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30 Spheroidal Wave FunctionsComputation

§30.16 Methods of Computation

  1. §30.16(i) Eigenvalues
  2. §30.16(ii) Spheroidal Wave Functions of the First Kind
  3. §30.16(iii) Radial Spheroidal Wave Functions

§30.16(i) Eigenvalues

For small |γ2| we can use the power-series expansion (30.3.8). Schäfke and Groh (1962) gives corresponding error bounds. If |γ2| is large we can use the asymptotic expansions in §30.9. Approximations to eigenvalues can be improved by using the continued-fraction equations from §30.3(iii) and §30.8; see Bouwkamp (1947) and Meixner and Schäfke (1954, §3.93).

Another method is as follows. Let nm be even. For d sufficiently large, construct the d×d tridiagonal matrix 𝐀=[Aj,k] with nonzero elements

30.16.1 Aj,j =(m+2j2)(m+2j1)2γ2(m+2j2)(m+2j1)1+m2(2m+4j5)(2m+4j1),
Aj,j+1 =γ2(2m+2j1)(2m+2j)(2m+4j1)(2m+4j+1),
Aj,j1 =γ2(2j3)(2j2)(2m+4j7)(2m+4j5),

and real eigenvalues α1,d, α2,d, , αd,d, arranged in ascending order of magnitude. Then

30.16.2 αj,d+1αj,d,


30.16.3 λnm(γ2)=limdαp,d,

The eigenvalues of 𝐀 can be computed by methods indicated in §§3.2(vi), 3.2(vii). The error satisfies

30.16.4 αp,dλnm(γ2)=O(γ4d42d+1((m+2d1)!(m+2d+1)!)2),


For m=2, n=4, γ2=10,

30.16.5 α2,2 =14.18833 246,
α2,3 =13.98002 013,
α2,4 =13.97907 459,
α2,5 =13.97907 345,
α2,6 =13.97907 345,

which yields λ42(10)=13.97907 345. If nm is odd, then (30.16.1) is replaced by

30.16.6 Aj,j =(m+2j1)(m+2j)2γ2(m+2j1)(m+2j)1+m2(2m+4j3)(2m+4j+1),
Aj,j+1 =γ2(2m+2j)(2m+2j+1)(2m+4j+1)(2m+4j+3),
Aj,j1 =γ2(2j2)(2j1)(2m+4j5)(2m+4j3).

§30.16(ii) Spheroidal Wave Functions of the First Kind

If |γ2| is large, then we can use the asymptotic expansions referred to in §30.9 to approximate 𝖯𝗌nm(x,γ2).

If λnm(γ2) is known, then we can compute 𝖯𝗌nm(x,γ2) (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions w(0)=1, w(0)=0 if nm is even, or w(0)=0, w(0)=1 if nm is odd.

If λnm(γ2) is known, then 𝖯𝗌nm(x,γ2) can be found by summing (30.8.1). The coefficients an,rm(γ2) are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5).

A fourth method, based on the expansion (30.8.1), is as follows. Let 𝐀 be the d×d matrix given by (30.16.1) if nm is even, or by (30.16.6) if nm is odd. Form the eigenvector [e1,d,e2,d,,ed,d]T of 𝐀 associated with the eigenvalue αp,d, p=12(nm)+1, normalized according to

30.16.7 j=1dej,d2(n+m+2j2p)!(nm+2j2p)!12n+4j4p+1=(n+m)!(nm)!12n+1.


30.16.8 an,km(γ2)=limdek+p,d,
30.16.9 𝖯𝗌nm(x,γ2)=limdj=1d(1)jpej,d𝖯n+2(jp)m(x).

For error estimates see Volkmer (2004a).

§30.16(iii) Radial Spheroidal Wave Functions

The coefficients an,km(γ2) calculated in §30.16(ii) can be used to compute Snm(j)(z,γ), j=1,2,3,4 from (30.11.3) as well as the connection coefficients Knm(γ) from (30.11.10) and (30.11.11).

For other methods see Van Buren and Boisvert (2002, 2004).