as

Part of the replay to this question is found when are considered the complex components Fi ¼ Pi � Ui, with i ¼ 1, 2, 3, 4, … , f; being Pi, P, P ∗ , and Ui, open subsets of i, belonging to the correct cohomology to the Penrose transform on

The idea is to obtain an image of the vector field as element of a cohomology on

homogeneous bundles of lines in each component of the field (that is to say, determine a cohomology for each line integral of each field component). Beforehand this is foreseen that will happen with the Penrose transform, which is an

defined as:

∪ : Hpþqþ<sup>1</sup>

and this demonstrates that

� : Hpþ<sup>1</sup>

image of the connecting mapping r, in:

of Eq. (22) that can be illustrated (**Figure 5**).

<sup>ð</sup>X∪Y‐<sup>F</sup> <sup>∩</sup> G, S <sup>⊗</sup> <sup>T</sup>Þ !<sup>r</sup> Hpþqþ<sup>2</sup>

*α* • *β* ¼ r

Hfþ<sup>d</sup> <sup>Π</sup>‐ℓ<sup>0</sup> ð Þ , , with an integral cohomology of vector fields?

integral transform on the homogeneous bundles of lines.

‐1

<sup>F</sup> ð Þ X, S <sup>⊗</sup> Hqþ<sup>1</sup>

elements of groups H1

relative cohomology:

Hpþqþ<sup>1</sup>

with

H1

**Figure 5.**

**90**

*Feynman boxes diagrams [1].*

ð Þ U, Oð Þ �**r** .

! <sup>H</sup>pþqþ<sup>2</sup>

*Advances in Complex Analysis and Applications*

or also the Einstein's operator

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

∇ð<sup>A</sup>

ðA0∇<sup>B</sup><sup>Þ</sup> B0

which, in this concrete case, are the Penrose transform.

a mapping of unitary modules.

<sup>Þ</sup> <sup>þ</sup> <sup>Φ</sup>ð ÞÞ AB A0

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

or the conformally invariant modification of the square of the wave operator <sup>O</sup>½ �! ‐<sup>1</sup> <sup>O</sup>½ � ‐<sup>4</sup> , that is to say, the wave operator that involves in its term Ricci tensors:

□<sup>2</sup> : *<sup>ϕ</sup>* ! <sup>∇</sup><sup>b</sup> <sup>∇</sup><sup>b</sup>∇<sup>a</sup> � 2Rab <sup>þ</sup> ð Þ <sup>2</sup>*=*<sup>3</sup> Rgab <sup>∇</sup><sup>a</sup>

Then, the integration of the partial differential equations corresponding to these linear invariant differential operators is realized due to integral transforms of the Penrose type since the irreducible unitary representation scheme to these operators is unitary representations of components of the group SL 4, ð Þ , such as SO 2n ð Þ. In fact, in the flat case, the invariant differential operator classifications were described to determine a problem of representation theory of Lie groups applied to the Lie group SL 4, ð Þ and its compact subgroups. Then, own vision to these operators through SL 4, ð Þ will be as equivariant operators between homogeneous vector bundles on M, considering to SL 4, ð Þ as homogeneous space or class space. The integrals in this case are realizations of these representations and are orbital integrals of the integral transform of the resolutions to these differential equations,

Then, the resolution problem of the partial differential equations is reduce to the use of representation theory, but for this case, no always can construct the curved analogues of conformally invariant differential operators of the flat space. This demonstrates that cannot be generated a curved analogous under an integral transform on homogeneous bundles of lines that are direct images of the operators Dn, of G, of G*=*L. However, yes is possible to obtain a complete list under this procedure as

Also, the scheme of the ‐modules in the quaternion analysis serves to compute and determine the properties of manifold through the scheme of fibers that can be in closed complex submanifolds. In fact, this is an alternative for the determination

Now well, cohomologically: How similar are these two methodologies for the study in field theory? Can the direct product of Lie groups SU 2ð Þ*T*, subjacent in the structure of a complex Riemannian manifold that models the space-time to its vector field study and its integration through the isometries of the space Lð Þ H be carried? Which are the integration limitations for the integral transforms on

The first question is related to the double fibration that can be realized on some complex projective spaces and their quaternion equivalent. Since it always exists this bijection due to this double fibration with some corresponding homotopy group that is frequently given for spheres, some real and complex projective spaces that are necessarily identified with some *<sup>n</sup>*‐dimensional sphere exist. Such is the case, for

can be determined isomorphic cohomological spaces via some integral transform of the mentioned for the double fibrations. Some of these integrals result be of Feynman type due to the complex projective bundles are spin bundles in some sphere that determines some state space in quantum mechanics. For example, for the complex case, is had in an infinite succession of non-trivial bundles, the infinite set

These with proper connections represent Dirac magnetic monopoles of charge k.

<sup>k</sup> ! , *<sup>ℓ</sup>*‐<sup>1</sup> with k<sup>∈</sup> , and k 6¼ *<sup>ℓ</sup>*, which represents the infinite

, <sup>1</sup> ffi S4, or <sup>7</sup> ffi Spin 2, 6 ð Þ. Then

of vector fields through line bundles, which defined these as spin bundles.

homogenous bundles in global descriptions of the vector fields?

set of corresponding monopoles bundles to the case *ℓ* ¼ 2.

example, of the projective spaces <sup>1</sup> ffi <sup>S</sup><sup>1</sup>

of bundles S<sup>1</sup> ! <sup>2</sup>*ℓ*�<sup>1</sup>

**93**

<sup>B</sup><sup>0</sup> ð Þ : <sup>O</sup>½ �! �<sup>1</sup> <sup>O</sup>ð ÞÞ AB

A0

, (31)

<sup>B</sup><sup>0</sup> ð Þ½ � �<sup>1</sup> , (30)

**Figure 6.**

*(a). Field state in a cohomology* H3,0 <sup>ℝ</sup><sup>3</sup> ð Þ ,<sup>L</sup> , *to the line bundle* <sup>L</sup>*:. (b).* P, *is a principal* SO 2ð Þ�*bundle. The fiber of the fiber bundle is the points* Ex <sup>¼</sup> <sup>P</sup>�SO 2ð Þℝ<sup>2</sup>*:*

Likewise we have demonstrated that ifð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> , is homotopic to ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup><sup>Λ</sup> , then the functionals on H1 ð Þ U1, Oð Þ �r1 ⊗⋯⊗ <sup>H</sup><sup>1</sup> Uf ð Þ , Oð Þ �rf , associated with the kernel h<sup>∈</sup> H3f,qð Þ <sup>Π</sup>‐ℓ, Oð Þ ‐**<sup>r</sup>** , are given for elements of the homology group Hfþ<sup>q</sup>ð Þ Λ � ℓ . Now, which of these contours are cohomological? A classes of contours are the classic or traditional contours. However, realizing extensions of these contour classes through twistor geometry, we can consider cohomological contours to all image elements of the generator of Hd <sup>Π</sup>‐ℓ<sup>0</sup> ð Þ <sup>0</sup>, , under two mappings of Mayer-Vietoris. Likewise, the box nonprojective diagram also engages three cohomological contours. Can this particular theory of contours to the spin context be understood?

The response is yes, for example, of the foreseen construction given in **Figure 6b**).

If f ∈C<sup>1</sup> ð Þ Ω ∩ C Ω <sup>∩</sup> kerD�<sup>a</sup>ð Þ <sup>Ω</sup> , with f <sup>¼</sup> <sup>K</sup>�af <sup>∈</sup> H0ð Þ <sup>Π</sup> � <sup>℘</sup>, , being <sup>℘</sup> <sup>¼</sup> f g �<sup>a</sup> , then h <sup>¼</sup> <sup>1</sup> ð Þ <sup>z</sup>�<sup>a</sup> <sup>∈</sup> H1 ð Þ Π � ℘, Oð Þ �2 . Then an integral formula in hypercomplex analysis of a vector field is an element of the integral cohomology H1 ð Þ Π � ℘, . We can realize more work in this sense until we can arrive to the Penrose transform on hypercomplex numbers.

#### **3. The main conjecture and some notes of integral cohomology in low dimension of a complex Riemannian manifold**

Using definitions and results exposed with before can be enunciated and demonstrated the following conjectures:

**Conjecture 3.1**. The cohomology of closed submanifolds of co-dimensions <sup>k</sup>‐1, n‐k, and n kð Þ ‐<sup>1</sup> , can be represented and evaluated by a function cohomology. The cohomology of contours is represented and evaluated by a complex functional cohomology. The cohomology of line bundles is represented and evaluated by a vector field cohomologies under the *<sup>∂</sup>*‐cohomology corresponding.

There are indicium of that the differential operator class that accepts a scheme of integral cohomology (integral cohomology) like due for the Penrose transform, twistor transform, and so on is the class conformally invariant differential operators, of fact the Penrose transform generates these conformally invariant operators. Some examples of these differential operators are for the massless field equations (for flat versions and some curved versions [20]) and the conformally invariant wave operator due to the mapping:

$$
\Box + -\mathsf{R}/\mathsf{G}: \mathsf{O}[-\mathsf{1}] \to \mathsf{O}[-\mathsf{3}],\tag{29}
$$

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

or also the Einstein's operator

$$\left(\nabla^{\mathbf{(A}}{}\_{(\mathbf{A}'}\nabla^{\mathbf{B})}{}\_{\mathbf{B}'}\right) + \boldsymbol{\Phi}^{(\mathbf{AB})}\_{(\mathbf{A}'\mathbf{B}')} : \mathbf{O}[-\mathbf{1}] \to \mathbf{O}^{(\mathbf{AB})}\_{(\mathbf{A}'\mathbf{B}')}[-\mathbf{1}],\tag{30}$$

or the conformally invariant modification of the square of the wave operator <sup>O</sup>½ �! ‐<sup>1</sup> <sup>O</sup>½ � ‐<sup>4</sup> , that is to say, the wave operator that involves in its term Ricci tensors:

$$
\Box^{\mathsf{Z}} : \phi \to \nabla\_{\mathsf{b}} \left[ \nabla^{\mathsf{b}} \nabla^{\mathsf{a}} - 2\mathsf{R}^{\mathsf{a}\mathsf{b}} + (2/3) \left( \mathsf{R} \mathsf{g}^{\mathsf{a}\mathsf{b}} \right) \nabla\_{\mathsf{a}} \right], \tag{31}
$$

Then, the integration of the partial differential equations corresponding to these linear invariant differential operators is realized due to integral transforms of the Penrose type since the irreducible unitary representation scheme to these operators is unitary representations of components of the group SL 4, ð Þ , such as SO 2n ð Þ.

