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33 Coulomb FunctionsVariables ρ,η

§33.9 Expansions in Series of Bessel Functions

Contents

§33.9(i) Spherical Bessel Functions

33.9.1 F(η,ρ)=ρk=0akj+k(ρ),

where the function j is as in §10.47(ii), a-1=0, a0=(2+1)!!C(η), and

33.9.2 k(k+2+1)2k+2+1ak-2ηak-1+(k-2)(k+2-1)2k+2-3ak-2=0,
k=1,2,.

The series (33.9.1) converges for all finite values of η and ρ.

§33.9(ii) Bessel Functions and Modified Bessel Functions

In this subsection the functions J, I, and K are as in §§10.2(ii) and 10.25(ii).

With t=2|η|ρ,

33.9.3 F(η,ρ)=C(η)(2+1)!(2η)2+1ρ-k=2+1bktk/2Ik(2t),
η>0,
33.9.4 F(η,ρ)=C(η)(2+1)!(2|η|)2+1ρ-k=2+1bktk/2Jk(2t),
η<0.

Here b2=b2+2=0, b2+1=1, and

33.9.5 4η2(k-2)bk+1+kbk-1+bk-2=0,
k=2+2,2+3,.

The series (33.9.3) and (33.9.4) converge for all finite positive values of |η| and ρ.

Next, as η+ with ρ (>0) fixed,

33.9.6 G(η,ρ)ρ-(+12)λ(η)C(η)k=2+1(-1)kbktk/2Kk(2t),

where

33.9.7 λ(η)k=2+1(-1)k(k-1)!bk.

For other asymptotic expansions of G(η,ρ) see Fröberg (1955, §8) and Humblet (1985).