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33 Coulomb FunctionsVariables r,ϵ

§33.20 Expansions for Small |ϵ|

Contents

§33.20(i) Case ϵ=0

33.20.1 f(0,;r) =(2r)1/2J2+1(8r),
h(0,;r) =-(2r)1/2Y2+1(8r),
r>0,
33.20.2 f(0,;r) =(-1)+1(2|r|)1/2I2+1(8|r|),
h(0,;r) =(-1)(2/π)(2|r|)1/2K2+1(8|r|),
r<0.

For the functions J, Y, I, and K see §§10.2(ii), 10.25(ii).

§33.20(ii) Power-Series in ϵ for the Regular Solution

33.20.3 f(ϵ,;r)=k=0ϵkFk(;r),

where

33.20.4 Fk(;r)=p=2k3k(2r)(p+1)/2Ck,pJ2+1+p(8r),
r>0,
33.20.5 Fk(;r)=p=2k3k(-1)+1+p(2|r|)(p+1)/2Ck,pI2+1+p(8|r|),
r<0.

The functions J and I are as in §§10.2(ii), 10.25(ii), and the coefficients Ck,p are given by C0,0=1, C1,0=0, and

33.20.6 Ck,p =0,
p<2k or p>3k,
Ck,p =(-(2+p)Ck-1,p-2+Ck-1,p-3)/(4p),
k>0, 2kp3k.

The series (33.20.3) converges for all r and ϵ.

§33.20(iii) Asymptotic Expansion for the Irregular Solution

As ϵ0 with and r fixed,

33.20.7 h(ϵ,;r)-A(ϵ,)k=0ϵkHk(;r),

where A(ϵ,) is given by (33.14.11), (33.14.12), and

33.20.8 Hk(;r)=p=2k3k(2r)(p+1)/2Ck,pY2+1+p(8r),
r>0,
33.20.9 Hk(;r)=(-1)+12πp=2k3k(2|r|)(p+1)/2Ck,pK2+1+p(8|r|),
r<0.

The functions Y and K are as in §§10.2(ii), 10.25(ii), and the coefficients Ck,p are given by (33.20.6).

§33.20(iv) Uniform Asymptotic Expansions

For a comprehensive collection of asymptotic expansions that cover f(ϵ,;r) and h(ϵ,;r) as ϵ0± and are uniform in r, including unbounded values, see Curtis (1964a, §7). These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders 2+1 and 2+2.