In fact, in the flat case, the invariant differential operator classifications were described to determine a problem of representation theory of Lie groups applied to the Lie group SL 4, ð Þ and its compact subgroups. Then, own vision to these operators through SL 4, ð Þ will be as equivariant operators between homogeneous vector bundles on M, considering to SL 4, ð Þ as homogeneous space or class space. The integrals in this case are realizations of these representations and are orbital integrals of the integral transform of the resolutions to these differential equations, which, in this concrete case, are the Penrose transform.

Then, the resolution problem of the partial differential equations is reduce to the use of representation theory, but for this case, no always can construct the curved analogues of conformally invariant differential operators of the flat space. This demonstrates that cannot be generated a curved analogous under an integral transform on homogeneous bundles of lines that are direct images of the operators Dn, of G, of G*=*L. However, yes is possible to obtain a complete list under this procedure as a mapping of unitary modules.

Also, the scheme of the ‐modules in the quaternion analysis serves to compute and determine the properties of manifold through the scheme of fibers that can be in closed complex submanifolds. In fact, this is an alternative for the determination of vector fields through line bundles, which defined these as spin bundles.

Now well, cohomologically: How similar are these two methodologies for the study in field theory? Can the direct product of Lie groups SU 2ð Þ*T*, subjacent in the structure of a complex Riemannian manifold that models the space-time to its vector field study and its integration through the isometries of the space Lð Þ H be carried? Which are the integration limitations for the integral transforms on homogenous bundles in global descriptions of the vector fields?

The first question is related to the double fibration that can be realized on some complex projective spaces and their quaternion equivalent. Since it always exists this bijection due to this double fibration with some corresponding homotopy group that is frequently given for spheres, some real and complex projective spaces that are necessarily identified with some *<sup>n</sup>*‐dimensional sphere exist. Such is the case, for example, of the projective spaces <sup>1</sup> ffi <sup>S</sup><sup>1</sup> , <sup>1</sup> ffi S4, or <sup>7</sup> ffi Spin 2, 6 ð Þ. Then can be determined isomorphic cohomological spaces via some integral transform of the mentioned for the double fibrations. Some of these integrals result be of Feynman type due to the complex projective bundles are spin bundles in some sphere that determines some state space in quantum mechanics. For example, for the complex case, is had in an infinite succession of non-trivial bundles, the infinite set of bundles S<sup>1</sup> ! <sup>2</sup>*ℓ*�<sup>1</sup> <sup>k</sup> ! , *<sup>ℓ</sup>*‐<sup>1</sup> with k<sup>∈</sup> , and k 6¼ *<sup>ℓ</sup>*, which represents the infinite set of corresponding monopoles bundles to the case *ℓ* ¼ 2.

These with proper connections represent Dirac magnetic monopoles of charge k.

Likewise we have demonstrated that ifð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup> <sup>F</sup> , is homotopic to

*(a). Field state in a cohomology* H3,0 <sup>ℝ</sup><sup>3</sup> ð Þ ,<sup>L</sup> , *to the line bundle* <sup>L</sup>*:. (b).* P, *is a principal* SO 2ð Þ�*bundle.*

associated with the kernel h<sup>∈</sup> H3f,qð Þ <sup>Π</sup>‐ℓ, Oð Þ ‐**<sup>r</sup>** , are given for elements of the homology group Hfþ<sup>q</sup>ð Þ Λ � ℓ . Now, which of these contours are cohomological? A classes of contours are the classic or traditional contours. However, realizing extensions of these contour classes through twistor geometry, we can consider cohomological contours to all image elements of the generator of Hd <sup>Π</sup>‐ℓ<sup>0</sup> ð Þ <sup>0</sup>, , under two mappings of Mayer-Vietoris. Likewise, the box nonprojective diagram also engages three cohomological contours. Can this particular theory of contours to the spin

The response is yes, for example, of the foreseen construction given in

hypercomplex analysis of a vector field is an element of the integral cohomology

ð Þ Π � ℘, . We can realize more work in this sense until we can arrive to the

**3. The main conjecture and some notes of integral cohomology in low**

Using definitions and results exposed with before can be enunciated and

**Conjecture 3.1**. The cohomology of closed submanifolds of co-dimensions <sup>k</sup>‐1, n‐k, and n kð Þ ‐<sup>1</sup> , can be represented and evaluated by a function cohomology. The cohomology of contours is represented and evaluated by a complex functional cohomology. The cohomology of line bundles is represented and evaluated by a

