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30 Spheroidal Wave FunctionsProperties

§30.2 Differential Equations


§30.2(i) Spheroidal Differential Equation

30.2.1 ddz((1-z2)dwdz)+(λ+γ2(1-z2)-μ21-z2)w=0.

This equation has regular singularities at z=±1 with exponents ±12μ and an irregular singularity of rank 1 at z= (if γ0). The equation contains three real parameters λ, γ2, and μ. In applications involving prolate spheroidal coordinates γ2 is positive, in applications involving oblate spheroidal coordinates γ2 is negative; see §§30.13, 30.14.

§30.2(ii) Other Forms

The Liouville normal form of equation (30.2.1) is

30.2.2 d2gdt2+(λ+14+γ2sin2t-μ2-14sin2t)g=0,
30.2.3 z =cost,
w(z) =(1-z2)-14g(t).

With ζ=γz Equation (30.2.1) changes to

30.2.4 (ζ2-γ2)d2wdζ2+2ζdwdζ+(ζ2-λ-γ2-γ2μ2ζ2-γ2)w=0.

§30.2(iii) Special Cases

If γ=0, Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). If μ2=14, Equation (30.2.2) reduces to the Mathieu equation; see (28.2.1). If γ=0, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).