Some properties of the gamma function including Stirling's formula

Abstract

The gamma function, introduced by the Swiss mathematician Leonhard Euler (1707-1783), plays a central role in modern mathematics. This remarkable brainchild was a result of generalizing the factorial to non integer values. Afterward, because of its great importance, it was discovered by great mathematicians as well as many others. The aim of this work is to give meticulous proofs of several important known properties of the gamma function. We start by defining gamma function as an integral form and it is extended by analytic continuation to all complex numbers except the non-positive integers, yielding a meromorphic function we henceforth continue to call the gamma function. After describing several important results of gamma function, we finalize our work with obtaining the Stirling's formula to observe the behavior of the gamma function for large values inasmuch as it indeed grows rapidly - faster than exponential function.