There are indicium of that the differential operator class that accepts a scheme of integral cohomology (integral cohomology) like due for the Penrose transform, twistor transform, and so on is the class conformally invariant differential operators, of fact the Penrose transform generates these conformally invariant operators. Some examples of these differential operators are for the massless field equations (for flat versions and some curved versions [20]) and the conformally invariant

□ þ �R*=*<sup>6</sup> : <sup>O</sup>½ �! �<sup>1</sup> <sup>O</sup>½ � �<sup>3</sup> , (29)

<sup>∩</sup> kerD�<sup>a</sup>ð Þ <sup>Ω</sup> , with f <sup>¼</sup> <sup>K</sup>�af <sup>∈</sup> H0ð Þ <sup>Π</sup> � <sup>℘</sup>, , being <sup>℘</sup> <sup>¼</sup>

ð Þ Π � ℘, Oð Þ �2 . Then an integral formula in

ð Þ U1, Oð Þ �r1 ⊗⋯⊗ <sup>H</sup><sup>1</sup> Uf ð Þ , Oð Þ �rf ,

ð Þ <sup>Π</sup> � <sup>ℓ</sup>, <sup>Π</sup>‐<sup>ℓ</sup> <sup>∪</sup><sup>Λ</sup> , then the functionals on H1

*The fiber of the fiber bundle is the points* Ex <sup>¼</sup> <sup>P</sup>�SO 2ð Þℝ<sup>2</sup>*:*

*Advances in Complex Analysis and Applications*

context be understood?

ð Þ Ω ∩ C Ω

ð Þ <sup>z</sup>�<sup>a</sup> <sup>∈</sup> H1

demonstrated the following conjectures:

wave operator due to the mapping:

Penrose transform on hypercomplex numbers.

**dimension of a complex Riemannian manifold**

vector field cohomologies under the *<sup>∂</sup>*‐cohomology corresponding.

**Figure 6b**). If f ∈C<sup>1</sup>

**Figure 6.**

H1

**92**

f g �<sup>a</sup> , then h <sup>¼</sup> <sup>1</sup>

The constitutive integrals of these monopoles are Cauchy integrals that for diagrams of a cohomology H1 ð Þ <sup>Π</sup>‐ℓ, , these are reduced to integrals of Feynman type on the diagrams-boxes corresponding to the state monopoles vertices. These are identified for the factors 1*=*Z*α*W*<sup>α</sup>* . Likewise, an integral of Cauchy type given for an integral for a *<sup>ϕ</sup>*4‐vertex representing the projective space , or its dual <sup>∗</sup> , comes given for

$$\mathcal{LTF}(\phi^4) = \oint \mathbf{D}^1(\mathbf{Z}) \mathbf{f}(\mathbf{Z}\_a) \mathbf{g}(\mathbf{W}^a) \mathbf{h}(\mathbf{X}\_a) \mathbf{j}(\mathbf{Y}^a) / \mathbf{Z}\_a \mathbf{W}^a \mathbf{X}\_a \mathbf{W}^a \mathbf{X}\_a \mathbf{Y}^a \mathbf{Z}\_a \mathbf{Y}^a,\tag{32}$$

which is not different to the Cauchy integral for a monopole in z ¼ z0, and representing the space <sup>1</sup> ð Þ .

The response to the second question also is positive since it is possible to determine a cohomology of the space–time based on light geodesics as orbits of a complex torus *T*, when we consider our Universe as a complex hyperbolic manifold. The corresponding integral operators on the corresponding orbits result to be n‐ dimensional Fourier transforms Fn, that can be calculated for the relation

$$\mathbf{f}, \ = \mathcal{F}\_{\mathbf{n}}^{-1} \mathcal{F}\_{1} \mathcal{R} \mathbf{f}, \tag{33}$$

domain off. This integral is not more different than the Cauchy integral, in fact, in certain sense, this is a generalization. Further, this is not more different than the John integrals, Conway integrals, and Penrose integrals on . The first two are used

The Penrose line integral in integral geometry has the interpretation as was mentioned in the Radon transform on lines of a flag manifold *<sup>F</sup>* <sup>¼</sup> L L<sup>⊂</sup> <sup>4</sup>

these integrals belong to a same cohomological class, which can be determined

Ep,q <sup>¼</sup> <sup>Γ</sup> U, *<sup>ν</sup><sup>q</sup>*

groups are sections of the sheaf with coefficients on the fibered bundle O p, q, r ð Þ. The space *<sup>μ</sup>*‐1O p, q, r ð Þ, represents the inverse image of said sheaf. U <sup>⊂</sup> M, and U<sup>0</sup> <sup>¼</sup> *<sup>ν</sup>*‐<sup>1</sup>ð Þ <sup>U</sup> <sup>⊂</sup>*F*, and *<sup>μ</sup>*‐<sup>1</sup> <sup>U</sup><sup>0</sup> ð Þ⊂, with *<sup>μ</sup>*, and *<sup>ν</sup>*, are the corresponding homomorphisms of the double fibration in integral geometry to relate objects in M, and , can be a

