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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).