What is pi factorial (or pi!)?

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Answer 1 · 2016-07-25T07:59:44

#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#