31 Heun Functions Properties 31.5 Solutions Analytic at Three Singularities: Heun Polynomials 31.7 Relations to Other Functions

§31.6 Path-Multiplicative Solutions

Keywords:: Heun’s equation
Referenced by:: §31.11(v), §31.8
Permalink:: http://dlmf.nist.gov/31.6

A further extension of the notation (31.4.1) and (31.4.3) is given by

31.6.1 $\mathop{(s_{1},s_{2})\mathit{Hf}_{{m}}^{{\nu}}\/}\nolimits\!\left(a,q_{m};% \alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$ ,

Symbols:: $\mathop{(s_{1},s_{2})\mathit{Hf}_{{m}}\/}\nolimits\!\left(a,q_{m};\alpha,\beta% ,\gamma,\delta;z\right)$ : Heun functions, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\nu$ : real or complex parameter, : nonnegative integer, : complex parameter, : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $s_{j}$ : parameters
Permalink:: http://dlmf.nist.gov/31.6.E1
Encodings:: TeX, pMML, png

with $(s_{1},s_{2})\in\{0,1,a\}$ , but with another set of $\{q_{m}\}$ . This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the -plane that encircles $s_{1}$ and $s_{2}$ once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor $e^{{2\nu\pi i}}$ . These solutions are called path-multiplicative. See Schmidt (1979).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 31.5 Solutions Analytic at Three Singularities: Heun Polynomials 31.7 Relations to Other Functions