**5.1 The action** L*<sup>ν</sup>*

**Lemma 5.1.** *We set z* <sup>¼</sup> *<sup>ρ</sup>eit, and I* ¼ �<sup>Ð</sup> <sup>2</sup>*<sup>π</sup>* <sup>0</sup> *<sup>e</sup>i k*ð Þ �*<sup>m</sup> <sup>θ</sup> log* <sup>j</sup>*<sup>z</sup>* � *re<sup>i</sup><sup>θ</sup>*<sup>j</sup> � � *<sup>d</sup><sup>θ</sup>* <sup>2</sup>*π, we have*

$$\begin{cases} I = -\log\left(\rho \wedge r\right) & k = m, \\\\ I = \frac{e^{i(k-m)t}}{2|m-k|} \left( \left(\frac{r}{\rho}\right)^{m-k} \wedge \left(\frac{r}{\rho}\right)^{m-k} \right) & k \neq m, \end{cases} \tag{34}$$

*Proof.* By ref. [3], it remains to prove that this lemma for *k*< *m*. We have

$$\int\_0^{2\pi} e^{i(k-m)\theta} \log\left(|\rho \epsilon^{it} - r \epsilon^{i\theta}|\right) d\theta = -\int\_0^{2\pi} e^{i(m-k)(-\theta)} \log\left(|re^{i(-t)} - \rho \epsilon^{i(-\theta)}|\right) d(-\theta) \tag{35}$$

The function *<sup>θ</sup>* ! *ei m*ð Þ� �*<sup>k</sup>* ð Þ*<sup>θ</sup> log rei*ð Þ �*<sup>t</sup>* � *<sup>ρ</sup>ei*ð Þ �*<sup>θ</sup>* � � � � � � is a periodic mapping with the period equal 2*π*, then

$$\int\_0^{2\pi} e^{i(k-m)\theta} \log\left(\left|\rho e^{it} - re^{i\theta}\right|\right) d\theta$$

$$= -\int\_0^{2\pi} e^{i(m-k)(-\theta)} \log\left(\left|re^{i(-t)} - \rho e^{i(-\theta)}\right|\right) d(-\theta)$$

$$= \frac{e^{i(k-m)t}}{2(m-k)} \times \left(\left(\frac{r}{\rho}\right)^{m-k} \wedge \left(\frac{r}{\rho}\right)^{m-k}\right).$$

$$R\_{\lambda} \phi\_k^{\nu, m}(z) = \lambda^{k-m} \phi\_k^{\nu, m}(z), \ \forall k \neq m.$$

$$\begin{aligned} &\left(\mathcal{L}\_{\nu}\left(\phi\_{k}^{\nu,m}\right),\mathcal{L}\_{\nu}\left(\phi\_{j}^{\nu,m}\right)\right) \\ &= \left(R\_{\lambda}\mathcal{L}\_{\nu}\left(\phi\_{k}^{\nu,m}\right),R\_{\lambda}\mathcal{L}\_{\nu}\left(\phi\_{j}^{\nu,m}\right)\right) \\ &= \left(\mathcal{L}\_{\nu}R\_{\lambda}\left(\phi\_{k}^{\nu,m}\right),\mathcal{L}\_{\nu}R\_{\lambda}\left(\phi\_{j}^{\nu,m}\right)\right) \\ &= \overline{\lambda^{j-k}}\left(\mathcal{L}\_{\nu}\left(\phi\_{k}^{\nu,m}\right),\mathcal{L}\_{\nu}\left(\phi\_{j}^{\nu,m}\right)\right),\text{ if }\ m>k. \end{aligned}$$

$$=\lambda^{k-j}\left(\mathcal{L}\_{\nu}\left(\phi\_k^{\nu,m}\right),\mathcal{L}\_{\nu}\left(\phi\_j^{\nu,m}\right)\right),\text{ if }\ m$$

