Heine's identity

Last updated February 07, 2022

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-{\frac {1}{2}}}(z)e^{im\psi }

where^[2] $Q_{m-{\frac {1}{2}}}$ 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-{\frac {1}{2}}}={\sqrt {\frac {2}{\pi }}}{\frac {(z^{2}-1)^{\frac {n}{2}}}{\Gamma ({\frac {1}{2}}-n)}}\sum _{m=-\infty }^{\infty }{\frac {\Gamma (m-n+{\frac {1}{2}})}{\Gamma (m+n+{\frac {1}{2}})}}Q_{m-{\frac {1}{2}}}^{n}(z)e^{im\psi },

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

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References

↑ Heine, Heinrich Eduard (1881). Handbuch der Kugelfunctionen, Theorie und Andwendungen. Wuerzburg: Physica-Verlag.(See page 286)
↑ 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.
↑ 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. Vol. 293. ISBN 1-58381-140-0.

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[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. Vol. 293. ISBN 1-58381-140-0.

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