An accurate approximation formula for gamma function

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*}