The gamma function is defined by the following integral that shows up frequently in many pure and applied mathematical settings:

See a graph | Some Fractional Values | Value of Gamma Function

Note that the gamma function with a negative argument is defined by utilizing the recursion formula explained in the next section.

Recursion Formula: Given the following formula, a gamma function at one point can be evaluated recursively in terms of its value at another point:

Generalized Factorial: When x is a positive integer, one can easily prove the following by using the recursive formula:

In fact, many formulas involving n! can be extended to non-integer cases by replacing n! with , and that is why the gamma function is also commonly referred to as the generalized factorial function.

Reflection Formula:

Multiplication Formula: