What's New
About the Project
NIST
30 Spheroidal Wave FunctionsProperties

§30.4 Functions of the First Kind

Contents

§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,
-1x1,

where

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),

where

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.