About the Project
33 Coulomb FunctionsVariables ρ,η

§33.2 Definitions and Basic Properties


§33.2(i) Coulomb Wave Equation

33.2.1 d2wdρ2+(1-2ηρ-(+1)ρ2)w=0,

This differential equation has a regular singularity at ρ=0 with indices +1 and -, and an irregular singularity of rank 1 at ρ= (§§2.7(i), 2.7(ii)). There are two turning points, that is, points at which d2w/dρ2=02.8(i)). The outer one is given by

33.2.2 ρtp(η,)=η+(η2+(+1))1/2.

§33.2(ii) Regular Solution F(η,ρ)

The function F(η,ρ) is recessive (§2.7(iii)) at ρ=0, and is defined by

33.2.3 F(η,ρ)=C(η)2--1(i)+1M±iη,+12(±2iρ),

or equivalently

33.2.4 F(η,ρ)=C(η)ρ+1eiρM(+1iη,2+2,±2iρ),

where Mκ,μ(z) and M(a,b,z) are defined in §§13.14(i) and 13.2(i), and

33.2.5 C(η)=2e-πη/2|Γ(+1+iη)|(2+1)!.

The choice of ambiguous signs in (33.2.3) and (33.2.4) is immaterial, provided that either all upper signs are taken, or all lower signs are taken. This is a consequence of Kummer’s transformation (§13.2(vii)).

F(η,ρ) is a real and analytic function of ρ on the open interval 0<ρ<, and also an analytic function of η when -<η<.

The normalizing constant C(η) is always positive, and has the alternative form

33.2.6 C(η)=2((2πη/(e2πη-1))k=1(η2+k2))1/2(2+1)!.

§33.2(iii) Irregular Solutions G(η,ρ),H±(η,ρ)

The functions H±(η,ρ) are defined by

33.2.7 H±(η,ρ)=(i)e(πη/2)±iσ(η)Wiη,+12(2iρ),

or equivalently

33.2.8 H±(η,ρ)=e±iθ(η,ρ)(2iρ)+1±iηU(+1±iη,2+2,2iρ),

where Wκ,μ(z), U(a,b,z) are defined in §§13.14(i) and 13.2(i),

33.2.9 θ(η,ρ)=ρ-ηln(2ρ)-12π+σ(η),


33.2.10 σ(η)=phΓ(+1+iη),

the branch of the phase in (33.2.10) being zero when η=0 and continuous elsewhere. σ(η) is the Coulomb phase shift.

H+(η,ρ) and H-(η,ρ) are complex conjugates, and their real and imaginary parts are given by

33.2.11 H+(η,ρ) =G(η,ρ)+iF(η,ρ),
H-(η,ρ) =G(η,ρ)-iF(η,ρ).

As in the case of F(η,ρ), the solutions H±(η,ρ) and G(η,ρ) are analytic functions of ρ when 0<ρ<. Also, eiσ(η)H±(η,ρ) are analytic functions of η when -<η<.

§33.2(iv) Wronskians and Cross-Product

With arguments η,ρ suppressed,

33.2.12 𝒲{G,F}=𝒲{H±,F}=1.
33.2.13 F-1G-FG-1=/(2+η2)1/2,