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30 Spheroidal Wave FunctionsProperties

§30.11 Radial Spheroidal Wave Functions


§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)=(1-z-2)12mAn-m(γ2)2km-nan,k-m(γ2)ψn+2k(j)(γz).

Here an,k-m(γ2) is defined by (30.8.2) and (30.8.6), and

30.11.4 An±m(γ2)=2km-n(-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|π-δ (<π),

30.11.6 Snm(j)(z,γ)={ψn(j)(γz)+O(z-2e|z|),j=1,2,ψn(j)(γz)(1+O(z-1)),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(z2-1).

§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)=(n-m)!(n+m)!(-1)m+1Qsnm(z,γ2)γKnm(γ)Anm(γ2)An-m(γ2),


30.11.10 Knm(γ)=π2(γ2)m(-1)man,12(m-n)-m(γ2)Γ(32+m)An-m(γ2)Psnm(0,γ2),
n-m even,


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

§30.11(vi) Integral Representations

When z\(-,1]

30.11.12 An-m(γ2)Snm(1)(z,γ)=12im+nγm(n-m)!(n+m)!zm(1-z-2)12m-11e-iγzt(1-t2)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).