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30 Spheroidal Wave FunctionsProperties

§30.11 Radial Spheroidal Wave Functions

  1. §30.11(i) Definitions
  2. §30.11(ii) Graphics
  3. §30.11(iii) Asymptotic Behavior
  4. §30.11(iv) Wronskian
  5. §30.11(v) Connection with the Ps and Qs Functions
  6. §30.11(vi) Integral Representations

§30.11(i) Definitions


30.11.1 ψk(j)(z)=(π2z)12𝒞k+12(j)(z),


30.11.2 𝒞ν(1) =Jν,
𝒞ν(2) =Yν,
𝒞ν(3) =Hν(1),
𝒞ν(4) =Hν(2),

with Jν, Yν, Hν(1), and Hν(2) as in §10.2(ii). Then solutions of (30.2.1) with μ=m and λ=λnm(γ2) are given by

30.11.3 Snm(j)(z,γ)=(1z2)12mAnm(γ2)2kmnan,km(γ2)ψn+2k(j)(γz).

Here an,km(γ2) is defined by (30.8.2) and (30.8.6), and

30.11.4 An±m(γ2)=2kmn(1)kan,k±m(γ2)(0).

In (30.11.3) z0 when j=1, and |z|>1 when j=2,3,4.

Connection Formulas

30.11.5 Snm(3)(z,γ) =Snm(1)(z,γ)+iSnm(2)(z,γ),
Snm(4)(z,γ) =Snm(1)(z,γ)iSnm(2)(z,γ).

§30.11(ii) Graphics

See accompanying text
Figure 30.11.1: Sn0(1)(x,2), n=0,1, 1x10. Magnify
See accompanying text
Figure 30.11.2: Sn0(1)(iy,2i), n=0,1, 0y10. Magnify
See accompanying text
Figure 30.11.3: Sn1(1)(x,2), n=1,2, 1x10. Magnify
See accompanying text
Figure 30.11.4: Sn1(1)(iy,2i), n=1,2, 0y10. Magnify

§30.11(iii) Asymptotic Behavior

For fixed γ, as z in the sector |phz|πδ (<π),

30.11.6 Snm(j)(z,γ)={ψn(j)(γz)+O(z2e|z|),j=1,2,ψn(j)(γz)(1+O(z1)),j=3,4.

For asymptotic expansions in negative powers of z see Meixner and Schäfke (1954, p. 293).

§30.11(iv) Wronskian

30.11.7 𝒲{Snm(1)(z,γ),Snm(2)(z,γ)}=1γ(z21).

§30.11(v) Connection with the Ps and Qs Functions

30.11.8 Snm(1)(z,γ)=Knm(γ)Psnm(z,γ2),
30.11.9 Snm(2)(z,γ)=(nm)!(n+m)!(1)m+1Qsnm(z,γ2)γKnm(γ)Anm(γ2)Anm(γ2),


30.11.10 Knm(γ)=π2(γ2)m(1)man,12(mn)m(γ2)Γ(32+m)Anm(γ2)Psnm(0,γ2),
nm even,


30.11.11 Knm(γ)=π2(γ2)m+1(1)man,12(mn+1)m(γ2)Γ(52+m)Anm(γ2)(dPsnm(z,γ2)/dz|z=0),
nm odd.

§30.11(vi) Integral Representations

When z(,1]

30.11.12 Anm(γ2)Snm(1)(z,γ)=12im+nγm(nm)!(n+m)!zm(1z2)12m11eiγzt(1t2)12mPsnm(t,γ2)dt.

For further relations see Arscott (1964b, §8.6), Connett et al. (1993), Erdélyi et al. (1955, §16.13), Meixner and Schäfke (1954), and Meixner et al. (1980, §3.1).