# How is the following related to the Gamma function?

There's a shortcut formula in my book:

$$\int_{0}^{\pi/2}\sin^{m}\left(x\right)\cos^{n}\left(x\right)\,{\rm d}x = {\left[\left(m - 1\right)\left(m - 3\right)\ldots\,2\ \mbox{or}\ 1\right] \left[\left(n - 1\right)\left(n - 3\right)\ldots\,2\ \mbox{or}\ 1\right] \over \left(m + n\right)\left(m + n - 2\right)\ldots\,2\ \mbox{or}\ 1}$$

On the topic it just says Gamma Function. Please answer using as simple terms as possible. I have aware only about elementary integration.

Gamma function at integer argument is the factorial function, and the right hand side of your formula is a ratio of products of factorials...

• sorry, my mistake, I said it should be in as simple terms as possible. – Shubham Dec 4 '13 at 4:17
• Does the RHS have double factorials? – Shubham Dec 4 '13 at 4:20
• @Shubham The RHS can be written as $(m-1)! (n-1)!/(m+n)!,$ and each factorial is a $\Gamma$ function... – Igor Rivin Dec 4 '13 at 4:24
• But isn't (m-1)!=(m-1)(m-2)...3.2.1 ? – Shubham Dec 4 '13 at 16:10

What you have here is basically the beta function $\mathbf{B(m,n)}$. Now, $\mathbf{B(m,n)} = \frac{\Gamma(m).\Gamma(n)}{\Gamma(m+n)}$. Also, for integer n, we have $\Gamma(n) = (n-1)!$. The result follows thereby.