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 where Q m − 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 , where Γ is the Gamma function.
