What is pi factorial (or pi!)?

Question

15562 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-07-25T07:59:44

$Γ (π + 1) \approx 7.188082729$ $Gamma(pi+1) ~~ 7.188082729$

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:

$Gamma(pi+1) ~~ 7.188082729$