Stirling's Formula and Its Extension for the Gamma Function

We present new short proofs for both Stirling’s formula and Stirling’s formula for the Gamma function. Our approach in the …rst case relies upon analysis of Wallis’formula, while the second result follows from the logconvexity property of the Gamma function. The well known formula of Stirling asserts that (1) n! p 2 ne n as n!1; in the sense that the ratio of the two sides tends to 1. This provides an e¢ cient estimation to the factorial, used widely in probability theory and in statistical physics. Articles treating Stirling’s formula account for hundreds of items in JSTOR. A few of the most relevant references may be found in [2], [3], [5], and [7]. As was noticed by Stirling himself, the presence of in the formula (1) is motivated by the Wallis formula, 2 = lim n!1 2 2 4 4 (2n) (2n) 1 3 3 5 5 (2n 1) (2n 1) (2n+ 1) ;