A Contour Integral and Complex Power Series Proof of Stirling Formula

Abstract

As a matter of fact, the Stirling formula is a very important formula for estimating the size of factorials, which effectively simplifies the calculation of factorials. Based on the convergency theorem, it is very accurate when n is very small, for example, when n=6, the error is only 1.4%. This formula was first discovered by Abraham de Moivre and Stirling, and mathematicians such as provided much proof of it. In addition, there is a famous proof that only relies on ordinary calculus. With this in mind, this paper attempts to independently solve this problem using simple complex analysis methods. To be specific, contour integral, Residue Theorem, complex series will be demonstrated directly and immediately. At the same time, the proof processing will be presented in detail based on the derivations of the formulae. In the meantime, the current limitations will be clarified and the prospects will be proposed according to the analysis. Overall, these results shed light on guiding further exploration of Stirling Formula proofing and applications.