31 Heun Functions Properties 31.9 Orthogonality 31.11 Expansions in Series of Hypergeometric Functions

§31.10 Integral Equations and Representations

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Keywords:: Heun functions, Heun’s equation, integral equations, integral equations and representations, integral representation of solutions
Permalink:: http://dlmf.nist.gov/31.10
See also:: Annotations for Ch.31

§31.10(i) Type I
§31.10(ii) Type II

§31.10(i) Type I

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Keywords:: Heun’s operator
Notes:: See Lambe and Ward (1934) and Erdélyi (1942a). See also Valent (1986).
Permalink:: http://dlmf.nist.gov/31.10.i
See also:: Annotations for §31.10 and Ch.31

If $w(z)$ is a solution of Heun’s equation, then another solution $W(z)$ (possibly a multiple of $w(z)$ ) can be represented as

31.10.1		$W(z)=\int_{C}\mathcal{K}(z,t)w(t)\rho(t)\,\mathrm{d}t$
ⓘ Defines: $W(z)$ : solution (locally) Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $C$ : contour, $\rho(t)$ : weight function and $\mathcal{K}(z,t)$ : kernel Referenced by: §31.10(i), §31.10(i) Permalink: http://dlmf.nist.gov/31.10.E1 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10 and Ch.31

for a suitable contour $C$ . The weight function is given by

31.10.2		$\rho(t)=t^{\gamma-1}(t-1)^{\delta-1}(t-a)^{\epsilon-1},$
ⓘ Defines: $\rho(t)$ : weight function (locally) Symbols: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter Permalink: http://dlmf.nist.gov/31.10.E2 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10 and Ch.31

and the kernel $\mathcal{K}(z,t)$ is a solution of the partial differential equation

31.10.3		$(\mathcal{D}_{z}-\mathcal{D}_{t})\mathcal{K}=0,$
ⓘ Defines: $\mathcal{K}(z,t)$ : kernel (locally) Symbols: $z$ : complex variable and $\mathcal{D}_{z}$ : Heun’s operator Permalink: http://dlmf.nist.gov/31.10.E3 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10 and Ch.31

where $\mathcal{D}_{z}$ is Heun’s operator in the variable $z$ :

31.10.4		$\mathcal{D}_{z}=z(z-1)(z-a)(\ifrac{{\partial}^{2}}{{\partial z}^{2}})+\left(% \gamma(z-1)(z-a)+\delta z(z-a)+\epsilon z(z-1)\right)(\ifrac{\partial}{% \partial z})+\alpha\beta z.$
ⓘ Defines: $\mathcal{D}_{z}$ : Heun’s operator (locally) Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter Referenced by: §31.10(ii) Permalink: http://dlmf.nist.gov/31.10.E4 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10 and Ch.31

The contour $C$ must be such that

31.10.5		$\left.p(t)\left(\frac{\partial\mathcal{K}}{\partial t}w(t)-\mathcal{K}\frac{% \mathrm{d}w(t)}{\mathrm{d}t}\right)\right\|_{C}=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $C$ : contour and $\mathcal{K}(z,t)$ : kernel Permalink: http://dlmf.nist.gov/31.10.E5 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10 and Ch.31

where

31.10.6		$p(t)=t^{\gamma}(t-1)^{\delta}(t-a)^{\epsilon}.$
ⓘ Symbols: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter Referenced by: §31.10(ii) Permalink: http://dlmf.nist.gov/31.10.E6 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10 and Ch.31

Kernel Functions

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Keywords:: Heun polynomials, Heun’s equation, integral equations and representations, integral representation of solutions, kernel equations, kernel functions, separation constant
See also:: Annotations for §31.10(i), §31.10 and Ch.31

