I have got some knowledge about gamma function.But when it comes to beta function I'm really stuck.I couldn't get anything by looking at http://en.wikipedia.org/wiki/Beta_function.I came to know that gamma function is used to find the factorials of real numbers.

Can anyone explain me in laymen terms what really does a beta function does.

 

The gamma function takes one number (n) and outputs another number \(\Gamma \left( n \right)\) according to its definition formula.
That is it has one parameter, n.

The beta function has uses two numbers (m) and (n) and outputs a number according to its definition formula.
That is it has two parameters, m & n.

Because these functions are constructed in such a way as to be useful, they have connections to each other and to certain trigonometric and other (definite) integrals.

The gamma function can readily be evaluated to simple numbers for many useful trig and other functions, including powers.
The link to the beta function extends this to products and quotients of two functions, particularly trig functions.

The Wiki article has general formula, using these

\(\int\limits_0^{2\pi } {{{\left( {\cos \theta } \right)}^{2m - 1}}} {\left( {\sin \theta } \right)^{2n - 1}}d\theta = \frac{{\Gamma \left( m \right)\Gamma \left( n \right)}}{{2\Gamma \left( {m + n} \right)}}\)

Now put r = 2m-1 and 0 = 2n-1


\(\int\limits_0^{2\pi } {{{\left( {\cos \theta } \right)}^r}} d\theta = \frac{{\Gamma \left( {r + \frac{1}{2}} \right)\sqrt \pi }}{{2\Gamma \left( {\frac{r}{2} + 1} \right)}}\)

 

The gamma function takes one number (n) and outputs another number \(\Gamma \left( n \right)\) according to its definition formula.

Could you tell me what really beta function does as gamma function computes the factorial for real numbers(non-negative).