- Heine's identity
In
mathematical analysis , Heine's identity, named afterHeinrich Eduard Heine [cite book
last = Heine
first = Heinrich Eduard
title = Handbuch der Kugelfunctionen, Theorie und Andwendungen
publisher =Physica-Verlag
date = 1881
place = Wuerzburg "(See [http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=01410002&seq=&view=50&frames=0&pagenum=286 page 286] )"] is aFourier expansion of a reciprocalsquare root which Heine presented as:frac{1}{sqrt{z-cospsi=frac{sqrt{2{pi}sum_{m=-infty}^infty Q_{m-frac12}(z) e^{impsi}
where [cite journal | last=Cohl | first=Howard S. |coauthors=J.E. Tohline, A.R.P. Rau;H.M. Srivastava | title=Developments in determining the gravitational potential using toroidal functions | year=2000 | journal=
Astronomische Nachrichten | issn=0004-6337 | volume=321 | issue=5/6 | pages=363–372 | doi=10.1002/1521-3994(200012)321:5/6<363::AID-ASNA363>3.0.CO;2-X | doilabel=10.1002/1521-3994(200012)321:5/6363::AID-ASNA3633.0.CO;2-X] Q_{m-frac12} is aLegendre 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 [cite conference
first = H. S.
last = Cohl
title = Portent of Heine's Reciprocal Square Root Identity
booktitle = 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
volume = 293
isbn = 1583811400
year = 2003 ] for arbitrary half-integer powers as follows:z-cospsi)^{n-frac12}=sqrt{frac{2}{pifrac{(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^{impsi},
where scriptstyle,Gamma is the
Gamma function .References
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