Finally, we can say that the descriptions in Section 3 are only few examples of our theory of integrals that we want to construct, and that are examples to enforce our conjecture on the integral geometry bases obtained from the geometry and

The idea to obtain an integral operator cohomology is develop a theory through

integral invariants, that is to say, explore the complex Riemannian manifolds though the value of its integrals along the cycles and the corresponding cocycles (submanifolds, contours, vertices, edges, complexes, and so on) of the manifold. The duality between these cycles obeys to the spectral transformation that follows much of these integrals as solution of the corresponding differential equations. For example, in some case, it is used the tomography of Riemannian manifold whose cocycles are submanifolds. However, this idea can be generalized and induced beyond the tomography, for example, the integral transforms that generate differential operators with certain property of invariance inside the manifold and establish solution classes through these properties as the case to the conformally invariant differential operators. Then, the representation of objects, such as differential operators, functions, hyperfunctions, and fields, through integrals also appears in a natural way using the cohomology groups of its cocycles as first, second, …, *n*th integrals for a problem of the differential or functional equations. Likewise, much of these solutions are given through the integral transforms that

search solution classes as equivalence classes in the dual problem. The inverse problems are developed in the geometrical analysis corresponding. The cohomological problem consists in developing a cohomology H•ð Þ M, <sup>J</sup> , the sufficiently general that means the solution to enlarge number of differential equations and that can be

The reinterpretation for physics phenomena in the case when said complex Riemannian manifold models the space-time, results interestingly, and let open the possibility of constructing an Universe theory that includes macroscopic and

applied in the solution of the field equations in exploring the Universe.

microscopic phenomena through a good integral theory.

calculating the cohomology of *<sup>μ</sup>*‐1O p, q, r ð Þ, using the spectral sequence:

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

Then it is possible to calculate the cohomology groups Hp,q U<sup>0</sup>

variables determining the solution of the wave equation in 4.

, the third, for obtaining the harmonic functions of three and four

<sup>∗</sup> <sup>p</sup> ð Þ , taking the meaning of Eq. (35). These cohomology

. All

, *<sup>μ</sup>* ð Þ ‐1O p, q, r ð Þ , in

<sup>∗</sup> <sup>p</sup> ð Þ , (35)

on the circle S<sup>1</sup>

terms of Hp,q U<sup>0</sup>

analysis.

**95**

**4. Conclusions**

, *ν q*

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

projective twistor space and *F*, the flag manifold.

in a n-dimensional manifold. The operator R, is the Radon transform calculated on the corresponding cycles. It is well known that F ∈Lð Þ H , and that the integral cohomology given forFn, is the *<sup>∂</sup>*‐ cohomology of one codimensional submanifolds in M.

A response to the last question could be the limitations that are observed when it is wanted to extend the integration on the orbits of M, to a global integration of vector fields, since it is required the global integration of a vector field without the necessity of calculating previously the integrals on orbits of sections of a homogeneous bundle.

However, certain feasibility exists to obtain a methodology in this respect, generalizing, in some sense, the concept of conformal generalized structure on the manifold M.

The existing equivalences between twistor spaces, quaternion spaces, and Riemannian manifolds establish isomorphisms between different cohomology classes whose geometrical invariants are with similar invariant properties in such different cohomology classes. Likewise, we have, for example, a John integral on a complex bundle of lines *F*, which includes the same integration invariants with respect to the line bundle of the linear concave domains in the space <sup>n</sup> (respectively, n) for the integral of the Radon transform. The cohomology of the singularities in the description of the massless fields can be done through a twistor description of the fields using a relative cohomology of sheaves on the massless fields distributed on a real Minkowski space. Likewise, we can have other examples of equivalences for different cohomology classes.

Much results in complex analysis in , or <sup>2</sup> , can be generalized on a context of analytic functions more extensive, using a holomorphic language of a *<sup>∂</sup>*‐ cohomology. Example of it is the use of hyperfunctions for generalizing some contour integrals. If F⊂ , where F, is a closed interval and the hyperfunctions of F, are given for the quotient space Oð Þ ‐<sup>F</sup> *<sup>=</sup>*Oð Þ , and iff, is an analytic complex function (analytic in a real sense), the sesquilinear coupling with a hyperfunction represented by the holomorphic function *<sup>φ</sup>* (which can be a hypercomplex function) on ð Þ ‐<sup>F</sup> , is given for the contour integral:

$$(\phi, \mathbf{f}) = \oint \rho(\mathbf{z}) \mathbf{f}(\mathbf{z}) \, \text{d}z,\tag{34}$$

where the function f, must be extended holomorphically to a little portion of F, and the contour in ð Þ ‐<sup>F</sup> , transits around of F, sufficiently near of the definition

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

domain off. This integral is not more different than the Cauchy integral, in fact, in certain sense, this is a generalization. Further, this is not more different than the John integrals, Conway integrals, and Penrose integrals on . The first two are used on the circle S<sup>1</sup> , the third, for obtaining the harmonic functions of three and four variables determining the solution of the wave equation in 4.

