Euler's Reflection Formula

Theorem
Let $\Gamma$ denote the gamma function.

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

Proof
We have the Weierstrass products:


 * $\displaystyle \map \sin {\pi z} = \pi z \prod_{n \mathop \ne 0} \paren {1 - \frac z n} \map \exp {\frac z n}$

From the Weierstrass form of the Gamma function:


 * $\displaystyle \frac 1 {\map \Gamma z} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {1 + \frac z n} \map \exp {-\frac z n}$

from which:

whence: