Computing the Barnes $G$-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 $Re(z)\ge 3/2$. These approximations involve only elementary functions and are easy to implement. The algorithm is based on a two-point Pad\'e approximation and we use it to provide two approximations to $\ln(G(z))$, accurate to $3 \times 10^{-16}$ and $3 \times 10^{-31}$ in the half-plane $Re(z)\ge 3/2$; 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.