Gamma Function Result

Question

I want to know the way to prove that

$$ \Gamma(n + 1/2)= \frac{(2n)! \sqrt{\pi}}{4^n n!}. $$

I tried writing term by term and it gives the result, but I want to know how to prove it without observing each term.

Answer 1

Use the Legendre Duplication Formula. A proof can be found here.

We have $$ \Gamma \left({n}\right) \Gamma \left (n + \dfrac 1 2 \right) = 2^{1 - 2 n} \sqrt \pi \Gamma \left({2 n}\right)$$

So

\begin{align} \Gamma \left (n + \dfrac 1 2 \right) &= \frac {2^{1 - 2 n} \sqrt \pi \Gamma \left({2 n}\right)} {\Gamma \left({n}\right) } \\ &= \frac {2 \sqrt \pi (2n - 1)!}{4^n(n-1)!} \\ &= \frac {\sqrt \pi (2n)!}{4^n n!}\end{align}

as required.

Answer 2

Hint

By induction:

The result is true for $n=0$: $\Gamma\left(\frac12\right)=\sqrt \pi$. We can show this result by several ways.

In the inductive step use the equality: $\Gamma(x+1)=x\Gamma(x)$.