How is the following related to the Gamma function?

Question

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.

The relation is via the Beta function : http://en.wikipedia.org/wiki/Beta_function — Prahlad Vaidyanathan, Dec 03 '13 at 17:34

score 1 · Answer 1 · answered Dec 03 '13 at 17:35

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

answered Dec 03 '13 at 17:35

Igor Rivin

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sorry, my mistake, I said it should be in as simple terms as possible. – Shubham Dec 04 '13 at 04:17
Does the RHS have double factorials? – Shubham Dec 04 '13 at 04:20
@Shubham The RHS can be written as $(m-1)! (n-1)!/(m+n)!,$ and each factorial is a $\Gamma$ function... – Igor Rivin Dec 04 '13 at 04:24
But isn't (m-1)!=(m-1)(m-2)...3.2.1 ? – Shubham Dec 04 '13 at 16:10

score 1 · Answer 2 · answered Apr 21 '14 at 19:22

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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.

answered Apr 21 '14 at 19:22

A.Chakraborty

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How is the following related to the Gamma function?

2 Answers2