**3. Asymptotics**

The letters *O* and *o* are order symbols. They are used to describe the rate at which functions approach limit values. We will consider the types of limit values, namely zero, a finite number but nonzero and infinite.

If a function *f x*ð Þ approaches a limiting value at the same rate of another function *g x*ð Þ as *x* ! *x*0, then we write

$$f(\mathbf{x}) = O(\mathbf{g}(\mathbf{x})), \quad \mathbf{ as } \mathbf{x} \to \mathbf{x}\_0 \tag{15}$$

The functions are said to be of the same order as *x* ! *x*0. The test for this is the limit of the ratio. Thus, if lim*<sup>x</sup>*!*x*<sup>0</sup> *f x*ð Þ *g x*ð Þ <sup>¼</sup> *<sup>C</sup>*, where *<sup>C</sup>* is finite, then we say (15) holds.

For example, we have the following functions:

$$\begin{aligned} \mathfrak{x}^2 &= O(\mathfrak{x}), & |\mathfrak{x}| &< 2, \\ \sin\left(\mathfrak{x}\right) &= O\left(\sqrt{\mathfrak{x}}\right), & \mathfrak{x} &\to \mathbf{0}, \\ \sin\left(\mathfrak{x}\right) &= O(\mathfrak{x}), & -\infty &< \mathfrak{x} < \infty. \end{aligned}$$

The expression

$$f(\mathbf{x}) = o(\mathbf{g}(\mathbf{x})), \quad \text{ as } \mathbf{x} \to \mathbf{x}\_0 \tag{16}$$

means that lim*<sup>x</sup>*!*x*<sup>0</sup> *f x*ð Þ *g x*ð Þ <sup>¼</sup> <sup>0</sup>*:* This is a stronger assertion that the corresponding *<sup>O</sup>*– formula. The relation (16) implies the relation (15), as convergence implies boundedness from a certain point onwards.

We have the following functions satisfy the *o*–relation:

$$\begin{aligned} \cos(\alpha) &= \mathbf{1} + o(\alpha), & |\alpha| &< 2, \\ \boldsymbol{\epsilon}^{\times} &= \mathbf{1} + o(\alpha), & \boldsymbol{\epsilon} &\to \mathbf{0} \\ n! &= \boldsymbol{\epsilon}^{-n} \cdot n^{n} \sqrt{2\pi n} (\mathbf{1} + o(\mathbf{1})), & n &\to \infty. \end{aligned}$$

#### **3.1 Asymptotic expansions**

Consider the expansion

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$$f(\mathbf{x}) = a\_0 + \frac{a\_1}{\mathbf{x}} + \frac{a\_2}{\mathbf{x}^2} + \dots + \frac{a\_N}{\mathbf{x}^N} + R\_N,\tag{17}$$

is an asymptotic expansion as *x* ! ∞, if, for any *N*,

$$R\_N = O\left(\frac{1}{\mathfrak{x}^{N+1}}\right), \quad \text{as } \mathfrak{x} \to \infty \tag{18}$$

The following expansion is used when (17) and (18) hold,

$$f(\mathbf{x}) \sim \sum\_{n=0}^{\infty} \frac{a\_n}{\mathbf{x}^n}, \quad \text{ as } \mathbf{x} \to \infty \tag{19}$$

Here, lim*n*!<sup>∞</sup>*RN* <sup>¼</sup> 0, for any value of *<sup>N</sup>*.