$$\left(\mathcal{L}\_{\nu}\left(\phi\_k^{\nu,m}\right), \mathcal{L}\_{\nu}\left(\phi\_j^{\nu,m}\right)\right) = 0 \,\,\,\sharp \,\,j \neq k.$$

$$
\mathcal{L}\_{\nu} \left( {}^{1} \phi\_{k}^{\nu, m} \right) (z) = \frac{\Gamma(k + 1) \Gamma(2\nu - m)}{\Gamma(m + 1) \Gamma(2\nu - k)} \mathcal{L}\_{\nu} \left( {}^{2} \phi\_{k}^{\nu, m} \right) (z) .
$$

$$\frac{\Gamma(m+1)}{\Gamma(m-s+1)}P\_{m}^{(-s,-a)}(u) = \frac{\Gamma(m+a+1)}{\Gamma(m-s+a+1)} \left(\frac{u-1}{2}\right)^{s} P\_{m-s}^{(s,a)}(u), \mathbb{1} \le s \le m. \tag{36}$$

$$\mathcal{L}\_{\boldsymbol{\nu}}(\boldsymbol{\phi}\_{k}^{\boldsymbol{\nu},m})(\boldsymbol{z}) = \frac{a\_{k}^{\boldsymbol{\nu},m}}{2(2\boldsymbol{\nu}-m+1)} \sqrt{\frac{2(\boldsymbol{\nu}-m)-1}{\pi}} (\boldsymbol{1}-\boldsymbol{\rho}^{2})^{2\boldsymbol{\nu}-m-1}{}\_{\boldsymbol{\lambda}}F\_{2} \begin{pmatrix} -m+1, \ 2\boldsymbol{\nu}-m, \ 2\boldsymbol{\nu}-m+1\\ 2(\boldsymbol{\nu}-m), \ 2\boldsymbol{\nu}-m+2 \end{pmatrix} \tag{37}$$

$$
\mathcal{L}\_{\nu} \left( \phi\_k^{\nu, m} \right)(z) = \frac{\pi \gamma\_k^{\nu, m} e^{i(k-m)t}}{2(k-m)} (I\_3 + I\_4),
$$

$$I\_3 = \frac{(1+k-m)\_m}{m!(k-m+1)} \rho^{k-m+2} \left(1-\rho^2\right)^{2\nu-m-1} \,\_2F\_1 \left( \begin{matrix} -m+1, 2(\nu-m)+k \\ 2+k-m \end{matrix} \; \middle| \; \rho^2 \right),$$

$$I\_4 = \frac{a\_k^{\nu, m}}{2\nu - m - 1} \left( 1 - \rho^2 \right)^{2\nu - m - 1} {}\_2F\_1 \left( \begin{matrix} -m + 1, 2\nu - m - 1 \\ & 2(\nu - m), \end{matrix} \bigg| \rho^2 \right).$$

$$\begin{split} \mathcal{L}\_{\nu}(\boldsymbol{\phi}\_{k}^{\boldsymbol{\nu},m})(\boldsymbol{z}) &= \frac{(-1)^{m}}{\pi} \sqrt{\frac{2(\boldsymbol{\nu}-\boldsymbol{m})-1}{\pi}} \Big|\_{\boldsymbol{D}} \left( 1 - |\boldsymbol{\xi}|^{2} \right)^{2\boldsymbol{\nu}-m-2} P\_{m}^{(0,\ 2(\boldsymbol{\nu}-\boldsymbol{m})-1)} \left( 1 - 2|\boldsymbol{\xi}|^{2} \right) \log \left( |\boldsymbol{z} - \boldsymbol{\xi}| \right) d\mu(\boldsymbol{\xi}) \\ &= (-1)^{m} \sqrt{\frac{2(\boldsymbol{\nu}-\boldsymbol{m})-1}{\pi}} \Big|\_{0}^{1} (1 - r^{2})^{2\boldsymbol{\nu}-m-2} P\_{m}^{(0,\ 2(\boldsymbol{\nu}-\boldsymbol{m})-1)} \left( 1 - 2r^{2} \right) \log \left( \boldsymbol{\rho} \wedge r \right) dr^{2} \\ &= \frac{(-1)^{m}}{2} \sqrt{\frac{2(\boldsymbol{\nu}-\boldsymbol{m})-1}{\pi}} \Big|\_{0}^{1} (1 - t)^{2\boldsymbol{\nu}-m-2} P\_{m}^{(0,\ 2(\boldsymbol{\nu}-\boldsymbol{m})-1)} (1 - 2t) \log \left( \boldsymbol{\rho}^{2} \vee t \right) dt \\ &= \frac{(-1)^{m}}{2} \sqrt{\frac{2(\boldsymbol{\nu}-\boldsymbol{m})-1}{\pi}} [I\_{1} + I\_{2}]. \end{split}$$

