# Gamma Function Extends Factorial

## Theorem

$\forall n \in \N: \map \Gamma {n + 1} = n!$

## Proof

For $n = 0$:

 $\ds \map \Gamma 1$ $=$ $\ds \int_0^\infty e^{-t} \rd t$ Definition of Gamma Function $\ds$ $=$ $\ds \bigintlimits {-e^{-t} } 0 \infty$ $\ds$ $=$ $\ds 0 - \paren {-1}$ $\ds$ $=$ $\ds 1$

Then by Gamma Difference Equation:

$\forall z \in \Z_{> 0}: \map \Gamma {z + 1} = z \, \map \Gamma z$

Hence the result.

$\blacksquare$