What is pi factorial (or pi!)?

##Gamma(pi+1) ~~ 7.188082729##

Strictly speaking, factorial is only defined for non-negative integers.

The usual recursive definition is:

##{ (0! = 1), (n! = n(n-1)! " for " n > 0) :}##

The normal way to extend the definition beyond non-negative integers is the Gamma function ##Gamma(x)##, which satisfies:

##Gamma(n) = (n-1)!##

for all positive integer values of ##n##

So using the Gamma function, "pi factorial" is ##Gamma(pi+1)##

For positive Real numbers (and Complex numbers with a positive Real part) we can define:

##Gamma(t) = int_(x=0)^oo x^(t-1) e^(-x) dx##

It is then possible to extend the definition to all Complex numbers, except the negative integers.

With this definition we find:

##Gamma(pi+1) ~~ 7.188082729##