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30 Spheroidal Wave FunctionsProperties

§30.4 Functions of the First Kind


§30.4(i) Definitions

The eigenfunctions of (30.2.1) that correspond to the eigenvalues λnm(γ2) are denoted by Psnm(x,γ2), n=m,m+1,m+2,. They are normalized by the condition

30.4.1 -11(Psnm(x,γ2))2dx=22n+1(n+m)!(n-m)!,

the sign of Psnm(0,γ2) being (-1)(n+m)/2 when n-m is even, and the sign of dPsnm(x,γ2)/dx|x=0 being (-1)(n+m-1)/2 when n-m is odd.

When γ2>0 Psnm(x,γ2) is the prolate angular spheroidal wave function, and when γ2<0 Psnm(x,γ2) is the oblate angular spheroidal wave function. If γ=0, Psnm(x,0) reduces to the Ferrers function Pnm(x):

30.4.2 Psnm(x,0)=Pnm(x);

compare §14.3(i).

§30.4(ii) Elementary Properties

30.4.3 Psnm(-x,γ2)=(-1)n-mPsnm(x,γ2).

Psnm(x,γ2) has exactly n-m zeros in the interval -1<x<1.

§30.4(iii) Power-Series Expansion

30.4.4 Psnm(x,γ2)=(1-x2)12mk=0gkxk,


30.4.5 αkgk+2+(βk-λnm(γ2))gk+γkgk-2=0

with αk, βk, γk from (30.3.6), and g-1=g-2=0, gk=0 for even k if n-m is odd and gk=0 for odd k if n-m is even. Normalization of the coefficients gk is effected by application of (30.4.1).

§30.4(iv) Orthogonality

30.4.6 -11Pskm(x,γ2)Psnm(x,γ2)dx=22n+1(n+m)!(n-m)!δk,n.

If f(x) is mean-square integrable on [-1,1], then formally

30.4.7 f(x)=n=mcnPsnm(x,γ2),


30.4.8 cn=(n+12)(n-m)!(n+m)!-11f(t)Psnm(t,γ2)dt.

The expansion (30.4.7) converges in the norm of L2(-1,1), that is,

30.4.9 limN-11|f(x)-n=mNcnPsnm(x,γ2)|2dx=0.

It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for -1x1.