Computing the Barnes G-function and the gamma function in the entire complex plane

2021/09/24 by Kuznetsov, Alexey · 1 citation
#33B15 #33E30 #FOS: Mathematics #Numerical Analysis (math.NA) #Primary 33F05 #Secondary 30E10

paper · doi:10.48550/arxiv.2109.12061

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.

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• Approximating functions on \mathbb R⁺ by exponential sums

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