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Numerical Evaluation of Special Functions

D. W. Lozier and F. W. J. Olver

5.2. Coulomb Wave Functions .

5.2.1. Real Arguments and Parameters.


[ CH70b] , [ She74] .

Software Packages:

[ Bar76, Fortran] , [ Bar81b, Fortran] , [ Bar82b, Fortran] , [ BDG+72, Fortran] , [ BS80, Fortran] , [ NT84, Fortran] , [ Sea82, Fortran] .

Intermediate Libraries:

[ Bak92] .

5.2.2. Complex Arguments and Parameters.

Software Packages:

[ TB85, Fortran] , [ TR69, Fortran] .

5.2.3. Articles.

[ AS92] , [ Bar81a] , [ Bar82a] , [ Bar82c] , [ Gau69b] , [ Kol72b, includes survey] , [ Nes84] , [ Pex70] , [ SG72] , [ TB86] .


J. Abad and J. Sesma, Computation of Coulomb wave functions at low energies, Comput. Phys. Comm. 71 (1992), 110--124.

L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New York, 1992, includes diskette.

A. R. Barnett, RCWFF---A modification of the real Coulomb wavefunction program RCWFN, Comput. Phys. Comm. 11 (1976), 141--142.

A. R. Barnett, An algorithm for regular and irregular Coulomb and Bessel functions of real order to machine accuracy, Comput. Phys. Comm. 21 (1981), 297--314.

A. R. Barnett, KLEIN : Coulomb functions for real and positive energy to high accuracy, Comput. Phys. Comm. 24 (1981), 141--159.

A. R. Barnett, Continued-fraction evaluation of Coulomb functions , and their derivatives, J. Comput. Phys. 46 (1982), 171--188.

A. R. Barnett, COULFG : Coulomb and Bessel functions and their derivatives, for real arguments, by Steed's method, Comput. Phys. Comm. 27 (1982), 147--166.

A. R. Barnett, High-precision evaluation of the regular and irregular Coulomb wavefunctions, J. Comput. Appl. Math. 8 (1982), 29--33.

C. Bardin, Y. Dandeu, L. Gauthier, J. Guillerman, T. Lena, and J.-M. Pernet, Coulomb functions in the entire --plane, Comput. Phys. Comm. 3 (1972), 73--87.

K. L. Bell and N. S. Scott, Coulomb functions (negative energies ), Comput. Phys. Comm. 20 (1980), 447--458.

W. J. Cody and K. E. Hillstrom, Chebyshev approximations for the Coulomb phase shift, Math. Comp. 24 (1970), 671--677, for corrigendum, see same journal v. 26 (1972), p. 1031.

W. Gautschi, An application of minimal solutions of three-term recurrences to Coulomb wave functions, Aequationes Math. 2 (1969), 171--176.

K. S. Kölbig, Remarks on the computation of Coulomb wavefunctions, Comput. Phys. Comm. 4 (1972), 214--220.

R. K. Nesbet, Algorithms for regular and irregular Coulomb and Bessel functions, Comput. Phys. Comm. 32 (1984), 341--347.

C. J. Noble and I. J. Thompson, COULN, A program for evaluating negative energy Coulomb functions, Comput. Phys. Comm. 33 (1984), 413--419.

R. L. Pexton, Computer investigation of Coulomb wave functions, Math. Comp. 24 (1970), 409--411.

M. J. Seaton, Coulomb functions analytic in the energy, Comput. Phys. Comm. 25 (1982), 87--95.

A. J. Strecok and J. A. Gregory, High precision evaluation of the irregular Coulomb wave functions, Math. Comp. 26 (1972), 955--961.

V. B. Sheorey, Chebyshev expansions for wave functions, Comput. Phys. Comm. 7 (1974), 1--12.

I. J. Thompson and A. R. Barnett, COULCC : A continued-fraction algorithm for Coulomb functions of complex order with complex arguments, Comput. Phys. Comm. 36 (1985), 363--372.

I. J. Thompson and A. R. Barnett, Coulomb and Bessel functions of complex arguments and order, J. Comput. Phys. 64 (1986), 490--509.

T. Tamura and F. Rybicki, Coulomb functions for complex energies, Comput. Phys. Comm. 1 (1969), 25--30, for erratum see same journal v. 3 (1972), p. 276.


This document is an excerpt from the current hypertext version of an article that appeared in Walter Gautschi (ed.), Mathematics of Computation 1943--1993: A Half-Century of Computational Mathematics, Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, Providence, RI 02940, 1994. The symposium was held at the University of British Columbia August 9--13, 1993, in honor of the fiftieth anniversary of the journal Mathematics of Computation.

The original abstract follows.

Higher transcendental functions continue to play varied and important roles in investigations by engineers, mathematicians, scientists and statisticians. The purpose of this paper is to assist in locating useful approximations and software for the numerical generation of these functions, and to offer some suggestions for future developments in this field.

Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Md 20899

E-mail address:

Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742

E-mail address:

The research of the second author has been supported by NSF Grant CCR 89-14933.

1991 Mathematics Subject Classification. Primary 65D20; Secondary 33-00.

Daniel W Lozier
Fri Apr 7 14:07:17 EDT 1995