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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).