Question

What is pi factorial (or pi!)?

Answer 1

`Gamma(pi+1) ~~ 7.188082729`

Explanation:

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`