3.4 Associate Legendre Functions with Complex Indices
61
Equation (3.23e); method 7: The radius of convergence is |z| > 1, and again, we
have two summands, Eq. (3.26),
ta(n)
(1)
=
(
ν−μ−1
2
+ n)(
ν−μ
2 + n)
n(ν + n +
1
2 )
and
ta(n)
(2)
=
(
−ν−μ
2
+ n − 1)(
−1−ν−μ
2
+ n)
n(−ν + n −
1
2 )
.
In some cases, the prefactor will diverge, e.g., for method 7 for ν half-integer.
In these cases, we will use either one of the other series expansions or the path
integration technique based on a different method, as appropriate. The same holds
in case the hypergeometric series expansion should not sufficiently well converge.
Legendre Function of Second Kind
The computation of the Legendre function of second kind, Q
μ
ν (z), is based on
hypergeometric series expansions, the path integration technique, or if necessary
on Eq. (3.12b).
Equation (3.24a); method 1: This series converges for |z| > 1 with ta(n) =
(
μ+ν
2 +n)(
μ+ν−1
2
+n)
n(n+
1
2 +ν)
.
Equation (3.24b); method 2: Here, we have a combination of two hypergeometric
series with the same nominator, ta(n) ± =
(−ν−1+n)(ν+n)
n(n±μ)
, and the denominator
differs by ±μ. The series converges for |1 − z| < 2 and is undefined for μ positive
or negative integer. This case will be covered by methods 1 and 3.
Equation (3.24c); method 3: For the two series of method 3, we get
ta(n)
(1)
=
(−
ν+μ
2 + n − 1)(
1+ν−μ
2
+ n − 1)
n(n −
1
2 )
, and
ta(n)
(2)
=
(−
ν+μ−1
2
+ n − 1)(
ν−μ
2 + n)
n(n +
1
2 )
,
with convergence domain |z| < 1.
Equation (3.24d); this expansion is restricted to real x-values, −1 < x < +1:
ta(n)
(1)
=
(
ν+μ+1
2
+ n − 1)(
μ−ν
2 + n − 1)
n(n −
1
2
, and
ta(n)
(2)
=
(
ν+μ
2 + n)(
μ−ν+1
2
+ n − 1)
n(n +
1
2
.