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

§31.10 Integral Equations and Representations

Keywords:: Heun functions, Heun’s equation, integral equations
Permalink:: http://dlmf.nist.gov/31.10

§31.10(i) Type I
§31.10(ii) Type II

§31.10(i) Type I

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

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

31.10.1 $W(z)=\int_{C}\mathcal{K}(z,t)w(t)\rho(t)dt$

Symbols:: : differential of , $\int$ : integral, : complex variable, : solution, : 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
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for a suitable contour . The weight function is given by

31.10.2 $\rho(t)=t^{{\gamma-1}}(t-1)^{{\delta-1}}(t-a)^{{\epsilon-1}},$

Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : complex parameter and $\rho(t)$ : weight function
Permalink:: http://dlmf.nist.gov/31.10.E2
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and the kernel $\mathcal{K}(z,t)$ is a solution of the partial differential equation

31.10.3 $(\mathcal{D}_{z}-\mathcal{D}_{t})\mathcal{K}=0,$

Symbols:: : complex variable, $\mathcal{K}(z,t)$ : kernel and $\mathcal{D}_{z}$ : Heun’s operator
Permalink:: http://dlmf.nist.gov/31.10.E3
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where $\mathcal{D}_{z}$ is Heun’s operator in the variable :

31.10.4 $\mathcal{D}_{z}=z(z-1)(z-a)(\ifrac{{\partial}^{2}}{{\partial z}^{2}})+\left(% \gamma(z-1)(z-a)+\delta z(z-a)+\epsilon z(z-1)\right)(\ifrac{\partial}{% \partial z})+\alpha\beta z.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $\mathcal{D}_{z}$ : Heun’s operator
Referenced by:: §31.10(ii)
Permalink:: http://dlmf.nist.gov/31.10.E4
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The contour must be such that

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

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : contour and $\mathcal{K}(z,t)$ : kernel
Permalink:: http://dlmf.nist.gov/31.10.E5
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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 : complex parameter
Referenced by:: §31.10(ii)
Permalink:: http://dlmf.nist.gov/31.10.E6
Encodings:: TeX, pMML, png

¶ Kernel Functions

Keywords:: Heun polynomials, Heun’s equation, kernel equations, kernel functions

Set

31.10.7

$\mathop{\cos\/}\nolimits\theta=\left(\frac{zt}{a}\right)^{{1/2}},$

$\mathop{\sin\/}\nolimits\theta\mathop{\cos\/}\nolimits\phi=i\left(\frac{(z-a)(% t-a)}{a(1-a)}\right)^{{1/2}},$

$\mathop{\sin\/}\nolimits\theta\mathop{\sin\/}\nolimits\phi=\left(\frac{(z-1)(t% -1)}{1-a}\right)^{{1/2}}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, : complex parameter, $\theta$ : angle and $\phi$ : angle
Permalink:: http://dlmf.nist.gov/31.10.E7
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The kernel $\mathcal{K}$ must satisfy

31.10.8 ${\mathop{\sin\/}\nolimits^{{2}}}\theta\left(\frac{{\partial}^{2}\mathcal{K}}{{% \partial\theta}^{2}}+\left((1-2\gamma)\mathop{\tan\/}\nolimits\theta+2(\delta+% \epsilon-\tfrac{1}{2})\mathop{\cot\/}\nolimits\theta\right)\frac{\partial% \mathcal{K}}{\partial\theta}-4\alpha\beta\mathcal{K}\right)+\frac{{\partial}^{% 2}\mathcal{K}}{{\partial\phi}^{2}}+\left((1-2\delta)\mathop{\cot\/}\nolimits% \phi-(1-2\epsilon)\mathop{\tan\/}\nolimits\phi\right)\frac{\partial\mathcal{K}% }{\partial\phi}=0.$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : 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
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The solutions of (31.10.8) are given in terms of the Riemann $\mathop{P\/}\nolimits$ -symbol (see §15.11(i)) as

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

Symbols:: $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}$ : Riemann’s $\mathop{P\/}\nolimits$ -symbol for solutions of the generalized hypergeometric differential equation, $\mathop{\cos\/}\nolimits z$ : 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
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where $\sigma$ is a separation constant. For integral equations satisfied by the Heun polynomial $\mathop{\mathit{Hp}_{{n,m}}\/}\nolimits\!\left(z\right)$ we have $\sigma=\frac{1}{2}-\delta-j$ , $j=0,1,\dots,n$ .

For suitable choices of the branches of the -symbols in (31.10.9) and the contour , 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 $\mathcal{K}(z,t)=(zt-a)^{{\frac{1}{2}-\delta-\sigma}}\*\mathop{{{}_{{2}}F_{{1}% }}\/}\nolimits\!\left({\frac{1}{2}-\delta-\sigma+\alpha,\frac{1}{2}-\delta-% \sigma+\beta\atop\gamma};\frac{zt}{a}\right)\*\mathop{{{}_{{2}}F_{{1}}}\/}% \nolimits\!\left({-\frac{1}{2}+\delta+\sigma,-\frac{1}{2}+\epsilon-\sigma\atop% \delta};\frac{a(z-1)(t-1)}{(a-1)(zt-a)}\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : 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

where $\realpart{\gamma}>0$ , $\realpart{\delta}>0$ , and 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 $w_{m}(z)=\mathop{(0,1)\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{m};\alpha,% \beta,\gamma,\delta;z\right)$ and $\kappa_{m}$ is the corresponding eigenvalue.

