Euler's Reflection Formula/Corollary

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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.


Sources