Set

31.10.7	$\displaystyle\cos\theta$	$\displaystyle=\left(\frac{zt}{a}\right)^{1/2},$
	$\displaystyle\sin\theta\cos\phi$	$\displaystyle=\mathrm{i}\left(\frac{(z-a)(t-a)}{a(1-a)}\right)^{1/2},$
	$\displaystyle\sin\theta\sin\phi$	$\displaystyle=\left(\frac{(z-1)(t-1)}{1-a}\right)^{1/2}.$
ⓘ Defines: $\theta$ : angle (locally) and $\phi$ : angle (locally) Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $a$ : complex parameter Permalink: http://dlmf.nist.gov/31.10.E7 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

The kernel $\mathcal{K}$ must satisfy

31.10.8		${\sin}^{2}\theta\left(\frac{{\partial}^{2}\mathcal{K}}{{\partial\theta}^{2}}+% \left((1-2\gamma)\tan\theta+2(\delta+\epsilon-\tfrac{1}{2})\cot\theta\right)% \frac{\partial\mathcal{K}}{\partial\theta}-4\alpha\beta\mathcal{K}\right)+% \frac{{\partial}^{2}\mathcal{K}}{{\partial\phi}^{2}}+\left((1-2\delta)\cot\phi% -(1-2\epsilon)\tan\phi\right)\frac{\partial\mathcal{K}}{\partial\phi}=0.$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\mathcal{K}(z,t)$ : kernel, $\theta$ : angle and $\phi$ : angle Referenced by: §31.10(i) Permalink: http://dlmf.nist.gov/31.10.E8 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

The solutions of (31.10.8) are given in terms of the Riemann $P$ -symbol (see §15.11(i)) as

31.10.9		$\mathcal{K}(\theta,\phi)=P\begin{Bmatrix}0&1&\infty&\\[1.0pt] 0&\frac{1}{2}-\delta-\sigma&\alpha&{\cos}^{2}\theta\\[1.0pt] 1-\gamma&\frac{1}{2}-\epsilon+\sigma&\beta&\end{Bmatrix}\*P\begin{Bmatrix}0&1&% \infty&\\[1.0pt] 0&0&-\frac{1}{2}+\delta+\sigma&{\cos}^{2}\phi\\[1.0pt] 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix},$
ⓘ Symbols: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $\cos\NVar{z}$ : cosine function, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\mathcal{K}(z,t)$ : kernel, $\theta$ : angle, $\phi$ : angle and $\sigma$ : separation constant Referenced by: §31.10(i) Permalink: http://dlmf.nist.gov/31.10.E9 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

where $\sigma$ is a separation constant. For integral equations satisfied by the Heun polynomial $\mathit{Hp}_{n,m}\left(z\right)$ we have $\sigma=\frac{1}{2}-\delta-j$ , $j=0,1,\dots,n$ .

For suitable choices of the branches of the $P$ -symbols in (31.10.9) and the contour $C$ , we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution).

Example 1

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See also:: Annotations for §31.10(i), §31.10 and Ch.31

Let

31.10.10		$\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma}\{{}_{2}F_{1}}\left({\frac% {1}{2}-\delta-\sigma+\alpha,\frac{1}{2}-\delta-\sigma+\beta\atop\gamma};\frac{% zt}{a}\right)\{{}_{2}F_{1}}\left({-\frac{1}{2}+\delta+\sigma,-\frac{1}{2}+% \epsilon-\sigma\atop\delta};\frac{a(z-1)(t-1)}{(a-1)(zt-a)}\right),$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\mathcal{K}(z,t)$ : kernel and $\sigma$ : separation constant Permalink: http://dlmf.nist.gov/31.10.E10 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

where $\Re\gamma>0$ , $\Re\delta>0$ , and $C$ be the Pochhammer double-loop contour about 0 and 1 (as in §31.9(i)). Then the integral equation (31.10.1) is satisfied by $w(z)=w_{m}(z)$ and $W(z)=\kappa_{m}w_{m}(z)$ , where $w_{m}(z)=(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ and $\kappa_{m}$ is the corresponding eigenvalue.

