# How do you find the factorial of negative numbers?

Feb 5, 2017

The factorial function in the traditional sense takes only non-negative integers as the domain, with the convention that 0! =1

However the function can be extended to the range of all real numbers using the Gamma function,

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

which is what you have graphed. The gamma function is not the same as the factorial function, however it does have the property that for positive numbers that:

 n! = Gamma(n+1)

Using the gamma function we can therefore put a meaning to fractional and negative values, so for example:

 Gamma(1/2) = (-1/2)! = sqrt(pi)

Feb 5, 2017

It depends...

#### Explanation:

Strictly speaking, the domain of the function y = x! is ${\mathbb{N}}_{0}$, i.e. the Natural numbers (including $0$).

There are various ways to extend the domain, but the one most generally used is the Gamma function:

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

Then for any non-negative integer:

n! = Gamma(n+1)

and we can write:

$y = \Gamma \left(x + 1\right)$

The integral definition given above converges for any positive Real value of $z$, or any Complex value with positive Real part.

The Gamma function can be analytically continued to be defined for all Complex numbers except $0$ and negative integers.

So this does not give you a definition of factorial for negative integers.

The only extension of the definition of factorial that I have encountered that does have values for negative integers is the Roman Factorial:

stackrel "" (|__n ~|!) = {(n! " if " n >= 0), ((-1)^(-n-1)/((-n-1)!) " if " n < 0) :}