Euler's Reflection Formula

Theorem
Let $\Gamma$ denote the gamma function.

Then:
 * $\forall z \notin \Z: \Gamma \left({z}\right) \Gamma \left({1 - z}\right) = \dfrac \pi {\sin \left({\pi z}\right)}$

Proof
We have the Weierstrass products:


 * $\displaystyle \sin \left({\pi z}\right) = \pi z \prod_{n \mathop \ne 0} \left({1 - \frac z n}\right) \exp \left({\frac z n}\right)$


 * $\displaystyle \frac 1 {\Gamma \left({z}\right)} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \left({1 + \frac z n}\right) \exp \left({-\frac z n}\right)$ : see Equivalence of Definitions of Gamma Function.

from which:


 * $\dfrac 1 {-z \Gamma \left({z}\right) \Gamma \left({-z}\right)} = \dfrac {\sin \left({\pi z}\right)} \pi$

whence: