# How do I find the factorial of a negative number?

Jul 24, 2015

It depends.

#### Explanation:

Factorial as such is only defined for non-negative integers with the familiar recursive definition:

0! = 1
(n+1)! = (n+1)n! for $n \ge 0$

There are a couple of extensions of the definition of factorial to cover a larger domain.

Euler's gamma function

The most mainstream extension of the definition of factorial is given by Euler's gamma function,

For positive integers:

Gamma(n) = (n-1)!

For any complex number $t$ with a positive real part:

$\Gamma \left(t\right) = {\int}_{0}^{\infty} {x}^{t - 1} {e}^{- x} \mathrm{dx}$

Then we can use the identity $\Gamma \left(t + 1\right) = t \Gamma \left(t\right)$ to extend the definition to all values except negative integers (which would entail division by $0$).

Roman factorial

This extends the definition of factorial to the negative integers as follows:

|__n~|! = n! for $n \ge 0$

|__n~|! = (-1)^(-n-1)/((-n-1)!) for $n < 0$