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33 Coulomb FunctionsNotation

§33.1 Special Notation

(For other notation see Notation for the Special Functions.)

k,

nonnegative integers.

r,x

real variables.

ρ

nonnegative real variable.

ϵ,η

real parameters.

ψ(x)

logarithmic derivative of Γ(x); see §5.2(i).

δ(x)

Dirac delta; see §1.17.

primes

derivatives with respect to the variable.

The main functions treated in this chapter are first the Coulomb radial functions F(η,ρ), G(η,ρ), H±(η,ρ) (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions f(ϵ,;r), h(ϵ,;r), s(ϵ,;r), c(ϵ,;r) (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions.

Alternative Notations

  1. Curtis (1964a):

    P(ϵ,r)=(2+1)!f(ϵ,;r)/2+1, Q(ϵ,r)=-(2+1)!h(ϵ,;r)/(2+1A(ϵ,)).

  2. Greene et al. (1979):

    f(0)(ϵ,;r)=f(ϵ,;r), f(ϵ,;r)=s(ϵ,;r), g(ϵ,;r)=c(ϵ,;r).