$$I\_1 = \int\_0^{\rho^2} (\mathbf{1} - t)^{2\nu - m - 2} P\_m^{(0, \ 2(\nu - m) - 1)} (\mathbf{1} - 2t) \log \left( \rho^2 \vee t \right) dt,$$

$$I\_2 = \int\_{\rho^2}^1 (1-t)^{2\nu-m-2} P\_m^{(0,\ 2(\nu-m)-1)}(1-2t) \log\left(t\right) dt.$$

$$I\_1 = \log \left(\rho^2\right) \int\_{\rho^2}^1 (\mathbf{1} - t)^{2\nu - m - 2} P\_m^{(0, \ 2(\nu - m) - 1)}(\mathbf{1} - \mathbf{2}t) dt.$$

$$P\_k^{(a,\beta)}(u) = \frac{(\mathbf{1}+a)\_k}{k!} \,\_2F\_1 \left( \begin{array}{c} -k, \; \mathbf{1}+a+\beta+k\\ \mathbf{1}+a \end{array} \mid \frac{\mathbf{1}-u}{2} \right).$$

$$I\_1 = \log \left(\rho^2\right) \int\_0^{\rho^2} (1-t)^{2\nu-m-2} \,\_2F\_1 \left( \begin{array}{c} -m, \ 2\nu-m \\ 1 \end{array} \middle| t \right) dt$$

$$\int \mathfrak{x}^{\varepsilon-1} (\mathbf{1} - \mathfrak{x})^{b-\varepsilon-1} \,\_2F\_1 \left( \begin{matrix} a, b \\ c \end{matrix} \; \middle| \; \mathfrak{x} \right) d\mathfrak{x} = \frac{1}{c} \mathfrak{x}^{\varepsilon} (\mathbf{1} - \mathfrak{x})^{b-\varepsilon} \,\_2F\_1 \left( \begin{matrix} a+\mathbf{1}, b \\ c+\mathbf{1} \end{matrix} \; \middle| \; \mathfrak{x} \right),$$

$$I\_1 = \log\left(\rho^2\right)\rho^2 \left(1-\rho^2\right)^{2\nu-m-1} \,\_2F\_1\left(\begin{array}{c} -m+1, 2\nu-m\\2 \end{array}\; |\rho^2\right).$$

$$I\_2 = \int\_{\rho^2}^1 (\mathbf{1} - \mathbf{t})^{2\nu - m - 2} P\_m^{(0, \ 2(\nu - m) - 1)}(\mathbf{1} - \mathbf{2t}) \log \left( t \right) dt.$$