The sequence 1, 1*=x*, 1*=x*2, <sup>⋯</sup> � � is an *asymptotic sequence* as *<sup>x</sup>* ! <sup>∞</sup>. The characteristic feature of such sequences is that each member is dominated by the previous member. In constructing examples it is easier to deal with the limit zero than any other. Thus, for the case *x* ! ∞, we let *ε* ¼ 1*=x*, which for *x* ! *x*0, we let *ε* ¼ *x* � *x*<sup>0</sup> so that without loss of generality we may confirm our attention to the limit *ε* ! 0. The standard asymptotic sequence is 1, *<sup>ε</sup>*, *<sup>ε</sup>*2, <sup>⋯</sup> � � as *<sup>ε</sup>* ! 0. If we let *<sup>δ</sup>n*ð Þ*<sup>ε</sup>* represent members of an asymptotic sequence f g *δ*0ð Þ*ε* , *δ*1ð Þ*ε* , ⋯ as *ε* ! 0, then the following condition must hold

$$
\delta\_{n+1}(\varepsilon) = \sigma(\delta\_n(\varepsilon)), \quad \text{as } \varepsilon \to 0.
$$

Some examples of asymptotic sequences are

i. 1, sin ð Þ*<sup>ε</sup>* , sin ð Þ ð Þ*<sup>ε</sup>* <sup>2</sup> , sin ð Þ ð Þ*<sup>ε</sup>* <sup>3</sup> , ⋯ n o, here we have lim*ε*!0 *δ<sup>n</sup>*þ<sup>1</sup> *δn* <sup>¼</sup> lim*<sup>ε</sup>*!<sup>0</sup> sin ð Þ¼ *ε* 0*:*

ii. 1, ln 1ð Þ <sup>þ</sup> *<sup>ε</sup>* , ln 1 <sup>þ</sup> *<sup>ε</sup>*<sup>2</sup> ð Þ, ln 1 <sup>þ</sup> *<sup>ε</sup>*<sup>3</sup> ð Þ, <sup>⋯</sup> � �, with *<sup>δ</sup>*<sup>0</sup> <sup>¼</sup> 1, *<sup>δ</sup><sup>n</sup>* <sup>¼</sup> ln 1 <sup>þ</sup> *<sup>ε</sup><sup>n</sup>* ð Þ*n*<sup>≥</sup> 1, we have

$$\begin{aligned} \lim\_{\varepsilon \to 0} \frac{\delta\_1}{\delta\_0} &= \lim\_{\varepsilon \to 0} \ln \left( \mathbf{1} + \varepsilon \right) = \mathbf{0}, \\\lim\_{\varepsilon \to 0} \frac{\delta\_{n+1}}{\delta\_n} &= \lim\_{\varepsilon \to 0} \frac{\ln \left( \mathbf{1} + \varepsilon^{n+1} \right)}{\ln \left( \mathbf{1} + \varepsilon^n \right)} = \lim\_{\varepsilon \to 0} \frac{\varepsilon^{n+1} + O(\varepsilon^{2n+2})}{\varepsilon^n + O(\varepsilon^{2n})} = \mathbf{0}. \end{aligned}$$

The general expression for an asymptotic expansion of a function *f*ð Þ*ε* , in terms of an asymptotic sequence *δn*ð Þ*ε* is

$$f(\mathbf{x}) \sim \sum\_{n=0}^{\infty} a\_n \delta\_n(\varepsilon), \quad \text{as } \varepsilon \to 0,\tag{20}$$

where the coefficients *an* are independent of *ε*. The expression (20) involving the symbol �, means that for all *N*,

$$f(\mathbf{x}) = \sum\_{n=0}^{N} a\_n \delta\_n(\boldsymbol{\varepsilon}) + R\_N,\tag{21}$$

*Perturbation Expansion to the Solution of Differential Equations DOI: http://dx.doi.org/10.5772/intechopen.94173*

where

$$R\_N = O[\delta\_{N+1}(\varepsilon)], \quad \text{as } \varepsilon \to 0,\tag{22}$$

$$a\_n = \lim\_{\varepsilon \to 0} \left( \frac{f(\varepsilon) - \sum\_{n=0}^{N-1} a\_n \delta\_n(\varepsilon)}{\delta\_N(\varepsilon)} \right). \tag{23}$$

If a function possesses an asymptotic expansion involving the sequence f g *δ*0ð Þ*ε* , *δ*1ð Þ*ε* , ⋯ then the coefficients *an* of the expansion (21) given by the expression (24) are unique. However, another function may share the same set of coefficients. Thus, while functions have unique expansions, an expansion does not correspond to a unique function.

