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30 Spheroidal Wave FunctionsProperties

§30.4 Functions of the First Kind

Contents
  1. §30.4(i) Definitions
  2. §30.4(ii) Elementary Properties
  3. §30.4(iii) Power-Series Expansion
  4. §30.4(iv) Orthogonality

§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)!(nm)!,

the sign of 𝖯𝗌nm(0,γ2) being (1)(n+m)/2 when nm is even, and the sign of d𝖯𝗌nm(x,γ2)/dx|x=0 being (1)(n+m1)/2 when nm 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)nm𝖯𝗌nm(x,γ2).

𝖯𝗌nm(x,γ2) has exactly nm zeros in the interval 1<x<1.

§30.4(iii) Power-Series Expansion

30.4.4 𝖯𝗌nm(x,γ2)=(1x2)12mk=0gkxk,
1x1,

where

30.4.5 αkgk+2+(βkλnm(γ2))gk+γkgk2=0

with αk, βk, γk from (30.3.6), and g1=g2=0, gk=0 for even k if nm is odd and gk=0 for odd k if nm 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)!(nm)!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)(nm)!(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 limN11|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.