# Prove $\Gamma\left(\frac{1}{2}\right)= \sqrt\pi$, using $\Gamma(p)\Gamma(1-p) = \frac{\pi}{\sin(\pi p)}$

Prove that $\Gamma\left(\frac{1}{2}\right)= \sqrt\pi$

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

• Hint: what is $\sin(\pi/2)$? – gammatester Aug 5 '13 at 12:35
• @Sid: Welcome to MSE! It really helps readability to format questions using MathJax (see FAQ). Regards – Amzoti Aug 5 '13 at 12:50

## 1 Answer

Let $p=\frac12$, we have $$\Gamma\left(\frac12\right)\Gamma\left(\frac12\right)=\frac{\pi}{\sin\frac{\pi}{2}}=\pi$$ therefore $$\Gamma\left(\frac12\right)=\sqrt \pi$$

• (Because $\Gamma>0$. This is trivial, but forgetting that the square is not injective is often cause of problems) – Clement C. Aug 5 '13 at 13:50