Defining the gamma function in the usual way, \begin{align} \Gamma(u)=\int^{\infty}_{0}t^{u-1}e^{-t}dt \end{align} We know that for the positive integers we get $(u-1)!$ and the property $\Gamma(u+1)= u\Gamma(u)$. I'am trying to prove that when $u = \frac{n}{2}$ where n is an odd integer positive number, we get the following expression:
\begin{align} \Gamma\left(\frac{n}{2}\right)=\frac{\sqrt{\pi}(n-1)!}{2^{n-1}(\frac{n-1}{2})!} \end{align}
I'd like suggestions as I'm not making any progress by myself, please.
For $n=1$, try the substitution $t=s^2$ in the integral.
Once you have $n=1$, you can get the others via the functional equation.