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.
