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30 Spheroidal Wave FunctionsProperties

§30.3 Eigenvalues

Contents
  1. §30.3(i) Definition
  2. §30.3(ii) Properties
  3. §30.3(iii) Transcendental Equation
  4. §30.3(iv) Power-Series Expansion

§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 (1x2)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(n2),
n,
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 βpλαp2γpβp2λαp4γp2βp4λ=αpγp+2βp+2λαp+2γp+4βp+4λ,

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 β0λ or β1λ.

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+2m1)(2k+2m+3),
γk =γ2(k1)k(2k+2m3)(2k+2m1).

§30.3(iv) Power-Series Expansion

30.3.8 λnm(γ2)=k=02kγ2k,
|γ2|<rnm.

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

30.3.9 0 =n(n+1),
22 =1(2m1)(2m+1)(2n1)(2n+3),
24 =(nm1)(nm)(n+m1)(n+m)(2n3)(2n1)3(2n+1)(nm+1)(nm+2)(n+m+1)(n+m+2)(2n+1)(2n+3)3(2n+5).
30.3.10 6=(4m21)((nm+1)(nm+2)(n+m+1)(n+m+2)(2n1)(2n+1)(2n+3)5(2n+5)(2n+7)(nm1)(nm)(n+m1)(n+m)(2n5)(2n3)(2n1)5(2n+1)(2n+3)),
30.3.11 8=2(4m21)2A+116B+18C+12D,
30.3.12 A =(nm1)(nm)(n+m1)(n+m)(2n5)2(2n3)(2n1)7(2n+1)(2n+3)2(nm+1)(nm+2)(n+m+1)(n+m+2)(2n1)2(2n+1)(2n+3)7(2n+5)(2n+7)2,
B =(nm3)(nm2)(nm1)(nm)(n+m3)(n+m2)(n+m1)(n+m)(2n7)(2n5)2(2n3)3(2n1)4(2n+1)(nm+1)(nm+2)(nm+3)(nm+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 =(nm+1)2(nm+2)2(n+m+1)2(n+m+2)2(2n+1)2(2n+3)7(2n+5)2(nm1)2(nm)2(n+m1)2(n+m)2(2n3)2(2n1)7(2n+1)2,
D =(nm1)(nm)(nm+1)(nm+2)(n+m1)(n+m)(n+m+1)(n+m+2)(2n3)(2n1)4(2n+1)2(2n+3)4(2n+5).

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