Reflection relation for gamma function

Statement

Let $\Gamma$ denote the gamma function. Then, for any $z$ that is not an integer, we have:

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

Facts used

• Recurrence relation for gamma function: This shows that it suffices to prove the result for $\operatorname {Re} (z)\in (0,1)$