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QUESTION

# 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