48
3 Legendre Polynomials and Legendre Functions
Q
μ
ν (z)
=
1
2
exp(iμπ)
Γ (1 + ν + μ)Γ (−μ)
Γ (1 + ν − μ)
z − 1
z + 1
μ/2
× 2 F 1
−ν, 1 + ν; 1 + μ;
1 − z
2
+ Γ (μ)
z + 1
z − 1
μ/2
2 F 1
−ν, 1 + ν; 1 − μ;
1 − z
2
, |1 − z| < 2,
(3.24b)
resp.
1 − x
1 + x
μ/2
and
1 + x
1 − x
μ/2
z → x;
note that this series will diverge for μ integer, which will be covered by the following
equation:
Q
μ
ν (z)
= exp(iμπ)2
μ √
π (z − 1)
−μ/2 (z + 1)
−μ/2
×
Γ (
1+ν+μ
2
)
2Γ (1 +
ν−μ
2 )
exp
±i
1
2
π(μ − ν − 1)
× 2 F 1
−ν − μ
2
,
1 + ν − μ
2
;
1
2
; z
2
+
zΓ (1 +
ν+μ
2 ) exp
±i
1
2 π(μ − ν)
Γ (
1+ν−μ
2
)
(3.24c)
× 2 F 1
1 − ν − μ
2
, 1 +
ν − μ
2
;
3
2
; z
2
with + for (z) > 0 and
− for (z) < 0
for |z| < 1 and for −1 ≤ x ≤ +1 by
Q
μ
ν (x) = −
√
π
2 1−μ
sin
1
2 (ν + μ)π
Γ (
ν+μ+1
2
)
Γ (
ν−μ
2 + 1)
(1 − x
2 )
μ/2
× 2 F 1
ν + μ + 1
2
,
μ − ν
2
;
1
2
; x
2