3.4 Associate Legendre Functions with Complex Indices
51
P
1
2
ν (cosh α) =
2
π sinh α
cosh
ν +
1
2
α
(3.28b)
P
−
1
2
ν (cos θ) =
2
π sin θ
sin
ν +
1
2
θ
ν +
1
2
(3.28c)
P
−
1
2
ν (cosh α) =
2
π sinh α
sinh
ν +
1
2
α
ν +
1
2
(3.28d)
Q
1
2
ν (cos θ) = −
π
2 sin θ
sin
ν +
1
2
θ
(3.28e)
Q
1
2
ν (cosh α) = i
π
2 sinh α
exp
−
ν +
1
2
α
(3.28f)
Q
−
1
2
ν (cos θ) =
2π
sin θ
cos
ν +
1
2
θ
ν +
1
2
(3.28g)
Q
−
1
2
ν (z) = −i
√
2π
(z 2 − 1)
−
1
4
2ν + 1
z +
z 2 − 1
−ν−
1
2 ;
(3.28h)
Q
μ
ν (z)
are not defined for ν + μ a negative integer . (3.28i)
The following recurrence relation is stable for backward recursion (decreasing μ)
and useful for computation when two starting values can be evaluated for starting
the recursive process,
P
μ+2
ν
(z) + 2(μ + 1)
z
√
z 2 − 1
P
μ+1
ν
(z) = (ν − μ)(ν + μ + 1)P
μ
ν (z) .
(3.29)
Together with the path integration, Chap. 8, of the hypergeometric differential
equation, all numerical methods used are listed in the remainder of this chapter.
3.4.2 Conical or Mehler Functions
Conical or Mehler functions are widely used in applications in applied physics,
astrophysics, or geophysics. The conical functions are the associate Legendre
functions with ν = −
1
2 + iτ , τ ∈ R ∗ and μ real,
P
μ
−
1
2 +iτ
(z)
and
Q
μ
−
1
2 +iτ
(z)
and can be evaluated with the class mehlerfun, >> [obj, res] = mehlerfun
(tau,mu,z,PorQ). The input variables “tau” and “mu” have to be real arrays,
and “z” could be an arbitrary complex array. If “tau” and “mu” both are either row