Example 2

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See also:: Annotations for §31.10(i), §31.10 and Ch.31

Fuchs–Frobenius solutions $W_{m}(z)=\tilde{\kappa}_{m}z^{-\alpha}\mathit{H\!\ell}\left(1/a,q_{m};\alpha,% \alpha-\gamma+1,\alpha-\beta+1,\delta;1/z\right)$ are represented in terms of Heun functions $w_{m}(z)=(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$ by (31.10.1) with $W(z)=W_{m}(z)$ , $w(z)=w_{m}(z)$ , and with kernel chosen from

31.10.11		$\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma}\left(\ifrac{zt}{a}\right)^% {-\frac{1}{2}+\delta+\sigma-\alpha}\{{}_{2}F_{1}}\left({\frac{1}{2}-\delta-% \sigma+\alpha,\frac{3}{2}-\delta-\sigma+\alpha-\gamma\atop\alpha-\beta+1};% \frac{a}{zt}\right)\P\begin{Bmatrix}0&1&\infty&\\ 0&0&-\frac{1}{2}+\delta+\sigma&\dfrac{(z-a)(t-a)}{(1-a)(zt-a)}\\ 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix}.$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\mathcal{K}(z,t)$ : kernel and $\sigma$ : separation constant Permalink: http://dlmf.nist.gov/31.10.E11 Encodings: TeX, pMML, png See also: Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

Here $\tilde{\kappa}_{m}$ is a normalization constant and $C$ is the contour of Example 1.

§31.10(ii) Type II

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Notes:: See Sleeman (1969). An error in this reference is corrected here.
Permalink:: http://dlmf.nist.gov/31.10.ii
See also:: Annotations for §31.10 and Ch.31

If $w(z)$ is a solution of Heun’s equation, then another solution $W(z)$ (possibly a multiple of $w(z)$ ) can be represented as

31.10.12		$W(z)=\int_{C_{1}}\int_{C_{2}}\mathcal{K}(z;s,t)w(s)w(t)\rho(s,t)\,\mathrm{d}s% \,\mathrm{d}t$
ⓘ Defines: $W(z)$ : solution (locally) Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $C_{1}$ , $C_{2}$ : contours, $\rho(s,t)$ : weight function, $\mathcal{K}(z;s,t)$ : kernel and $w$ Permalink: http://dlmf.nist.gov/31.10.E12 Encodings: TeX, pMML, png See also: Annotations for §31.10(ii), §31.10 and Ch.31

for suitable contours $C_{1}$ , $C_{2}$ . The weight function is

31.10.13		$\rho(s,t)=(s-t)(st)^{\gamma-1}\left((1-s)(1-t)\right)^{\delta-1}\*\left((1-(s/% a))(1-(t/a))\right)^{\epsilon-1},$
ⓘ Defines: $\rho(s,t)$ : weight function (locally) Symbols: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter Permalink: http://dlmf.nist.gov/31.10.E13 Encodings: TeX, pMML, png See also: Annotations for §31.10(ii), §31.10 and Ch.31

and the kernel $\mathcal{K}(z;s,t)$ is a solution of the partial differential equation

31.10.14		$\left((t-z)\mathcal{D}_{s}+(z-s)\mathcal{D}_{t}+(s-t)\mathcal{D}_{z}\right)% \mathcal{K}=0,$
ⓘ Defines: $\mathcal{K}(z;s,t)$ : kernel (locally) Symbols: $z$ : complex variable and $\mathcal{D}_{z}$ : Heun’s operator Permalink: http://dlmf.nist.gov/31.10.E14 Encodings: TeX, pMML, png See also: Annotations for §31.10(ii), §31.10 and Ch.31

where $\mathcal{D}_{z}$ is given by (31.10.4). The contours $C_{1}$ , $C_{2}$ must be chosen so that

