Numerical computation of Tricomi’s psi function by the trapezoidal rule. (English) Zbl 0618.65009

The trapezoidal rule is applied to the numerical calculation of the integral representation of Tricomi’s psi function \(\Psi (a,c;x)=2x^{1- c}/\Gamma (a)\int ^{+\infty}_{0}e^{-u^ 2}(u^{2a-1}/(x+u^ 2)^{1-c+a})du\) for \(a,x\in {\mathbb{R}}^ +\), \(c\in {\mathbb{R}}\). The unexpectedly high accuracy is explained by means of a careful investigation in the complex field of the Euler-Maclaurin formula, and particularly of its remainder terms, considered as an extension of the trapezoidal rule. Since the same method has been used previously to evaluate the complementary incomplete gamma function [the authors, Numer. Math. 50, 419-428 (1987; Zbl 0593.65017)], the Euler gamma function and the digamma function, the present paper limits itself to quote the main features of the method and describes fully the peculiarities of this application.
A simple and efficient numerical procedure for obtaining values of the psi function is given; moreover, to reduce the amount of calculation, an iterative algorithm for the evaluation of the trapezoidal rule, similar to Horner’s scheme for polynomials, is suggested.


65D20 Computation of special functions and constants, construction of tables
33B15 Gamma, beta and polygamma functions
65B15 Euler-Maclaurin formula in numerical analysis


Zbl 0593.65017
Full Text: DOI


[1] Tricomi, F. G.: Funzioni Ipergeometriche Confluenti. Roma: Cremonese 1954. · Zbl 0068.28005
[2] Slater, L. J.: Confluent Hypergeometric Functions. Cambridge: Cambridge Univ. Press 1960. · Zbl 0086.27502
[3] Luke, Y. L.: The Special Functions and their Approximations, Vols. I, II. New York: Academic Press 1969. · Zbl 0193.01701
[4] Davis, P. J., Rabinowitz, P.: Methods of Numerical Integration, 2nd ed. New York: Academic Press 1984. · Zbl 0537.65020
[5] Allasia, G., Besenghi, R.: Numerical calculation of incomplete gamma functions by the trapezoidal rule. Num. Math.50, 419–428 (1987). · Zbl 0593.65017 · doi:10.1007/BF01396662
[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.
[7] Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. New York: Dover Publications 1970. · Zbl 0171.38503
[8] Wimp, J.: On the computation of Tricomi’s {\(\Psi\)} function. Computing13, 195–203 (1974). · Zbl 0294.65010 · doi:10.1007/BF02241712
[9] Temme, N. M.: The numerical computation of the confluent hypergeometric functionU(a, b; z). Num. Math.41, 63–82 (1983). · Zbl 0489.33001 · doi:10.1007/BF01396306
[10] Luke, Y. L.: Mathematical Functions and their Approximations. New York: Academic Press 1975. · Zbl 0318.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.