Improving Stirling's formula (Q5946712)

Stirling formula for real values of \(x\) \[ \Gamma(x+1) =2\pi x^{x +{1\over 2}}e^{-x} e^{\mu_0(x)}, \quad 0<\mu_0(x) <{1\over 12x},\;x>0, \] is compared with four other formulas. Two of these are \[ \Gamma(x+ \tfrac 12)= \sqrt {2\pi} x^xe^{-x} e^{\mu(\tfrac 12,x)},\;x>0, \] \[ \Gamma(x+\tfrac 12)= \sqrt {2\pi} (x^2-\tfrac {1}{12})^{x/2} e^{-x}e^{\mu_1(x)},\;x>\tfrac{1} {\sqrt {12}}. \] Bounds for the functions \(\mu(\tfrac 12,x)\) and \(\mu_1(x)\) (as well as for the corresponding functions in the other two formulas) are given, and it is shown that all four of the author's formulas are better than Stirling's formula.