Digital Library of Mathematical Functions
About the Project
NIST
30 Spheroidal Wave FunctionsProperties

§30.6 Functions of Complex Argument

The solutions

30.6.1 Psnm(z,γ2),
Qsnm(z,γ2),

of (30.2.1) with μ=m and λ=λnm(γ2) are real when z(1,), and their principal values (§4.2(i)) are obtained by analytic continuation to \(-,1].

Relations to Associated Legendre Functions

Wronskian

30.6.3 𝒲{Psnm(z,γ2),Qsnm(z,γ2)}=(-1)m(n+m)!(1-z2)(n-m)!Anm(γ2)An-m(γ2),

with An±m(γ2) as in (30.11.4).

Values on (-1,1)

For further properties see Arscott (1964b).

For results for Equation (30.2.1) with complex parameters see Meixner and Schäfke (1954).