Heine's identity

In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine 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 $Q_{m-\frac12}$ 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 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.