Gamma Function Extends Factorial

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Theorem

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


Proof

For $n = 0$:

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


Then by Gamma Difference Equation:

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

Hence the result.

$\blacksquare$


Sources