¶ Example 2

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

31.10.11 $\mathcal{K}(z,t)=(zt-a)^{{\frac{1}{2}-\delta-\sigma}}\left(\ifrac{zt}{a}\right% )^{{-\frac{1}{2}+\delta+\sigma-\alpha}}\*\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits% \!\left({\frac{1}{2}-\delta-\sigma+\alpha,\frac{3}{2}-\delta-\sigma+\alpha-% \gamma\atop\alpha-\beta+1};\frac{a}{zt}\right)\*\mathop{P\/}\nolimits\!\begin{% Bmatrix}0&1&\infty&\\ 0&0&-\frac{1}{2}+\delta+\sigma&\dfrac{(z-a)(t-a)}{(1-a)(zt-a)}\\ 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix}.$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}$ : Riemann’s $\mathop{P\/}\nolimits$ -symbol for solutions of the generalized hypergeometric differential equation, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : 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
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Here $\tilde{\kappa}_{m}$ is a normalization constant and is the contour of Example 1.

§31.10(ii) Type II

Notes:: See Sleeman (1969). An error in this reference is corrected here.
Permalink:: http://dlmf.nist.gov/31.10.ii

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

31.10.12 $W(z)=\int_{{C_{1}}}\int_{{C_{2}}}\mathcal{K}(z;s,t)w(s)w(t)\rho(s,t)dsdt$

Symbols:: : differential of , $\int$ : integral, : complex variable, : solution, $C_{1}$ , $C_{2}$ : contours, $\rho(s,t)$ : weight function, $\mathcal{K}(z;s,t)$ : kernel and
Permalink:: http://dlmf.nist.gov/31.10.E12
Encodings:: TeX, pMML, png

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

31.10.13 $\rho(s,t)=(s-t)(st)^{{\gamma-1}}\left((1-s)(1-t)\right)^{{\delta-1}}\*\left((1% -(s/a))(1-(t/a))\right)^{{\epsilon-1}},$

Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, : complex parameter and $\rho(s,t)$ : weight function
Permalink:: http://dlmf.nist.gov/31.10.E13
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and the kernel $\mathcal{K}(z;s,t)$ is a solution of the partial differential equation

31.10.14 $\left((t-z)\mathcal{D}_{s}+(z-s)\mathcal{D}_{t}+(s-t)\mathcal{D}_{z}\right)% \mathcal{K}=0,$

Symbols:: : complex variable, $\mathcal{K}(z;s,t)$ : kernel and $\mathcal{D}_{z}$ : Heun’s operator
Permalink:: http://dlmf.nist.gov/31.10.E14
Encodings:: TeX, pMML, png

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

31.10.15 $\left.p(t)\left(\frac{\partial\mathcal{K}}{\partial t}w(t)-\mathcal{K}\frac{dw% (t)}{dt}\right)\right|_{{C_{1}}}=0,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $C_{1}$ , $C_{2}$ : contours, $\mathcal{K}(z;s,t)$ : kernel and
Permalink:: http://dlmf.nist.gov/31.10.E15
Encodings:: TeX, pMML, png

and

31.10.16 $\left.p(s)\left(\frac{\partial\mathcal{K}}{\partial s}w(s)-\mathcal{K}\frac{dw% (s)}{ds}\right)\right|_{{C_{2}}}=0,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $C_{1}$ , $C_{2}$ : contours, $\mathcal{K}(z;s,t)$ : kernel and
Permalink:: http://dlmf.nist.gov/31.10.E16
Encodings:: TeX, pMML, png

where is given by (31.10.6).

¶ Kernel Functions

Keywords:: Heun’s equation, kernel equations, kernel functions

Set

31.10.17

$u=\frac{(stz)^{{1/2}}}{a},$

$v=\left(\frac{(s-1)(t-1)(z-1)}{1-a}\right)^{{1/2}},$

$w=i\left(\frac{(s-a)(t-a)(z-a)}{a(1-a)}\right)^{{1/2}}.$

Symbols:: : complex variable, : complex parameter, , and
Permalink:: http://dlmf.nist.gov/31.10.E17
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

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:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathcal{K}(z;s,t)$ : kernel, , and
Permalink:: http://dlmf.nist.gov/31.10.E18
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathcal{K}(z;s,t)$ : kernel, , , and $\sigma_{1}$ , $\sigma_{2}$ : separation constants
Permalink:: http://dlmf.nist.gov/31.10.E19
Encodings:: TeX, pMML, png

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

¶ Transformation of Independent Variable

Keywords:: Heun functions, Heun’s equation, integral equations

A further change of variables, to spherical coordinates,

31.10.20

$u=r\mathop{\cos\/}\nolimits\theta,$

$v=r\mathop{\sin\/}\nolimits\theta\mathop{\sin\/}\nolimits\phi,$

$w=r\mathop{\sin\/}\nolimits\theta\mathop{\cos\/}\nolimits\phi,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, , , , : radius, $\theta$ : angle and $\phi$ : angle
Permalink:: http://dlmf.nist.gov/31.10.E20
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

leads to the kernel equation

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

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathcal{K}(z;s,t)$ : kernel, : radius, $\theta$ : angle and $\phi$ : angle
Permalink:: http://dlmf.nist.gov/31.10.E21
Encodings:: TeX, pMML, png

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

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