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33 Coulomb FunctionsVariables r,ϵ

§33.14 Definitions and Basic Properties


§33.14(i) Coulomb Wave Equation

Another parametrization of (33.2.1) is given by

33.14.1 d2wdr2+(ϵ+2r-(+1)r2)w=0,


33.14.2 r =-ηρ,
ϵ =1/η2.

Again, there is a regular singularity at r=0 with indices +1 and -, and an irregular singularity of rank 1 at r=. When ϵ>0 the outer turning point is given by

33.14.3 rtp(ϵ,)=(1+ϵ(+1)-1)/ϵ;

compare (33.2.2).

§33.14(ii) Regular Solution f(ϵ,;r)

The function f(ϵ,;r) is recessive (§2.7(iii)) at r=0, and is defined by

33.14.4 f(ϵ,;r)=κ+1Mκ,+12(2r/κ)/(2+1)!,

or equivalently

33.14.5 f(ϵ,;r)=(2r)+1e-r/κM(+1-κ,2+2,2r/κ)/(2+1)!,

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

33.14.6 κ={(-ϵ)-1/2,ϵ<0,r>0,-(-ϵ)-1/2,ϵ<0,r<0,±iϵ-1/2,ϵ>0.

The choice of sign in the last line of (33.14.6) is immaterial: the same function f(ϵ,;r) is obtained. This is a consequence of Kummer’s transformation (§13.2(vii)).

f(ϵ,;r) is real and an analytic function of r in the interval -<r<, and it is also an analytic function of ϵ when -<ϵ<. This includes ϵ=0, hence f(ϵ,;r) can be expanded in a convergent power series in ϵ in a neighborhood of ϵ=033.20(ii)).

§33.14(iii) Irregular Solution h(ϵ,;r)

For nonzero values of ϵ and r the function h(ϵ,;r) is defined by

33.14.7 h(ϵ,;r)=Γ(+1-κ)πκ(Wκ,+12(2r/κ)+(-1)S(ϵ,r)Γ(+1+κ)2(2+1)!Mκ,+12(2r/κ)),

where κ is given by (33.14.6) and

33.14.8 S(ϵ,r)={2cos(π|ϵ|-1/2),ϵ<0,r>0,0,ϵ<0,r<0,eπϵ-1/2,ϵ>0,r>0,e-πϵ-1/2,ϵ>0,r<0.

(Again, the choice of the ambiguous sign in the last line of (33.14.6) is immaterial.)

h(ϵ,;r) is real and an analytic function of each of r and ϵ in the intervals -<r< and -<ϵ<, except when r=0 or ϵ=0.

§33.14(iv) Solutions s(ϵ,;r) and c(ϵ,;r)

The functions s(ϵ,;r) and c(ϵ,;r) are defined by

33.14.9 s(ϵ,;r) =(B(ϵ,)/2)1/2f(ϵ,;r),
c(ϵ,;r) =(2B(ϵ,))-1/2h(ϵ,;r),


33.14.10 B(ϵ,)={A(ϵ,)(1-exp(-2π/ϵ1/2))-1,ϵ>0,A(ϵ,),ϵ0,


33.14.11 A(ϵ,)=k=0(1+ϵk2).

An alternative formula for A(ϵ,) is

33.14.12 A(ϵ,)=Γ(1++κ)Γ(κ-)κ-2-1,

the choice of sign in the last line of (33.14.6) again being immaterial.

When ϵ<0 and >(-ϵ)-1/2 the quantity A(ϵ,) may be negative, causing s(ϵ,;r) and c(ϵ,;r) to become imaginary.

The function s(ϵ,;r) has the following properties:

33.14.13 0s(ϵ1,;r)s(ϵ2,;r)dr=δ(ϵ1-ϵ2),

where the right-hand side is the Dirac delta (§1.17). When ϵ=-1/n2, n=+1,+2,, s(ϵ,;r) is exp(-r/n) times a polynomial in r/n, and

33.14.14 ϕn,(r)=(-1)+1+n(2/n3)1/2s(-1/n2,;r)=(-1)+1+nn+2((n--1)!(n+)!)1/2(2r)+1e-r/nLn--1(2+1)(2r/n)


33.14.15 0ϕm,(r)ϕn,(r)dr=δm,n.

Note that the functions ϕn,, n=,+1,, do not form a complete orthonormal system.

§33.14(v) Wronskians