Introduction to the Gamma Function Pascal Sebah and Xavier Gourdon numbers.computation.free.fr/Constants/constants.html February 4, 2002 Abstract An elementary introduction to the celebrated gamma function Γ(x) and its various representations. Some of its most important properties are described. 1 Introduction The gamma function was first introduced by the Swiss mathematician Leon- hard Euler (1707-1783) in his goal to generalize the factorial to non integer values. Later, because of its great importance, it was studied by other eminent mathematicians like Adrien-Marie Legendre (1752-1833), Carl Friedrich Gauss (1777-1855), Christoph Gudermann (1798-1852), Joseph Liouville (1809-1882), Karl Weierstrass (1815-1897), Charles Hermite (1822-1901), ... as well as many others. The gamma function belongs to the category of the special transcendental functions and we will see that some famous mathematical constants are occur- ring in its study. It also appears in various area as asymptotic series, definite integration, hypergeometric series, Riemann zeta function, number theory ... Some of the historical background is due to Godefroy’s beautiful essay on this function [9] and the more modern textbook [3] is a complete study. 2 Definitions of the gamma function 2.1 Definite integral During the years 1729 and 1730 ([9], [12]), Euler introduced an analytic function which has the property to interpolate the factorial whenever the argument of the function is an integer. In a letter from January 8, 1730 to Christian Goldbach he proposed the following definition : Definition 1 (Euler, 1730) Let x> 0 Γ(x)=  1 0 (− log(t)) x−1 dt. (1) 1