The Penrose line integral in integral geometry has the interpretation as was mentioned in the Radon transform on lines of a flag manifold *<sup>F</sup>* <sup>¼</sup> L L<sup>⊂</sup> <sup>4</sup> . All these integrals belong to a same cohomological class, which can be determined calculating the cohomology of *<sup>μ</sup>*‐1O p, q, r ð Þ, using the spectral sequence:

$$\mathbf{E}^{\mathsf{p},\mathsf{q}} = \Gamma\Big(\mathsf{U}, \mathsf{\nu}\_{\mathsf{\*}}^{\mathsf{q}}\left(\mathbb{R}^{\mathsf{p}}\right)\Big),\tag{35}$$

Then it is possible to calculate the cohomology groups Hp,q U<sup>0</sup> , *<sup>μ</sup>* ð Þ ‐1O p, q, r ð Þ , in terms of Hp,q U<sup>0</sup> , *ν q* <sup>∗</sup> <sup>p</sup> ð Þ , taking the meaning of Eq. (35). These cohomology groups are sections of the sheaf with coefficients on the fibered bundle O p, q, r ð Þ. The space *<sup>μ</sup>*‐1O p, q, r ð Þ, represents the inverse image of said sheaf. U <sup>⊂</sup> M, and U<sup>0</sup> <sup>¼</sup> *<sup>ν</sup>*‐<sup>1</sup>ð Þ <sup>U</sup> <sup>⊂</sup>*F*, and *<sup>μ</sup>*‐<sup>1</sup> <sup>U</sup><sup>0</sup> ð Þ⊂, with *<sup>μ</sup>*, and *<sup>ν</sup>*, are the corresponding homomorphisms of the double fibration in integral geometry to relate objects in M, and , can be a projective twistor space and *F*, the flag manifold.

Finally, we can say that the descriptions in Section 3 are only few examples of our theory of integrals that we want to construct, and that are examples to enforce our conjecture on the integral geometry bases obtained from the geometry and analysis.

#### **4. Conclusions**

The constitutive integrals of these monopoles are Cauchy integrals that for dia-

the diagrams-boxes corresponding to the state monopoles vertices. These are identi-

for a *<sup>ϕ</sup>*4‐vertex representing the projective space , or its dual <sup>∗</sup> , comes given for

which is not different to the Cauchy integral for a monopole in z ¼ z0, and

The response to the second question also is positive since it is possible to determine a cohomology of the space–time based on light geodesics as orbits of a complex torus *T*, when we consider our Universe as a complex hyperbolic manifold. The corresponding integral operators on the corresponding orbits result to be n‐ dimen-

ð Þ .

of equivalences for different cohomology classes. Much results in complex analysis in , or <sup>2</sup>

sional Fourier transforms Fn, that can be calculated for the relation

f, ¼ F‐<sup>1</sup>

on the corresponding cycles. It is well known that F ∈Lð Þ H , and that the integral cohomology given forFn, is the *<sup>∂</sup>*‐ cohomology of one codimensional

necessity of calculating previously the integrals on orbits of sections of a

in a n-dimensional manifold. The operator R, is the Radon transform calculated

A response to the last question could be the limitations that are observed when it is wanted to extend the integration on the orbits of M, to a global integration of vector fields, since it is required the global integration of a vector field without the

However, certain feasibility exists to obtain a methodology in this respect, generalizing, in some sense, the concept of conformal generalized structure on the manifold M. The existing equivalences between twistor spaces, quaternion spaces, and Riemannian manifolds establish isomorphisms between different cohomology classes whose geometrical invariants are with similar invariant properties in such different cohomology classes. Likewise, we have, for example, a John integral on a complex bundle of lines *F*, which includes the same integration invariants with respect to the line bundle of the linear concave domains in the space <sup>n</sup> (respectively, n) for the integral of the Radon transform. The cohomology of the singularities in the description of the massless fields can be done through a twistor description of the fields using a relative cohomology of sheaves on the massless fields distributed on a real Minkowski space. Likewise, we can have other examples

analytic functions more extensive, using a holomorphic language of a *<sup>∂</sup>*‐ cohomology. Example of it is the use of hyperfunctions for generalizing some contour integrals. If F⊂ , where F, is a closed interval and the hyperfunctions of F, are given for the quotient space Oð Þ ‐<sup>F</sup> *<sup>=</sup>*Oð Þ , and iff, is an analytic complex function (analytic in a real sense), the sesquilinear coupling with a hyperfunction represented by the

holomorphic function *<sup>φ</sup>* (which can be a hypercomplex function) on ð Þ ‐<sup>F</sup> , is given

where the function f, must be extended holomorphically to a little portion of F,

and the contour in ð Þ ‐<sup>F</sup> , transits around of F, sufficiently near of the definition

ð Þ <sup>Π</sup>‐ℓ, , these are reduced to integrals of Feynman type on

. Likewise, an integral of Cauchy type given for an integral

<sup>n</sup> F1Rf, (33)

, can be generalized on a context of

ð Þ¼ *ϕ*, f ∮ *φ*ð Þz f zð Þdz, (34)

ð Þ <sup>Z</sup> f Zð Þ*<sup>α</sup>* g W*<sup>α</sup>* ð Þh Xð Þ*<sup>α</sup>* j Y*<sup>α</sup>* ð Þ*=*Z*α*W*<sup>α</sup>*X*α*W*<sup>α</sup>*X*α*Y*<sup>α</sup>*Z*α*Y*<sup>α</sup>*, (32)

grams of a cohomology H1

*Advances in Complex Analysis and Applications*

fied for the factors 1*=*Z*α*W*<sup>α</sup>*

IF *<sup>ϕ</sup>*<sup>4</sup> <sup>¼</sup> <sup>∮</sup> <sup>D</sup><sup>1</sup>

representing the space <sup>1</sup>

submanifolds in M.

homogeneous bundle.

for the contour integral:

**94**

The idea to obtain an integral operator cohomology is develop a theory through integral invariants, that is to say, explore the complex Riemannian manifolds though the value of its integrals along the cycles and the corresponding cocycles (submanifolds, contours, vertices, edges, complexes, and so on) of the manifold. The duality between these cycles obeys to the spectral transformation that follows much of these integrals as solution of the corresponding differential equations. For example, in some case, it is used the tomography of Riemannian manifold whose cocycles are submanifolds. However, this idea can be generalized and induced beyond the tomography, for example, the integral transforms that generate differential operators with certain property of invariance inside the manifold and establish solution classes through these properties as the case to the conformally invariant differential operators. Then, the representation of objects, such as differential operators, functions, hyperfunctions, and fields, through integrals also appears in a natural way using the cohomology groups of its cocycles as first, second, …, *n*th integrals for a problem of the differential or functional equations.

Likewise, much of these solutions are given through the integral transforms that search solution classes as equivalence classes in the dual problem. The inverse problems are developed in the geometrical analysis corresponding. The cohomological problem consists in developing a cohomology H•ð Þ M, <sup>J</sup> , the sufficiently general that means the solution to enlarge number of differential equations and that can be applied in the solution of the field equations in exploring the Universe.

The reinterpretation for physics phenomena in the case when said complex Riemannian manifold models the space-time, results interestingly, and let open the possibility of constructing an Universe theory that includes macroscopic and microscopic phenomena through a good integral theory.

*Advances in Complex Analysis and Applications*

**References**

USA: SCIRP; 2016

[1] Bulnes F, Shapiro M. On general theory of integral operators to analysis and geometry. In: Cladwell JP, editor. Monograph in Mathematics. 1st ed. México: IM/UNAM and IPN/SEPI; 2007

*DOI: http://dx.doi.org/10.5772/intechopen.92969*

editor. Vol. 4(30) 1 of Advanced Course in Applied Mathematics, National Polytechnique Institute. 2006;**8**(1):

electrodynamics: Groups, cohomology classes, unitary representations, orbits

electrophysics. American Journal of Electromagnetics and Applications. 2015;**3**(6):43-52. DOI: 1011648/j.

[10] Bulnes F, Fominko S. Dx-schemes and jets in conformal gravity using integral transforms. International Journal of Mathematical Research. 2016;

[11] Bulnes F. lectromagnetic waves in conformal actions of the group SU(2, 2) on a dimensional flat model of the space-time. In: VI International Conference on Geometry, Dynamics, Integrable Systems-(GDIS 2016), 2–5

[12] Dunne EG, Zierau R. Twistor theory for indefinite Kahler symmetric spaces. In: Contemporary Mathematics. Vol.

[13] Eastwood M, Penrose R, Wells RO Jr. Cohomology and massless fields. Communications in Mathematical

[14] Baston RJ. Local Cohomology, Elementary States and Evaluation. Vol. 22. Twistor, Newsletter (Oxford Preprint). United Kingdom: Oxford;

[15] Kobayashi K, Nomizu K. Foundations of Differential Geometry. (Vol. II). New

[16] Gindikin SG, Henkin GM. Integral geometry for ∂-cohomology in q-linear

June; Izhevsk, Russia; 2016

154. USA: AMS; 1993

Physics. 1981;**78**:26-30

York: Wiley and Sons; 1969

1986. pp. 8-13

[9] Bulnes F. Mathematical

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**5**(2):154-165

398-447

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex…*

[2] Bulnes F. Integral Geometry Methods in the Geometrical Langlands Program.

[4] F. Bulnes, "Conferences of lie groups (representation theory of reductive lie

[5] Bateman H. The solution of partial differential equations by means of definite integrals. Proceedings of the London Mathematical Society. 1904;

generalizations and penrose transform. In: Proceedings of 4th International Congress in Applied Mathematics, Special Section in Functional Analysis and Differential Equations. Mexico: IMUNAM and IPN/SEPI; 2008.

Kingdom: Cambridge University Press;

mathematical electrodynamics. Journal of Conferences and Advanced Courses of Applied Math II. SEPI-ESIME/IPN,

[3] Bulnes F. On the last progress of cohomological induction in the problem

of classification of lie groups representations. In: International Conference of Infinite Dimensional Analysis and Topology, Book of Plenary Conferences. Ukraine: Precarpatinan

National University; 2009

groups)", Monograph in Pure Mathematics, SEPI-ESIME/IPN, 2nd Edition by Paul Cladwell, Mexico, 2005.

