An ultimate extremely accurate formula for approximation of the factorial function

Abstract

We prove in this paper that for every x ≥ 0,

$$\sqrt{2\pi e}\cdot e^{-\omega}\left( \frac{x+\omega}{e}\right) ^{x+\frac {1}{2}} < \Gamma(x+1)\leq\alpha\cdot\sqrt{2\pi e}\cdot e^{-\omega}\left( \frac{x+\omega}{e}\right)^{x+\frac{1}{2}}$$

where ${\omega=(3-\sqrt{3})/6}$ and α = 1.072042464..., then

$$\beta\cdot\sqrt{2\pi e}\cdot e^{-\zeta}\left(\frac{x+\zeta}{e}\right)^{x+\frac{1}{2}}\leq\Gamma(x+1) < \sqrt{2\pi e}\cdot e^{-\zeta}\left( \frac{x+\zeta}{e}\right)^{x+\frac{1}{2}},$$

where ${\zeta=(3+\sqrt{3})/6}$ and β = 0.988503589... Besides the simplicity, our new formulas are very accurate, if we take into account that they are much stronger than Burnside’s formula, which is considered one of the best approximation formulas ever known having a simple form.