31.10.15	$\displaystyle\left.p(t)\left(\frac{\partial\mathcal{K}}{\partial t}w(t)-% \mathcal{K}\frac{\mathrm{d}w(t)}{\mathrm{d}t}\right)\right\|_{C_{1}}$	$\displaystyle=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $C_{1}$ , $C_{2}$ : contours, $\mathcal{K}(z;s,t)$ : kernel and $w$ Permalink: http://dlmf.nist.gov/31.10.E15 Encodings: TeX, pMML, png See also: Annotations for §31.10(ii), §31.10 and Ch.31
and
31.10.16	$\displaystyle\left.p(s)\left(\frac{\partial\mathcal{K}}{\partial s}w(s)-% \mathcal{K}\frac{\mathrm{d}w(s)}{\mathrm{d}s}\right)\right\|_{C_{2}}$	$\displaystyle=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $C_{1}$ , $C_{2}$ : contours, $\mathcal{K}(z;s,t)$ : kernel and $w$ Permalink: http://dlmf.nist.gov/31.10.E16 Encodings: TeX, pMML, png See also: Annotations for §31.10(ii), §31.10 and Ch.31

where $p(t)$ is given by (31.10.6).

Kernel Functions

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Keywords:: Heun’s equation, integral representation of solutions, kernel equations, kernel functions
See also:: Annotations for §31.10(ii), §31.10 and Ch.31

Set

31.10.17	$\displaystyle u$	$\displaystyle=\frac{(stz)^{1/2}}{a},$
	$\displaystyle v$	$\displaystyle=\left(\frac{(s-1)(t-1)(z-1)}{1-a}\right)^{1/2},$
	$\displaystyle w$	$\displaystyle=i\left(\frac{(s-a)(t-a)(z-a)}{a(1-a)}\right)^{1/2}.$
ⓘ Defines: $u$ (locally), $v$ (locally) and $w$ (locally) Symbols: $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $a$ : complex parameter Permalink: http://dlmf.nist.gov/31.10.E17 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

The kernel $\mathcal{K}$ must satisfy

31.10.18		$\frac{{\partial}^{2}\mathcal{K}}{{\partial u}^{2}}+\frac{{\partial}^{2}% \mathcal{K}}{{\partial v}^{2}}+\frac{{\partial}^{2}\mathcal{K}}{{\partial w}^{% 2}}+\frac{2\gamma-1}{u}\frac{\partial\mathcal{K}}{\partial u}+\frac{2\delta-1}% {v}\frac{\partial\mathcal{K}}{\partial v}+\frac{2\epsilon-1}{w}\frac{\partial% \mathcal{K}}{\partial w}=0.$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathcal{K}(z;s,t)$ : kernel, $u$ , $v$ and $w$ Permalink: http://dlmf.nist.gov/31.10.E18 Encodings: TeX, pMML, png See also: Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

This equation can be solved in terms of cylinder functions $\mathscr{C}_{\nu}\left(z\right)$ (§10.2(ii)):

31.10.19		$\mathcal{K}(u,v,w)=u^{1-\gamma}v^{1-\delta}w^{1-\epsilon}\mathscr{C}_{1-\gamma% }\left(u\sqrt{\sigma_{1}}\right)\*\mathscr{C}_{1-\delta}\left(v\sqrt{\sigma_{2% }}\right)\mathscr{C}_{1-\epsilon}\left(\mathrm{i}w\sqrt{\sigma_{1}+\sigma_{2}}% \right),$
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\mathrm{i}$ : imaginary unit, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathcal{K}(z;s,t)$ : kernel, $u$ , $v$ , $w$ and $\sigma_{1}$ , $\sigma_{2}$ : separation constants Permalink: http://dlmf.nist.gov/31.10.E19 Encodings: TeX, pMML, png See also: Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

where $\sigma_{1}$ and $\sigma_{2}$ are separation constants.