$$\begin{split} I\_{2} &=& \left[ t\mathbf{1} - t^{2\nu-m-1} \, \_2F\_1 \left( \begin{matrix} -m+1, 2\nu-m \\ 2 \end{matrix} \bigg| t \right) \log t \right]\_{\rho^2}^{1} - \int\_{\rho^2}^{1} \mathbf{1} - t^{2\nu-m} \, \_2F\_1 \left( \begin{matrix} -m+1, 2\nu-m \\ 2 \end{matrix} \bigg| t \right) dt \\ &=& -\rho^2 \log \rho^2 \mathbf{1} - \rho^{2\nu-m-1} \, \_2F\_1 \left( \begin{matrix} -m+1, 2\nu-m \\ 2 \end{matrix} \bigg| \rho^2 \right) - \int\_{\rho^2}^{1} \mathbf{1} - t^{2\nu-m} \, \_2F\_1 \left( \begin{matrix} -m+1, 2\nu-m \\ 2 \end{matrix} \bigg| t \right) dt. \end{split}$$

$$\int\_{\rho^2}^1 (1-t)^{2\nu-m} \,\_2F\_1 \left( \begin{array}{c} -m+1, \ 2\nu-m \\ 2 \end{array} \bigg| t \right) dt.$$

$$\begin{aligned} {}\_2F\_1\left( \begin{matrix} a,b \\ c \end{matrix} \Big| t \right) &= \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} {}\_2F\_1\left( \begin{matrix} a,b \\ a+b-c+1 \end{matrix} \Big| 1-t \right) \\ &+ \frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)} (1-t)^{c-a-b} {}\_2F\_1\left( \begin{matrix} a,b \\ a+b-c+1 \end{matrix} \Big| 1-t \right). \end{aligned}$$

$$\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod\_{n=1}^{\infty} \left( 1 + \frac{z}{n} \right)^{-1} e^{-\frac{z}{n}},$$

$${}\_{2}F\_{1}\left(\begin{array}{c} -m+1, \, 2\nu-m\\ 2 \end{array}\Big|t\right) = \frac{2\Gamma(2(m-\nu)+1)}{m!\Gamma(2+m-2\nu)\_{2}}F\_{1}\left(\begin{array}{c} -m+1, \, 2\nu-m\\ 2(\nu-m) \end{array}\Big|1-t\right),$$

$$\int\_{\rho^2}^1 (1-t)^{2\nu-m} \,\_2F\_1 \left( \begin{matrix} -m+1, 2\nu-m \\ 2 \end{matrix} \bigg| t \right) dt = \frac{2\Gamma(2(m-\nu)+1)}{m!\Gamma(2+m-2\nu)} \int\_{\rho^2}^1 (1-t)^{2\nu-m} \,\_2F\_1 \left( \begin{matrix} -m+1, 2\nu-m \\ 2(\nu-m) \end{matrix} \bigg| 1-t \right) dt.$$

$$\int\_{\rho^2}^1 (1-t)^{2\nu-m} \,\_2F\_1 \left( \begin{matrix} -m+1, 2\nu-m \\ 2 \end{matrix} \bigg| t \right) dt = \frac{2\Gamma(2(m-\nu)+1)}{m!\Gamma(2+m-2\nu)} \int\_0^{1-\rho^2} t^{2\nu-m} \,\_2F\_1 \left( \begin{matrix} -m+1, 2\nu-m \\ 2(\nu-m) \end{matrix} \bigg| t \right) dt.$$

$$\int \mathfrak{x}^{a-1} {}\_2F\_1 \left( \begin{matrix} a,b \\ c \end{matrix} \; \middle| \; -t \right) d\mathfrak{x} = \frac{\mathfrak{x}^a}{a \mathfrak{z}} F\_2 \left( \begin{matrix} a,b,a \\ c,a+1 \end{matrix} \; \middle| \; -t \right) + \frac{\Gamma(a)\Gamma(a-a)\Gamma(b-a)\Gamma(c)}{\Gamma(a)\Gamma(b)\Gamma(c-a)} \right.$$

$$\frac{\Gamma(a)\Gamma(a-a)\Gamma(b-a)\Gamma(c)}{\Gamma(a)\Gamma(b)\Gamma(c-a)} = 0$$

