Heine's identity

{{short description|Fourier expansion of a reciprocal square root}}

In mathematical analysis, Heine's identity, named after Heinrich 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 a Fourier expansion of a reciprocal square root which Heine presented as

$$\backslash frac\{1\}\{\backslash sqrt\{z-\backslash cos\backslash psi\}\}\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{\backslash pi\}\backslash sum\_\{m=-\backslash infty\}^\backslash infty\; Q\_\{m-\backslash frac12\}(z)\; e^\{im\backslash psi\}$$

where{{cite journal | last=Cohl | first=Howard S. |author2=J.E. Tohline |author3=A.R.P. Rau |author4=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|bibcode = 2000AN....321..363C }}

$Q\_\{m-\backslash frac12\}$ 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{{cite conference

| first = H. S.

| last = Cohl

| title = Portent of Heine's Reciprocal Square Root Identity

| book-title = 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 = 1-58381-140-0

| year = 2003 }} for arbitrary half-integer powers as follows

$$(z-\backslash cos\backslash psi)^\{n-\backslash frac12\}\; =\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\backslash frac\{(z^2-1)^\{\backslash frac\{n\}\{2\}\}\}\{\backslash Gamma(\backslash frac12-n)\}$$

\sum_{m=-\infty}^{\infty} \frac{\Gamma(m-n+\frac12)}{\Gamma(m+n+\frac12)}Q_{m-\frac12}^n(z)e^{im\psi},

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