Heine's identity

In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine^[1] is a Fourier expansion of a reciprocal square root which Heine presented as

$\frac{1}{\sqrt{z-\cos\psi}}=\frac{\sqrt{2}}{\pi}\sum_{m=-\infty}^\infty Q_{m-\frac12}(z) e^{im\psi}$

where^[2] $Q_{m-\frac12}$ is a Legendre function of the second kind, which has degree, m − ¹/₂, a half-integer, and argument, z, real and greater than one. This expression can be generalized^[3] for arbitrary half-integer powers as follows

$(z-\cos\psi)^{n-\frac12}=\sqrt{\frac{2}{\pi}}\frac{(z^2-1)^{\frac{n}{2}}}{\Gamma(\frac12-n)} \sum_{m=-\infty}^{\infty}\frac{\Gamma(m-n+\frac12)}{\Gamma(m+n+\frac12)}Q_{m-\frac12}^n(z)e^{im\psi},$

where $\scriptstyle\,\Gamma$ is the Gamma function.

References

1. Heine, Heinrich Eduard (1881). Handbuch der Kugelfunctionen, Theorie und Andwendungen. Wuerzburg: Physica-Verlag. (See page 286)
2. Cohl, Howard S.; J.E. Tohline; A.R.P. Rau; H.M. Srivastava (2000). "Developments in determining the gravitational potential using toroidal functions". Astronomische Nachrichten 321 (5/6): 363–372. Bibcode:2000AN....321..363C. doi:10.1002/1521-3994(200012)321:5/6<363::AID-ASNA363>3.0.CO;2-X. ISSN 0004-6337.
3. Cohl, H. S. (2003). "Portent of Heine's Reciprocal Square Root Identity". 3D Stellar Evolution, ASP Conference Proceedings, held 22-26 July 2002 at University of California Davis, Livermore, California, USA. Edited by Sylvain Turcotte, Stefan C. Keller and Robert M. Cavallo. ISBN 1-58381-140-0.