Computing the Barnes G-function in the entire complex plane

09/24/2021

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by   Alexey Kuznetsov, et al.

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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.