Euler's Reflection Formula

Theorem
Let $\Gamma$ denote the gamma function.

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

Proof
We have the Weierstrass products:


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


 * $\displaystyle \frac 1 {\Gamma \left({z}\right)} = z e^{\gamma z} \prod_{n=1}^\infty \left({ 1 + \frac z n}\right) \exp\left({-\frac z n}\right)$

From which we see that:


 * $\displaystyle \frac{1}{-z \Gamma(z) \Gamma(-z)} = \frac{\sin( \pi z)} \pi$

whence