ln_gamma

Computes the logarithm of the Gamma function in the entire complex plane.

This function computes the logarithm of the gamma function in the entire complex plane. The algorithm is based on approximating the log(Gamma(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(Gamma(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