Numerical calculation of incomplete gamma functions by the trapezoidal rule. (English) Zbl 0593.65017

Allasia, Giampietro; Besenghi, Renata

Numer. Math. 50, 419-428 (1987).

The trapezoidal rule is applied to the numerical calculation of the known integral representation of the complementary incomplete gamma function \[ \Gamma (a,x)=(2e^{-x}x^ a/\Gamma (1-a))\int^{+\infty}_{0}e^{- u^ 2}u^{-2a+1}/x+\quad u^ 2\quad du \] in the region \(a<-1\) and \(x>0\). This application of the trapezoidal rule is not standard since the function \(u^{-2a+1}(x+u^ 2)^{-1}\) in the integrand is not even, but numerical tests show that using the rule may still be convenient. The explanation of this surprising fact requires a careful investigation of the related Euler-Maclaurin formula and expecially of its remainder term. The outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with good accuracy in the stated region. The explained method can be applied to the numerical evaluation of other important special functions.

MSC:

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

Keywords:

complementary incomplete gamma function; trapezoidal rule; Euler- Maclaurin formula

Software:

Algorithm 542

References:

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