Consider a function *f x*ð Þ ; *ε* , which depends on both an independent variable *x*, and a small parameter *ε*. Suppose that *f x*ð Þ ; *ε* is expanded using an asymptotic sequence f g *δn*ð Þ*ε* ,

$$f(\mathbf{x}; \boldsymbol{\varepsilon}) = \sum\_{n=0}^{N} a\_n(\mathbf{x}) \delta\_n(\boldsymbol{\varepsilon}) + R\_N(\mathbf{x}; \boldsymbol{\varepsilon}). \tag{24}$$

The coefficients of the gauge functions *δn*ð Þ*ε* are functions of *x*, and the remainder after *N* terms is a function of both *x* and *ε*. For this to be an asymptotic expansion, we require

$$R\_N(\mathfrak{x}; \varepsilon) = O[\delta\_{N+1}(\varepsilon)], \quad \text{as } \varepsilon \to 0. \tag{25}$$

Refer [3, 4] for more details. For (24) to be a uniform asymptotic expansion the ultimate proportionality between *RN* and *δ<sup>N</sup>*þ<sup>1</sup> must be bounded by a number independent of *x*, i.e.,

$$|\mathcal{R}\_N(\varkappa;\varepsilon)| \le K|\delta\_{N+1}(\varepsilon)|,\tag{26}$$

for *ε* in the neighborhood near zero, where *K* is a fixed constant. An example of a uniform asymptotic expansion is *f x*ð Þ¼ ; *<sup>ε</sup>* <sup>1</sup> <sup>1</sup>�*<sup>ε</sup>* sin ð Þ *<sup>x</sup> :* An example of a nonuniform expansion is

$$f(\boldsymbol{x}; \boldsymbol{\varepsilon}) \sim \sum\_{n=0}^{N} \boldsymbol{\pi}^{n} \boldsymbol{\varepsilon}^{n} + R\_{N}(\boldsymbol{\kappa}; \boldsymbol{\varepsilon}), \quad \text{ as } \boldsymbol{\varepsilon} \to \mathbf{0}. \tag{27}$$

Here, one cannot find a fixed *K* which satisfy ∣*RN*∣ ≤*K*∣*ε<sup>N</sup>*þ<sup>1</sup>∣, because for any choice of *K*, *x* can be chosen so that *x<sup>N</sup>*þ<sup>1</sup> exceeds this value.

#### **3.2 Nonuniformity**

The expansion (27) becomes nonuniform when subsequent terms are no longer small corrections to previous terms. This occurs when subsequent terms are of the same order or of dominant order than previous terms. Subsequent terms dominate previous terms for larger *<sup>x</sup>*, for example, when *<sup>x</sup>* <sup>¼</sup> *<sup>O</sup>* <sup>1</sup>*=ε*<sup>2</sup> ð Þ. The expansion is valid for *x* ¼ *O*ð Þ1 since then subsequent terms decrease by a factor of *ε*. The expansion remains valid for large *x*, provided *x* is not as large as 1*=ε*. For instance, the expansion is valid for *<sup>x</sup>* <sup>¼</sup> *<sup>O</sup>* <sup>1</sup>*<sup>=</sup>* ffiffi *<sup>ε</sup>* <sup>p</sup> ð Þ, as *<sup>ε</sup>* ! 0.

The critical case is such that subsequent terms are of the same order. This determines the region of nonuniformity. In (27), the region of nonuniformity occurs when *<sup>ε</sup><sup>x</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ<sup>1</sup> , i.e., *<sup>x</sup>* <sup>¼</sup> *<sup>O</sup> <sup>ε</sup>*�<sup>1</sup> ð Þ, as *<sup>ε</sup>* ! 0.
