Numerische Mathematik

, Volume 50, Issue 4, pp 419–428 | Cite as

Numerical calculation of incomplete gamma functions by the trapezoidal rule

  • Giampietro Allasia
  • Renata Besenghi
Convergence of the SSOR Method for Nonlinear Systems of Simultaneous Equations


The trapezoidal rule is applied to the numerical calculation of a known integral representation of the complementary incomplete gamma function Г (a,x) in the regiona<−1 andx>0. Since this application of the rule is not standard, a careful investigation of the remainder terms using the Euler-Maclaurin formula is carried out. The outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with surprising accuracy in the stated region.

Subject Classifications

AMS(MOS): 65B15 65D20 65D30 CR: G1.2 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allasia, G., Besenghi, R.: Sul calcolo numerico delle funzioni gamma e digamma mediante la formula del trapezio. Boll. Unione Mat. Ital. (to appear)Google Scholar
  2. 2.
    Davis, P.J., Rabinowitz, P.: Methods of numerical integration (2nd Ed.). New York: Academic Press 1984Google Scholar
  3. 3.
    Erdély, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions. New York: McGraw-Hill 1953Google Scholar
  4. 4.
    Gatteschi, L.: Funzioni speciali. Torino: U.T.E.T. 1973Google Scholar
  5. 5.
    Gautschi, W.: Un procedimento di calcolo per le funzioni gamma incomplete. Rend. Semin. Mat. Torino37, 1–9 (1979)Google Scholar
  6. 6.
    Gautschi, W.: A computational procedure for incomplete gamma functions. ACM Trans. Math. Software5, 466–481 (1979)Google Scholar
  7. 7.
    Gautschi, W.: Algorithm 542-Incomplete gamma functions. ACM Trans. Math. Software5, 482–489 (1979)Google Scholar
  8. 8.
    Hardy, G.H.: Divergent series. Oxford: University Press 1949Google Scholar
  9. 9.
    Hunter, D.B.: The calculation of certain Bessel functions. Math. Comput.18, 123–128 (1964)Google Scholar
  10. 10.
    Hunter, D.B.: The evaluation of a class of functions defined by an integral. Math. Comput.22, 440–444 (1968)Google Scholar
  11. 11.
    Lindelöf, E.: Le calcul des résidus. New York: Chelsea 1947Google Scholar
  12. 12.
    Luke, Y.L.: The special functions and their approximations. New York: Academic Press 1969Google Scholar
  13. 13.
    Luke, Y.L.: Mathematical functions and their approximations. New York: Academic Press 1975Google Scholar
  14. 14.
    McNamee, J.: Error bounds for the evaluation of integrals by the Euler-Maclaurin formula and by Gauss-type formulae. Math. Comput.18, 368–381 (1964)Google Scholar
  15. 15.
    Pagurova, V.I.: Tables of the Exponential integral\(E_r (x) = \int\limits_1^x {e^{ - xu} u^{ - r} du} \). New York: Pergamon Press 1961Google Scholar
  16. 16.
    Tricomi, F.G.: Sulla funzione gamma incompleta. Ann. Mat. Pura Appl.31, 263–279 (1950)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Giampietro Allasia
    • 1
  • Renata Besenghi
    • 1
  1. 1.Dipartimento di MatematicaUniversità di TorinoTorinoItaly

Personalised recommendations