Leonhard Euler solved the problem of finding a function of a continuous variable $x$ which for integer values of $x=n$ agrees with the factorial function $n\mapsto n!$. The gamma function is a shift by one of the solution to this problem.
For a complex variable $x\neq -1,-2,\ldots$, we define $\Gamma(x)$ by the formula
where $(x)_0 =1$ and for positive integer $k = 1,2,\ldots$,
is the “Pochhammer symbol” (or rising factorial). It easily follows that $\Gamma(n+1) = n!$ for natural numbers $n = 0, 1, 2, \ldots$.
As a function of a complex variable, the Gamma function $\Gamma(x)$ is a meromorphic function with simple poles at $x = 0, -1, -2, \ldots$.
Extending the recursive definition of the ordinary factorial function, the Gamma function satisfies the following translation formula:
away from $x = 0, -1, -2, \ldots$.
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 $(0, \infty)$ is the unique function $f$ such that $f(x+1) = x f(x)$, $f(1) = 1$, and $\log f$ is convex.
A number of other representations of the Gamma function are known and frequently utilized, e.g.,
Product representation:
where $\gamma$ is Euler's constant?.
Integral representation: