If is a solution of Heun’s equation, then another solution (possibly a multiple of ) can be represented as
31.10.1 | |||
for a suitable contour . The weight function is given by
31.10.2 | |||
and the kernel is a solution of the partial differential equation
31.10.3 | |||
where is Heun’s operator in the variable :
31.10.4 | |||
The contour must be such that
31.10.5 | |||
where
31.10.6 | |||
Set
31.10.7 | ||||
The kernel must satisfy
31.10.8 | |||
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).
Fuchs–Frobenius solutions are represented in terms of Heun functions by (31.10.1) with , , and with kernel chosen from
31.10.11 | |||
Here is a normalization constant and is the contour of Example 1.
If is a solution of Heun’s equation, then another solution (possibly a multiple of ) can be represented as
31.10.12 | |||
for suitable contours , . The weight function is
31.10.13 | |||
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
31.10.15 | ||||
and | ||||
31.10.16 | ||||
where is given by (31.10.6).
Set
31.10.17 | ||||
The kernel must satisfy
31.10.18 | |||
This equation can be solved in terms of cylinder functions (§10.2(ii)):
31.10.19 | |||
where and are separation constants.
A further change of variables, to spherical coordinates,
31.10.20 | ||||
leads to the kernel equation
31.10.21 | |||
This equation can be solved in terms of hypergeometric functions (§15.11(i)):
31.10.22 | |||
with
31.10.23 | ||||
and and are separation constants.