§31.6 Path-Multiplicative Solutions

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Keywords:: Heun’s equation, path-multiplicative solutions
Referenced by:: §31.11(v), §31.8
Permalink:: http://dlmf.nist.gov/31.6
See also:: Annotations for Ch.31

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

31.6.1		$(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z% \right),$
$m=0,1,2,\dots$ ,
ⓘ Symbols: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\nu$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : 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 See also: Annotations for §31.6 and Ch.31

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 $z$ -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 ${\mathrm{e}}^{2\nu\pi i}$ . These solutions are called path-multiplicative. See Schmidt (1979).