An accurate approximation formula for gamma function

Abstract

Abstract In this paper, we present a very accurate approximation for the gamma function: Γ(x+1)∼2πx(xe)x(xsinh1x)x/2exp(73241x3(35x2+33))=W2(x) $$ \Gamma( x+1 ) \thicksim\sqrt{2\pi x} \biggl( \frac{x}{e} \biggr) ^{x} \biggl( x\sinh\frac{1}{x} \biggr) ^{x/2}\exp \biggl( \frac{7}{324}\frac{1}{x^{3} ( 35x^{2}+33 ) } \biggr) =W_{2} ( x ) $$ as x→∞ $x\rightarrow\infty$, and we prove that the function x↦lnΓ(x+1)−lnW2(x) $x\mapsto\ln \Gamma ( x+1 ) -\ln W_{2} ( x ) $ is strictly decreasing and convex from (1,∞) $( 1,\infty ) $ onto (0,β) $( 0,\beta ) $, where β=22,02522,032−ln2πsinh1≈0.00002407. $$ \beta=\frac{22{,}025}{22{,}032}-\ln\sqrt{2\pi\sinh1}\approx0.00002407. $$

Keywords

• Gamma function
• Monotonicity
• Convexity
• Approximation