Computing the Barnes [formula omitted]-function and the gamma function in the entire complex plane

Abstract

We present an algorithm for generating approximations for the logarithm of Barnes G-function in the half-plane Re(z)≥3/2. These approximations involve only elementary functions and are easy to implement. The algorithm is based on a two-point Padé approximation and we use it to provide two approximations to ln(G(z)), accurate to 3×10−16 and 3×10−31 in the half-plane Re(z)≥3/2; a reflection formula is then used to compute Barnes G-function in the entire complex plane. A by-product of our algorithm is that it also produces accurate approximations to the gamma function.