Heine's identity explained

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$$\backslash frac\; =\; \backslash frac\backslash sum\_^\backslash infty\; Q\_(z)\; e^$$where^[2]

Q _{m - (12)/()} is a Legendre function of the second kind, which has degree, m − , a half-integer, and argument, z, real and greater than one. This expression can be generalized^[3] for arbitrary half-integer powers as follows$$(z-\backslash cos\backslash psi)^\; =\; \backslash sqrt\backslash frac\backslash sum\_^\; \backslash fracQ\_^n(z)e^,$$where \scriptstyle\Gamma is the Gamma function.

References

1. Book: Heine , Heinrich Eduard . Handbuch der Kugelfunctionen, Theorie und Andwendungen . . 1881 . Wuerzburg . (See page 286)
2. Cohl . Howard S. . J.E. Tohline . A.R.P. Rau . H.M. Srivastava . Developments in determining the gravitational potential using toroidal functions . 2000 . . 0004-6337 . 321 . 5/6 . 363–372 . 10.1002/1521-3994(200012)321:5/6<363::AID-ASNA363>3.0.CO;2-X. 2000AN....321..363C .
3. H. S. . Cohl . 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 . 293 . 1-58381-140-0 . 2003 .