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Chebyshev approximations for the Coulomb phase shift


Authors: W. J. Cody and K. E. Hillstrom
Journal: Math. Comp. 24 (1970), 671-677
MSC: Primary 65.25
DOI: https://doi.org/10.1090/S0025-5718-1970-0273785-4
Corrigendum: Math. Comp. 26 (1972), 1031.
MathSciNet review: 0273785
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Abstract: This note presents nearly-best rational approximations for the Coulomb phase shift ${\sigma _0}(\eta ) = \arg \Gamma (1 + i\eta )$. Maximal relative errors range down to between $4.24 \times {10^{ - 19}}$ and $1.09 \times {10^{ - 20}}$. The nontrivial zero of ${\sigma _0}(\eta )$ is also given.


References [Enhancements On Off] (What's this?)

    M. Abramowitz, "Coulomb wave functions," Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, M. Abramowitz & I. A. Stegun (Editors), Nat. Bur. Standards Appl. Math. Series, 55, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964, chap. 14, pp. 537–554. MR 29 #4914.
  • Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 213062, DOI https://doi.org/10.1137/1009002
  • W. Gautschi, "Algorithm 292, regular Coulomb wave functions," Comm. ACM, v. 9, 1966, pp. 793–795. H. F. Lutz & M. D. Karvelis, "Numerical calculation of Coulomb wave functions for repulsive Coulomb fields," Nuclear Phys., v. 43, 1963, pp. 31–44. J. H. Gunn, "Algorithm 300, Coulomb wave functions," Comm. ACM, v. 10, 1967, pp. 244–245.
  • W. J. Cody, Handbook Series Methods of Approximation: Rational Chebyshev approximation using linear equations, Numer. Math. 12 (1968), no. 4, 242–251. MR 1553964, DOI https://doi.org/10.1007/BF02162506
  • H. Werner, J. Stoer, and W. Bommas, Handbook Series Methods of Approximation: Rational Chebyshev approximation, Numer. Math. 10 (1967), no. 4, 289–306. MR 1553955, DOI https://doi.org/10.1007/BF02162028

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Keywords: Rational Chebyshev approximations, Coulomb phase shift, complex gamma function
Article copyright: © Copyright 1970 American Mathematical Society