Leonhard Euler solved the problem of finding a function of a continuous variable which for integer values of agrees with the factorial function . In fact, gamma function is a shift by one of the solution of this problem.
For a complex variable , we define by the formula
where and for positive integer ,
is the “Pochhammer symbol” (or rising factorial). It easily follows that for natural numbers .
As a function of a complex variable, the Gamma function is a meromorphic function with simple poles at .
Extending the recursive definition of the ordinary factorial function, the Gamma function satisfies the following translation formula:
away from .
It also satisfies a reflection formula, due to Euler:
Quite remarkably, the Gamma function (this time as a function of a real variable) is uniquely characterized in the following theorem:
The restriction of the Gamma function to the interval is the unique function such that , , and is convex.
A number of other representations of the Gamma function are known and frequently utilized, e.g.,
where is Euler's constant?.