[6] Bulnes F. Radon transform,

[7] Dunne EG, Eastwood MG. The Twistor Transform. London Mathematical Society Lecture Note Series, 156. Cambridge, United

[8] Bulnes F. Doctoral course of

**1**(2):451-458

pp. 63-76

1990

**97**

### **Author details**

Francisco Bulnes IINAMEI, Research Department in Mathematics and Engineering, TESCHA, Chalco, Mexico

\*Address all correspondence to: francisco.bulnes@tesch.edu.mx

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex… DOI: http://dx.doi.org/10.5772/intechopen.92969*

#### **References**

[1] Bulnes F, Shapiro M. On general theory of integral operators to analysis and geometry. In: Cladwell JP, editor. Monograph in Mathematics. 1st ed. México: IM/UNAM and IPN/SEPI; 2007

[2] Bulnes F. Integral Geometry Methods in the Geometrical Langlands Program. USA: SCIRP; 2016

[3] Bulnes F. On the last progress of cohomological induction in the problem of classification of lie groups representations. In: International Conference of Infinite Dimensional Analysis and Topology, Book of Plenary Conferences. Ukraine: Precarpatinan National University; 2009

[4] F. Bulnes, "Conferences of lie groups (representation theory of reductive lie groups)", Monograph in Pure Mathematics, SEPI-ESIME/IPN, 2nd Edition by Paul Cladwell, Mexico, 2005.

[5] Bateman H. The solution of partial differential equations by means of definite integrals. Proceedings of the London Mathematical Society. 1904; **1**(2):451-458

[6] Bulnes F. Radon transform, generalizations and penrose transform. In: Proceedings of 4th International Congress in Applied Mathematics, Special Section in Functional Analysis and Differential Equations. Mexico: IMUNAM and IPN/SEPI; 2008. pp. 63-76

[7] Dunne EG, Eastwood MG. The Twistor Transform. London Mathematical Society Lecture Note Series, 156. Cambridge, United Kingdom: Cambridge University Press; 1990

[8] Bulnes F. Doctoral course of mathematical electrodynamics. Journal of Conferences and Advanced Courses of Applied Math II. SEPI-ESIME/IPN,

editor. Vol. 4(30) 1 of Advanced Course in Applied Mathematics, National Polytechnique Institute. 2006;**8**(1): 398-447

[9] Bulnes F. Mathematical electrodynamics: Groups, cohomology classes, unitary representations, orbits and integral transforms in electrophysics. American Journal of Electromagnetics and Applications. 2015;**3**(6):43-52. DOI: 1011648/j. ajea.20150306.12

[10] Bulnes F, Fominko S. Dx-schemes and jets in conformal gravity using integral transforms. International Journal of Mathematical Research. 2016; **5**(2):154-165

[11] Bulnes F. lectromagnetic waves in conformal actions of the group SU(2, 2) on a dimensional flat model of the space-time. In: VI International Conference on Geometry, Dynamics, Integrable Systems-(GDIS 2016), 2–5 June; Izhevsk, Russia; 2016

[12] Dunne EG, Zierau R. Twistor theory for indefinite Kahler symmetric spaces. In: Contemporary Mathematics. Vol. 154. USA: AMS; 1993

[13] Eastwood M, Penrose R, Wells RO Jr. Cohomology and massless fields. Communications in Mathematical Physics. 1981;**78**:26-30

[14] Baston RJ. Local Cohomology, Elementary States and Evaluation. Vol. 22. Twistor, Newsletter (Oxford Preprint). United Kingdom: Oxford; 1986. pp. 8-13

[15] Kobayashi K, Nomizu K. Foundations of Differential Geometry. (Vol. II). New York: Wiley and Sons; 1969

[16] Gindikin SG, Henkin GM. Integral geometry for ∂-cohomology in q-linear

**Author details**

Francisco Bulnes

Chalco, Mexico

**96**

IINAMEI, Research Department in Mathematics and Engineering, TESCHA,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: francisco.bulnes@tesch.edu.mx

provided the original work is properly cited.

*Advances in Complex Analysis and Applications*

concave domains in CPn. Funktsional. Anal. i Prilozhen. 1978;**12**(4):6-23

[17] Gel'fand IM, Graev MI, Shapiro IIP. Generalized Functions. New York, USA: Academic Press; 1952

[18] S. Huggett and Singer, "Cohomology of contours and residues", Transactions of the American Mathematical Society. 1982. pp. 308-316

[19] Derbyshire S. An illustration to Cauchy's integral formula in complex analysis. Available from: http://en. wikipedia.org/wiki/File:ComplexResid uesExample.png

[20] Bulnes F. Cohomology of moduli spaces in differential operators classification to the field theory (II). In: Proceedings of FSDONA-11 (Function Spaces, Differential Operators and Non-linear Analysis, 2011); Tabarz Thur. Vol. 1, No. 12; 2011. pp. 1-22

Section 4

Complex Functional

Transforms and Their

Applications

**99**

### Section 4
