The Lanczos Approximation for the $Γ$-Function with Complex Coefficients

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.