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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 ${\Psi }\left(a,c;x\right)=2{x}^{1-c}/{\Gamma }\left(a\right){\int }_{0}^{+\infty }{e}^{-{u}^{2}}\left({u}^{2a-1}/{\left(x+{u}^{2}\right)}^{1-c+a}\right)du$ for $a,x\in {ℝ}^{+}$, $c\in ℝ$. 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.

##### MSC:
 65D20 Computation of special functions, construction of tables 33B15 Gamma, beta and polygamma functions 65B15 Euler-Maclaurin formula (numerical analysis)
##### References:
 [1] Tricomi, F. G.: Funzioni Ipergeometriche Confluenti. Roma: Cremonese 1954. [2] Slater, L. J.: Confluent Hypergeometric Functions. Cambridge: Cambridge Univ. Press 1960. [3] Luke, Y. L.: The Special Functions and their Approximations, Vols. I, II. New York: Academic Press 1969. [4] Davis, P. J., Rabinowitz, P.: Methods of Numerical Integration, 2nd ed. New York: Academic Press 1984. [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. [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.