Transformation of Independent Variable

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Keywords:: Heun functions, Heun’s equation, confluent forms, integral equations, integral equations and representations, integral representation of solutions
See also:: Annotations for §31.10(ii), §31.10 and Ch.31

A further change of variables, to spherical coordinates,

31.10.20	$\displaystyle u$	$\displaystyle=r\cos\theta,$
	$\displaystyle v$	$\displaystyle=r\sin\theta\sin\phi,$
	$\displaystyle w$	$\displaystyle=r\sin\theta\cos\phi,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $u$ , $v$ , $w$ , $r$ : radius, $\theta$ : angle and $\phi$ : angle Permalink: http://dlmf.nist.gov/31.10.E20 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

leads to the kernel equation

31.10.21		$\frac{{\partial}^{2}\mathcal{K}}{{\partial r}^{2}}+\frac{2(\gamma+\delta+% \epsilon)-1}{r}\frac{\partial\mathcal{K}}{\partial r}+\frac{1}{r^{2}}\frac{{% \partial}^{2}\mathcal{K}}{{\partial\theta}^{2}}+\frac{(2(\delta+\epsilon)-1)% \cot\theta-(2\gamma-1)\tan\theta}{r^{2}}\frac{\partial\mathcal{K}}{\partial% \theta}+\frac{1}{r^{2}{\sin}^{2}\theta}\frac{{\partial}^{2}\mathcal{K}}{{% \partial\phi}^{2}}+\frac{(2\delta-1)\cot\phi-(2\epsilon-1)\tan\phi}{r^{2}{\sin% }^{2}\theta}\frac{\partial\mathcal{K}}{\partial\phi}=0.$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathcal{K}(z;s,t)$ : kernel, $r$ : radius, $\theta$ : angle and $\phi$ : angle Permalink: http://dlmf.nist.gov/31.10.E21 Encodings: TeX, pMML, png See also: Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

This equation can be solved in terms of hypergeometric functions (§15.11(i)):

31.10.22		$\mathcal{K}(r,\theta,\phi)=r^{m}{\sin}^{2p}\theta P\begin{Bmatrix}0&1&\infty&% \\ 0&0&a&{\cos}^{2}\theta\\ \tfrac{1}{2}(3-\gamma)&c&b&\end{Bmatrix}\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&a^{\prime}&{\cos}^{2}\phi\\ 1-\epsilon&1-\delta&b^{\prime}&\end{Bmatrix},$
ⓘ Symbols: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $\mathcal{K}(z;s,t)$ : kernel, $r$ : radius, $\theta$ : angle, $\phi$ : angle, $b$ and $c$ Permalink: http://dlmf.nist.gov/31.10.E22 Encodings: TeX, pMML, png See also: Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

with

31.10.23	$\displaystyle m^{2}+2(\alpha+\beta)m-\sigma_{1}=0,$
	$\displaystyle p^{2}+(\alpha+\beta-\gamma-\tfrac{1}{2})p-\tfrac{1}{4}\sigma_{2}% =0,$
	$\displaystyle a+b$	$\displaystyle=2(\alpha+\beta+p)-1,$
	$\displaystyle ab$	$\displaystyle=p^{2}-p(1-\alpha-\beta)-\tfrac{1}{4}\sigma_{1},$
	$\displaystyle c$	$\displaystyle=\gamma-\tfrac{1}{2}-2(\alpha+\beta+p),$
	$\displaystyle a^{\prime}+b^{\prime}$	$\displaystyle=\delta+\epsilon-1,$
	$\displaystyle a^{\prime}b^{\prime}$	$\displaystyle=-\tfrac{1}{4}\sigma_{2},$
ⓘ Defines: $b$ (locally) and $c$ (locally) Symbols: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $\sigma_{1}$ , $\sigma_{2}$ : separation constants Permalink: http://dlmf.nist.gov/31.10.E23 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png See also: Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

and $\sigma_{1}$ and $\sigma_{2}$ are separation constants.

For integral equations for special confluent Heun functions (§31.12) see Kazakov and Slavyanov (1996).

31.9 Orthogonality 31.11 Expansions in Series of Hypergeometric Functions

NIST

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§31.10 Integral Equations and Representations

Contents

§31.10(i) Type I

Kernel Functions

Example 1

Example 2

§31.10(ii) Type II

Kernel Functions

Transformation of Independent Variable