Computing

, Volume 39, Issue 3, pp 271–279 | Cite as

Numerical computation of Tricomi's psi function by the trapezoidal rule

  • G. Allasia
  • R. Besenghi
Short Communication

Abstract

The trapezoidal rule is applied to the numerical calculation of an integral representation of Tricomi's psi function Ψ(a, c; x) fora, x ε ℝ+ andc ε ℝ. The unexpectedly high accuracy is explained by means of a careful investigation of the remainder terms of the Euler-Maclaurin formula. A simple and efficient numerical procedure for obtaining values of the psi function is given.

AMS Subject Classifications

65B15 65D20 65D30 

Key words

Triconi's psi function trapezoidal rule 

Numerische Berechnung von Tricomis Psi-Funktion mit der Trapezregel

Zusammenfassung

Die Trapezregel wird zur numerischen Auswertung einer Integraldarstellung von Tricomis Psi-Funktion Ψ(a, c; x) füra, x ε ℝ+ undc ε ℝ verwendet. Die unvermutet hohe Genauigkeit wird durch eine gründliche Untersuchung des Restglieds der Euler-Maclaurin-Formel erklärt. Außerdem wird eine einfache und effektive numerische Prozedur angegeben, durch die man explizite Zahlenwerte der Psi-Funktion erhält.

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References

  1. [1]
    Tricomi, F. G.: Funzioni Ipergeometriche Confluenti. Roma: Cremonese 1954.Google Scholar
  2. [2]
    Slater, L. J.: Confluent Hypergeometric Functions. Cambridge: Cambridge Univ. Press 1960.Google Scholar
  3. [3]
    Luke, Y. L.: The Special Functions and their Approximations, Vols. I, II. New York: Academic Press 1969.Google Scholar
  4. [4]
    Davis, P. J., Rabinowitz, P.: Methods of Numerical Integration, 2nd ed. New York: Academic Press 1984.Google Scholar
  5. [5]
    Allasia, G., Besenghi, R.: Numerical calculation of incomplete gamma functions by the trapezoidal rule. Num. Math.50, 419–428 (1987).MathSciNetGoogle Scholar
  6. [6]
    Allasia, G., Besenghi, R.: Sul calcolo numerico delle funzioni gamma e digamma mediante la formula del trapezio. To appear on Boll Unione Mat. Italiana.Google Scholar
  7. [7]
    Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. New York: Dover Publications 1970.Google Scholar
  8. [8]
    Wimp, J.: On the computation of Tricomi's Ψ function. Computing13, 195–203 (1974).CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    Temme, N. M.: The numerical computation of the confluent hypergeometric functionU(a, b; z). Num. Math.41, 63–82 (1983).CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    Luke, Y. L.: Mathematical Functions and their Approximations. New York: Academic Press 1975.Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • G. Allasia
    • 1
  • R. Besenghi
    • 1
  1. 1.Dipartimento di MatematicaUniversità di TorinoTorinoItaly

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