The Lanczos Approximation for the $\Gamma$-Function with Complex Coefficients

TL;DR

This paper analyzes the Lanczos approximation for the gamma function with complex coefficients, revealing that using complex coefficients increases computational cost and decreases accuracy, thus recommending real coefficients for practical use.

Contribution

It provides a detailed study of the properties and convergence of Lanczos coefficients with complex parameters, offering practical guidance for numerical gamma function evaluation.

Findings

Complex coefficients increase computational cost.

Complex coefficients decrease approximation accuracy.

Real coefficients are recommended for practical applications.

Abstract

We examined the properties of the coefficients of the \cite{lanczos1964} approximation of the $\Gamma$-function with complex values of the free parameter together with the convergence properties of the approximation when using these coefficients. We report that for fixed real parts of the free parameter that using complex coefficients both increases the computational cost of the Lanczos approximation while drecreasing the accuracy. We conclude that in practical applications of numerical evaluation of the $\Gamma$-function only coefficients generated with real values of the free parameter should be used.