ln_Barnes_G

Computes the logarithm of the Barnes G-function

This function computes the logarithm of the Barnes G-function (dlmf.nist.gov/5.17) in the entire complex plane. The algorithm is based on approximating the log(G(z)) in the half-plane Re(z)>1.5 by an algorithm from [1] (with improved coefficients obtained in [2]). This approximation is correct to 10^{-16} (though round-off errors may decrease the accuracy). The approximation in the half-plane Re(z)>1.5 is then used to compute ln(G(z)) in the entire complex plane by the use of reflection formula and the functional equation.

References:

[1] A. Kuznetsov, "Computing the Barnes G-function and the gamma function in the entire complex plane", Journal of Computational and Applied Mathematics, Vol. 411, 2022, 114270. https://doi.org/10.1016/j.cam.2022.114270

[2] A. Kuznetsov, A. Mohammadioroojeh, "Approximating functions on R^+ by exponential sums", 2025, preprint, https://arxiv.org/abs/2508.19095