$$\begin{aligned} &\int\_0^{1-\rho^2} t^{2\nu-m+1} \,\_2F\_1 \binom{-m+1,\ 2\nu-m}{2(\nu-m)} dt \\ &= (-1)^m \int\_0^{\rho^2} t^{2\nu-m} \,\_2F\_1 \binom{-m+1,\ 2\nu-m}{2(\nu-m)} dt \end{aligned}$$

$$=(-1)^{m}\frac{(\rho^{2}-1)^{2\nu-m+1}}{2\nu-m+1}{}\_{3}F\_{2}\left(\begin{array}{c}-m+1,\ 2\nu-m,\ 2\nu-m+1\\2(\nu-m),\ 2\nu-m+2\end{array}\Big|1-\rho^{2}\Big|\right).$$

$$\begin{aligned} & -\rho^2 \log \rho^2 \mathbf{1} - \rho^{2\nu - m - 1} \,\_2F\_1 \left( \begin{matrix} -m + 1, 2\nu - m \\ 2 \end{matrix} \vert \rho^2 \right) \\ & + - \mathbf{1}^m a\_k^{\nu, m} \frac{\mathbf{1} - \rho^{2\nu - m - 1}}{2\nu - m + 1} \,\_3F\_2 \left( \begin{matrix} -m + 1, 2\nu - m, 2\nu - m + 1 \\ 2\nu - m, 2\nu - m + 2 \end{matrix} \vert \mathbf{1} - \rho^2 \right). \end{aligned}$$

$$
\mathcal{L}\_{\boldsymbol{\nu}}(\phi\_{k}^{\boldsymbol{\nu},m})(\boldsymbol{z}) = \frac{a\_{k}^{\boldsymbol{\nu},m}}{2(2\boldsymbol{\nu}-m+1)} \sqrt{\frac{2(\boldsymbol{\nu}-m)-\mathbf{1}}{\boldsymbol{\pi}}} (\mathbf{1}-\boldsymbol{\rho}^{2})^{2\boldsymbol{\nu}-m-1}{}\_{\boldsymbol{\mathcal{B}}}F\_{2} \begin{pmatrix} -m+1, 2\boldsymbol{\nu}-m, 2\boldsymbol{\nu}-m+1\\ 2(\boldsymbol{\nu}-m), 2\boldsymbol{\nu}-m+2 \end{pmatrix} \mathbf{1} \\
$$

$$
\begin{split}
\mathscr{L}\_{\nu}(\delta\_{k}^{\omega,m})(x) &= \gamma\_{k}^{\omega,m} \int\_{\mathbb{D}} \left(1-|\boldsymbol{\varepsilon}|^{2}\right)^{2\boldsymbol{\varepsilon}-m-2} \boldsymbol{\xi}^{k-m} \log\left(\frac{1}{|\boldsymbol{z}-\boldsymbol{\varepsilon}|}\right) P\_{m}^{(k-m,\,2(\boldsymbol{\varepsilon}-m)-1)}\left(1-2|\boldsymbol{\varepsilon}|^{2}\right) d\mu(\boldsymbol{\xi}) \\
&= \gamma\_{k}^{\omega,m} \int\_{0}^{1} (1-r^{2})^{2\boldsymbol{\varepsilon}-m-2} r^{k-m+1} P\_{m}^{(k-m,\,2(\boldsymbol{\varepsilon}-m)-1)}\left(1-2r^{2}\right) \int\_{0}^{2\pi} \epsilon^{i(k-m)\theta} \log\left(\frac{1}{|\boldsymbol{z}-\boldsymbol{\varepsilon}|^{\theta}}\right) d\theta dr \\
&= \frac{\pi \gamma\_{k}^{\omega,m} e^{i(k-m)\theta}}{2(k-m)} \int\_{0}^{1} (1-r^{2})^{2\boldsymbol{\varepsilon}-m-2} r^{k-m} P\_{m}^{(k-m,\,2(\boldsymbol{\varepsilon}-m)-1)}\left(1-2r^{2}\right) \left(\left(\frac{r}{\rho}\right)^{k-m} \wedge \left(\frac{\rho}{r}\right)^{k-m}\right) dr \\
&= \frac{\pi \gamma\_{k}^{\omega,m} e^{i(k-m)\theta}}{2(k-m)} \left(\int\_{0}^{\rho} (1-r^{2})^{2\boldsymbol{\varepsilon}-m-2} r^{k-m} P\_{m}^{(k-m,\,2(\boldsymbol{\varepsilon}-m)-1)}\left(1-2r^{2}\right) \left(\left(\frac{r}{\rho}\right)^{k-m} \wedge \left(\frac{\rho}{r}\right)^{k-m}\right) dr^{2} \\
&\quad +$$

