Heine's identity

Short description: Fourier expansion of a reciprocal square root

In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as $$\frac{1}{\sqrt{z-\cos \psi }}=\frac{\sqrt{2}}{\pi }\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{Q}_{m-\frac{1}{2}}(z)e^{im\psi }$$ where $Q_{m-\frac{1}{2}}$ 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 for arbitrary half-integer powers as follows $$(z-\cos \psi {)}^{n-\frac{1}{2}}=\sqrt{\frac{2}{\pi }}\frac{(z^2-1{)}^{\frac{n}{2}}}{\mathrm{\Gamma }(\frac{1}{2}-n)}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }(m-n+\frac{1}{2})}{\mathrm{\Gamma }(m+n+\frac{1}{2})}{Q}_{m-\frac{1}{2}}^n(z)e^{im\psi },$$ where $\mathrm{\Gamma }$ is the Gamma function.