The Coulomb wave function is a special case of the confluent hypergeometric function of the first kind. It gives the solution to the radial
Schrödinger equation in the Coulomb potential () of a point nucleus
(1)
(Abramowitz and Stegun 1972; Zwillinger 1997, p. 122). The complete solution is
where ,
,
and
are defined in Abramowitz and Stegun (1972, p. 538).
The Coulomb wave functions of the first and second kind are implemented in the Wolfram Language as CoulombF[l,
eta, r] and CoulombG[l,
eta, r], respectively.
Abramowitz, M. and Antosiewicz, H. A. "Coulomb Wave Functions in the Transition Region." Phys. Rev.96, 75-77,
1954.Abramowitz, M. and Rabinowitz, P. "Evaluation of Coulomb Wave
Functions along the Transition Line." Phys. Rev.96, 77-79, 1954.Abramowitz,
M. and Stegun, I. A. (Eds.). "Coulomb Wave Functions." Ch. 14
in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 537-544, 1972.Biedenharn, L. C.; Gluckstern,
R. L.; Hull, M. H. Jr.; and Breit, G. "Coulomb Wave Functions for
Large Charges and Small Velocities." Phys. Rev.97, 542-554, 1955.Bloch,
I.; Hull, M. H. Jr.; Broyles, A. A.; Bouricius, W. G.; Freeman, B. E.;
and Breit, G. "Coulomb Functions for Reactions of Protons and Alpha-Particles
with the Lighter Nuclei." Rev. Mod. Phys.23, 147-182, 1951.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 631-633,
1953.National Bureau of Standards. Tables of Coulomb Wave Functions,
Vol. 1, Applied Math Series 17. Washington, DC: U.S. Government Printing
Office, 1952.Stegun, I. A. and Abramowitz, M. "Generation
of Coulomb Wave Functions by Means of Recurrence Relations." Phys. Rev.98,
1851-1852, 1955.Zwillinger, D. Handbook
of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997.
) of a point nucleus
![(d^2W)/(drho^2)+[1-(2eta)/rho-(L(L+1))/(rho^2)]W=0](/images/equations/CoulombWaveFunction/NumberedEquation1.svg)
is the confluent hypergeometric
function of the first kind, 
is the gamma function.
This function
![G_L(eta,rho)=(2eta)/(C_0^2(eta))F_L(eta,rho)[ln(2rho)+(q_L(eta))/(p_L(eta))]
+1/((2L+1)C_L(eta))rho^(-L)sum_(K=-L)^inftya_k^L(eta)rho^(K+L),](/images/equations/CoulombWaveFunction/NumberedEquation5.svg)
,
,
and 
are defined in Abramowitz and Stegun (1972, p. 538).
