About the Project
30 Spheroidal Wave FunctionsProperties

§30.6 Functions of Complex Argument

The solutions

30.6.1 𝑃𝑠nm(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


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