How do you find the factorial of negative numbers?

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(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) #

Strictly speaking, the domain of the function #y = x!# is #NN_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(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 #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) :}#