An accurate approximation formula for gamma function

TL;DR

This paper introduces a highly accurate approximation formula for the gamma function that improves existing methods, providing a decreasing and convex error function with a very small bound as x approaches infinity.

Contribution

The paper proposes a new approximation formula for the gamma function with proven decreasing and convex error properties, enhancing accuracy over previous approximations.

Findings

The approximation formula closely matches the gamma function for large x.

The error function between the log gamma and the approximation is strictly decreasing and convex.

The maximum error bound is approximately 0.00002407 as x approaches infinity.

Abstract

In this paper, we present a very accurate approximation for gamma function: \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi x}\left( \dfrac{x}{e}\right) ^{x}\left( x\sinh \frac{1}{x}\right) ^{x/2}\exp \left( \frac{7}{324}\frac{1}{ x^{3}\left( 35x^{2}+33\right) }\right) =W_{2}\left( x\right) \end{equation*} as $x\rightarrow \infty $, and prove that the function $x\mapsto \ln \Gamma \left( x+1\right) -\ln W_{2}\left( x\right) $ is strictly decreasing and convex from $\left( 1,\infty \right) $ onto $\left( 0,\beta \right) $, where \begin{equation*} \beta =\frac{22\,025}{22\,032}-\ln \sqrt{2\pi \sinh 1}\approx 0.00002407. \end{equation*}