Gamma function $\Gamma (\alpha)$ is for $\alpha>0$ but how do we calculate value of gamma function at $-1/2$? Is not contradicting the definition?

Question

Gamma function $\Gamma(\alpha)$ is defined for $\alpha>0$ but how do we calculate value of gamma function at $-1/2$? Is not contradicting the definition?

Answer 1

You could extend the definition of $\Gamma(s)$ by the functional equation $\Gamma(s + 1) = s\Gamma(s)$. Setting $s = -1/2$, we get $\Gamma(1/2) = (-1/2)\Gamma(-1/2)$, or $\Gamma(-1/2) = -2\Gamma(1/2)$. Since $\Gamma(1/2) = \sqrt{\pi}$, then $\Gamma(-1/2) = -2\sqrt{\pi}$.