Euler's Reflection Formula

Theorem

Let $\Gamma$ denote the gamma function.

Then:

$\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$

Corollary

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

Proof

We have the Weierstrass products:

$\ds \map \sin {\pi z} = \pi z \prod_{n \mathop \ne 0} \paren {1 - \frac z n} \map \exp {\frac z n}$
$\ds \frac 1 {\map \Gamma z} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {1 + \frac z n} \map \exp {-\frac z n}$

from which:

 $\ds \dfrac 1 {-z \, \map \Gamma z \map \Gamma {-z} }$ $=$ $\ds \frac {-z^2 \map \exp {\gamma z} \map \exp {-\gamma z} } {-z} \prod_{n \mathop = 1}^\infty \paren {1 + \frac z n} \paren {1 - \frac z n} \map \exp {\frac z n} \map \exp {-\frac z n}$ $\ds$ $=$ $\ds z \prod_{n \mathop = 1}^\infty \paren {1 - \frac {z^2} {n^2} }$ $\ds$ $=$ $\ds \dfrac {\map \sin {\pi z} } \pi$ Euler Formula for Sine Function

whence:

 $\ds \map \Gamma z \map \Gamma {1 - z}$ $=$ $\ds -z \, \map \Gamma z \map \Gamma {-z}$ Gamma Difference Equation $\ds$ $=$ $\ds \frac \pi {\map \sin {\pi z} }$

$\blacksquare$

Source of Name

This entry was named for Leonhard Paul Euler.