New approximations of the gamma function in terms of the digamma function (Q1049266)

The author proves the following asymptotic formula \[ \Gamma(x)\approx\sqrt{2\pi}e^{-b}\,(x+b)^x\,\exp\big(-x-\frac{1}{2}\,\psi(x+c)\big),\;\;\text{as}\,\,x\to\infty,\;\;x\in\mathbb{N}\,, \] where \(\Gamma(x)\) is Euler's gamma function and \(\psi(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}\). Moreover, optimal values of parameters \(b,\,c\) are calculated in such a way that this asymptotic convergence is the best possible.