# §31.10 Integral Equations and Representations

## §31.10(i) Type I

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 $\mathcal{D}_{z}=z(z-1)(z-a)(\ifrac{{\partial}^{2}}{{\partial z}^{2}})+\left(% \gamma(z-1)(z-a)+\delta z(z-a)+\epsilon z(z-1)\right)(\ifrac{\partial}{% \partial z})+\alpha\beta z.$

The contour $C$ must be such that

 31.10.5 $\left.p(t)\left(\frac{\partial\mathcal{K}}{\partial t}w(t)-\mathcal{K}\frac{% \mathrm{d}w(t)}{\mathrm{d}t}\right)\right|_{C}=0,$

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

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 ${\sin^{2}}\theta\left(\frac{{\partial}^{2}\mathcal{K}}{{\partial\theta}^{2}}+% \left((1-2\gamma)\tan\theta+2(\delta+\epsilon-\tfrac{1}{2})\cot\theta\right)% \frac{\partial\mathcal{K}}{\partial\theta}-4\alpha\beta\mathcal{K}\right)+% \frac{{\partial}^{2}\mathcal{K}}{{\partial\phi}^{2}}+\left((1-2\delta)\cot\phi% -(1-2\epsilon)\tan\phi\right)\frac{\partial\mathcal{K}}{\partial\phi}=0.$

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

 31.10.9 $\mathcal{K}(\theta,\phi)=P\begin{Bmatrix}0&1&\infty&\\ 0&\frac{1}{2}-\delta-\sigma&\alpha&{\cos^{2}}\theta\\ 1-\gamma&\frac{1}{2}-\epsilon+\sigma&\beta&\end{Bmatrix}\*P\begin{Bmatrix}0&1&% \infty&\\ 0&0&-\frac{1}{2}+\delta+\sigma&{\cos^{2}}\phi\\ 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix},$

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

Let

 31.10.10 $\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma}\*{{}_{2}F_{1}}\left({\frac% {1}{2}-\delta-\sigma+\alpha,\frac{1}{2}-\delta-\sigma+\beta\atop\gamma};\frac{% zt}{a}\right)\*{{}_{2}F_{1}}\left({-\frac{1}{2}+\delta+\sigma,-\frac{1}{2}+% \epsilon-\sigma\atop\delta};\frac{a(z-1)(t-1)}{(a-1)(zt-a)}\right),$

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

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 $\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma}\left(\ifrac{zt}{a}\right)^% {-\frac{1}{2}+\delta+\sigma-\alpha}\*{{}_{2}F_{1}}\left({\frac{1}{2}-\delta-% \sigma+\alpha,\frac{3}{2}-\delta-\sigma+\alpha-\gamma\atop\alpha-\beta+1};% \frac{a}{zt}\right)\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&-\frac{1}{2}+\delta+\sigma&\dfrac{(z-a)(t-a)}{(1-a)(zt-a)}\\ 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix}.$

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

## §31.10(ii) Type II

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 $\displaystyle\left.p(t)\left(\frac{\partial\mathcal{K}}{\partial t}w(t)-% \mathcal{K}\frac{\mathrm{d}w(t)}{\mathrm{d}t}\right)\right|_{C_{1}}$ $\displaystyle=0,$ and 31.10.16 $\displaystyle\left.p(s)\left(\frac{\partial\mathcal{K}}{\partial s}w(s)-% \mathcal{K}\frac{\mathrm{d}w(s)}{\mathrm{d}s}\right)\right|_{C_{2}}$ $\displaystyle=0,$

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

### Kernel Functions

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 $\frac{{\partial}^{2}\mathcal{K}}{{\partial u}^{2}}+\frac{{\partial}^{2}% \mathcal{K}}{{\partial v}^{2}}+\frac{{\partial}^{2}\mathcal{K}}{{\partial w}^{% 2}}+\frac{2\gamma-1}{u}\frac{\partial\mathcal{K}}{\partial u}+\frac{2\delta-1}% {v}\frac{\partial\mathcal{K}}{\partial v}+\frac{2\epsilon-1}{w}\frac{\partial% \mathcal{K}}{\partial w}=0.$

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

 31.10.19 $\mathcal{K}(u,v,w)=u^{1-\gamma}v^{1-\delta}w^{1-\epsilon}\mathscr{C}_{1-\gamma% }\left(u\sqrt{\sigma_{1}}\right)\*\mathscr{C}_{1-\delta}\left(v\sqrt{\sigma_{2% }}\right)\mathscr{C}_{1-\epsilon}\left(\mathrm{i}w\sqrt{\sigma_{1}+\sigma_{2}}% \right),$

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

### Transformation of Independent Variable

A further change of variables, to spherical coordinates,

 31.10.20 $\displaystyle u$ $\displaystyle=r\cos\theta,$ $\displaystyle v$ $\displaystyle=r\sin\theta\sin\phi,$ $\displaystyle w$ $\displaystyle=r\sin\theta\cos\phi,$

 31.10.21 $\frac{{\partial}^{2}\mathcal{K}}{{\partial r}^{2}}+\frac{2(\gamma+\delta+% \epsilon)-1}{r}\frac{\partial\mathcal{K}}{\partial r}+\frac{1}{r^{2}}\frac{{% \partial}^{2}\mathcal{K}}{{\partial\theta}^{2}}+\frac{(2(\delta+\epsilon)-1)% \cot\theta-(2\gamma-1)\tan\theta}{r^{2}}\frac{\partial\mathcal{K}}{\partial% \theta}+\frac{1}{r^{2}{\sin^{2}}\theta}\frac{{\partial}^{2}\mathcal{K}}{{% \partial\phi}^{2}}+\frac{(2\delta-1)\cot\phi-(2\epsilon-1)\tan\phi}{r^{2}{\sin% ^{2}}\theta}\frac{\partial\mathcal{K}}{\partial\phi}=0.$

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

 31.10.22 $\mathcal{K}(r,\theta,\phi)=r^{m}{\sin^{2p}}\theta P\begin{Bmatrix}0&1&\infty&% \\ 0&0&a&{\cos^{2}}\theta\\ \tfrac{1}{2}(3-\gamma)&c&b&\end{Bmatrix}\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&a^{\prime}&{\cos^{2}}\phi\\ 1-\epsilon&1-\delta&b^{\prime}&\end{Bmatrix},$

with

 31.10.23 $\displaystyle m^{2}+2(\alpha+\beta)m-\sigma_{1}$ $\displaystyle=0,$ $\displaystyle p^{2}+(\alpha+\beta-\gamma-\tfrac{1}{2})p-\tfrac{1}{4}\sigma_{2}$ $\displaystyle=0,$ $\displaystyle a+b$ $\displaystyle=2(\alpha+\beta+p)-1,$ $\displaystyle ab$ $\displaystyle=p^{2}-p(1-\alpha-\beta)-\tfrac{1}{4}\sigma_{1},$ $\displaystyle c$ $\displaystyle=\gamma-\tfrac{1}{2}-2(\alpha+\beta+p),$ $\displaystyle a^{\prime}+b^{\prime}$ $\displaystyle=\delta+\epsilon-1,$ $\displaystyle a^{\prime}b^{\prime}$ $\displaystyle=-\tfrac{1}{4}\sigma_{2},$ ⓘ 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).