In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as
1 z − cos ψ = 2 π ∑ m = − ∞ ∞ Q m − 1 2 ( z ) e i m ψ {\displaystyle {\frac {1}{\sqrt {z-\cos \psi }}}={\frac {\sqrt {2}}{\pi }}\sum _{m=-\infty }^{\infty }Q_{m-{\frac {1}{2}}}(z)e^{im\psi }}
where Q m − 1 2 {\displaystyle 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 ψ ) n − 1 2 = 2 π ( z 2 − 1 ) n 2 Γ ( 1 2 − n ) ∑ m = − ∞ ∞ Γ ( m − n + 1 2 ) Γ ( m + n + 1 2 ) Q m − 1 2 n ( z ) e i m ψ , {\displaystyle (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 Γ {\displaystyle \scriptstyle \,\Gamma } is the Gamma function.
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