# Euler's Reflection Formula/Corollary

## Corollary to Euler's Reflection Formula

Let $\Gamma$ denote the gamma function.

Then:

$\forall z \notin \Z: \paren {-z}! \, \map \Gamma z = \dfrac \pi {\map \sin {\pi z} }$

## Proof

 $\ds \dfrac \pi {\map \sin {\pi z} }$ $=$ $\ds \map \Gamma z \map \Gamma {1 - z}$ Euler's Reflection Formula $\ds$ $=$ $\ds \map \Gamma {-z + 1} \map \Gamma z$ $\ds$ $=$ $\ds \paren {-z}! \, \map \Gamma z$ Gamma Function Extends Factorial

$\blacksquare$

## Source of Name

This entry was named for Leonhard Paul Euler.