# §31.10 Integral Equations and Representations

## §31.10(i) Type I

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

for a suitable contour . The weight function is given by

and the kernel is a solution of the partial differential equation

31.10.3

where is Heun’s operator in the variable :

The contour must be such that

where

### ¶ Kernel Functions

Set

31.10.7

The kernel must satisfy

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

31.10.9

where is a separation constant. For integral equations satisfied by the Heun polynomial we have , .

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

where , , 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 and , where and is the corresponding eigenvalue.

### ¶ Example 2

Fuchs–Frobenius solutions are represented in terms of Heun functions by (31.10.1) with , , and with kernel chosen from

Here is a normalization constant and is the contour of Example 1.

## §31.10(ii) Type II

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

for suitable contours , . The weight function is

and the kernel is a solution of the partial differential equation

31.10.14

where is given by (31.10.4). The contours , must be chosen so that

where is given by (31.10.6).

### ¶ Kernel Functions

Set

31.10.17

The kernel must satisfy

This equation can be solved in terms of cylinder functions 10.2(ii)):

where and are separation constants.

### ¶ Transformation of Independent Variable

A further change of variables, to spherical coordinates,

31.10.20