# How do you find the factorial of negative numbers?

##### 2 Answers

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

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

# Gamma(z) = int_0^oo t^(z-1)e^-t 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) #

It depends...

#### Explanation:

Strictly speaking, the domain of the function

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

#Gamma(z) = int_0^oo x^(z-1) e^(-x) dx#

Then for any non-negative integer:

#n! = Gamma(n+1)#

and we can write:

#y = Gamma(x+1)#

The integral definition given above converges for any positive Real value of

The Gamma function can be analytically continued to be defined for all Complex numbers except

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) :}#