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30 Spheroidal Wave FunctionsProperties

§30.3 Eigenvalues


§30.3(i) Definition

With μ=m=0,1,2,, the spheroidal wave functions Psnm(x,γ2) are solutions of Equation (30.2.1) which are bounded on (-1,1), or equivalently, which are of the form (1-x2)12mg(x) where g(z) is an entire function of z. These solutions exist only for eigenvalues λnm(γ2), n=m,m+1,m+2,, of the parameter λ.

§30.3(ii) Properties

The eigenvalues λnm(γ2) are analytic functions of the real variable γ2 and satisfy

30.3.1 λmm(γ2)<λm+1m(γ2)<λm+2m(γ2)<,
30.3.2 λnm(γ2)=n(n+1)-12γ2+O(n-2),
30.3.3 λnm(0)=n(n+1),
30.3.4 -1<dλnm(γ2)d(γ2)<0.

§30.3(iii) Transcendental Equation

If p is an even nonnegative integer, then the continued-fraction equation

30.3.5 βp-λ-αp-2γpβp-2-λ-αp-4γp-2βp-4-λ-=αpγp+2βp+2-λ-αp+2γp+4βp+4-λ-,

where αk, βk, γk are defined by

30.3.6 αk =-(k+1)(k+2),
βk =(m+k)(m+k+1)-γ2,
γk =γ2,

has the solutions λ=λm+2jm(γ2), j=0,1,2,. If p is an odd positive integer, then Equation (30.3.5) has the solutions λ=λm+2j+1m(γ2), j=0,1,2,. If p=0 or p=1, the finite continued-fraction on the left-hand side of (30.3.5) equals 0; if p>1 its last denominator is β0-λ or β1-λ.

In equation (30.3.5) we can also use

30.3.7 αk =γ2(k+2m+1)(k+2m+2)(2k+2m+3)(2k+2m+5),
βk =(k+m)(k+m+1)-2γ2(k+m)(k+m+1)-1+m2(2k+2m-1)(2k+2m+3),
γk =γ2(k-1)k(2k+2m-3)(2k+2m-1).

§30.3(iv) Power-Series Expansion

30.3.8 λnm(γ2)=k=02kγ2k,

For values of rnm see Meixner et al. (1980, p. 109).

30.3.9 0 =n(n+1),
22 =-1-(2m-1)(2m+1)(2n-1)(2n+3),
24 =(n-m-1)(n-m)(n+m-1)(n+m)(2n-3)(2n-1)3(2n+1)-(n-m+1)(n-m+2)(n+m+1)(n+m+2)(2n+1)(2n+3)3(2n+5).
30.3.10 6=(4m2-1)((n-m+1)(n-m+2)(n+m+1)(n+m+2)(2n-1)(2n+1)(2n+3)5(2n+5)(2n+7)-(n-m-1)(n-m)(n+m-1)(n+m)(2n-5)(2n-3)(2n-1)5(2n+1)(2n+3)),
30.3.11 8=2(4m2-1)2A+116B+18C+12D,
30.3.12 A =(n-m-1)(n-m)(n+m-1)(n+m)(2n-5)2(2n-3)(2n-1)7(2n+1)(2n+3)2-(n-m+1)(n-m+2)(n+m+1)(n+m+2)(2n-1)2(2n+1)(2n+3)7(2n+5)(2n+7)2,
B =(n-m-3)(n-m-2)(n-m-1)(n-m)(n+m-3)(n+m-2)(n+m-1)(n+m)(2n-7)(2n-5)2(2n-3)3(2n-1)4(2n+1)-(n-m+1)(n-m+2)(n-m+3)(n-m+4)(n+m+1)(n+m+2)(n+m+3)(n+m+4)(2n+1)(2n+3)4(2n+5)3(2n+7)2(2n+9),
C =(n-m+1)2(n-m+2)2(n+m+1)2(n+m+2)2(2n+1)2(2n+3)7(2n+5)2-(n-m-1)2(n-m)2(n+m-1)2(n+m)2(2n-3)2(2n-1)7(2n+1)2,
D =(n-m-1)(n-m)(n-m+1)(n-m+2)(n+m-1)(n+m)(n+m+1)(n+m+2)(2n-3)(2n-1)4(2n+1)2(2n+3)4(2n+5).

Further coefficients can be found with the Maple program SWF9; see §30.18(i).