$$I\_3 = \int\_0^\rho \left(1 - r^2\right)^{2\nu - m - 2} r^{k - m} P\_m^{(k - m, \ 2(\nu - m) - 1)} \left(1 - 2r^2\right) \left(\left(\frac{r}{\rho}\right)^{k - m} \wedge \left(\frac{\rho}{r}\right)^{k - m}\right) dr^2 \dots$$

$$I\_4 = \int\_{\rho}^{1} \left(\mathbf{1} - r^2\right)^{2\nu - m - 2} r^{k - m} P\_m^{(k - m, \ 2(\nu - m) - 1)} \left(\mathbf{1} - 2r^2\right) \left(\left(\frac{r}{\rho}\right)^{k - m} \wedge \left(\frac{\rho}{r}\right)^{k - m}\right) dr^2 \dots$$

$$I\_3 = \frac{\rho^{m-k}(\mathbf{1} + k - m)\_m}{m!} \int\_0^{\rho^2} t^{k-m} (\mathbf{1} - t)^{2\nu - m - 2} \,\_2F\_1 \begin{pmatrix} -m, 2(\nu - m) + k & |t| \, d\nu \\ \mathbf{1} + k - m & |t| \, d\nu \end{pmatrix} dt.$$

$$
\int \mathfrak{x}^{\varepsilon-1} (\mathbf{1} - \mathfrak{x})^{b-\varepsilon-1} \,\_2F\_1 \left( \begin{matrix} a, b \\ c \end{matrix} \; \middle| \; \mathbf{x} \right) d\mathbf{x} = \frac{\mathbf{1}}{c} \mathfrak{x}^{\varepsilon} (\mathbf{1} - \mathfrak{x})^{b-\varepsilon} \,\_2F\_1 \left( \begin{matrix} a+\mathbf{1}, b \\ c+\mathbf{1} \end{matrix} \; \middle| \; \mathbf{x} \right),
$$

$$I\_3 = \frac{(1+k-m)\_m}{m!(k-m+1)} \rho^{k-m+2} (1-\rho^2)^{2\nu-m-1} \,\_2F\_1 \left( \begin{matrix} -m+1, \ 2(\nu-m)+k \\ 2+k-m \end{matrix} \; | \rho^2 \right)$$

$$\begin{split} I\_{4} &= \int\_{\rho}^{1} \left( \mathbf{1} - r^{2} \right)^{2\nu - m - 2} r^{k - m} P\_{m}^{(k - m, \ 2(\nu - m) - 1)} \left( \mathbf{1} - 2r^{2} \right) \left( \left( \frac{r}{\rho} \right)^{k - m} \wedge \left( \frac{\rho}{r} \right)^{k - m} \right) dr^{2} \\ &= \frac{\rho^{k - m} (\mathbf{1} + k - m)\_{m}}{2m!} \int\_{\rho^{2}}^{1} (\mathbf{1} - t)^{2\nu - m - 2} \,\_{2}F\_{1} \begin{pmatrix} -m, \ 2(\nu - m) + k \\ \mathbf{1} + k - m \end{pmatrix} dt. \end{split}$$

*The Singular Values of the Logarithmic Potential Transform on Bound States Spaces DOI: http://dx.doi.org/10.5772/intechopen.107090*
