Find .

First, consider the beta function, and note that

and, using the Euler reflection formula,

.

How does this relate to our integral? One may recall that

.

Now, .

Thus,

.

In fact,

>.

(See here for a similar use of the beta function and Euler reflection formula.)

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Tags: Beta Function, Euler Reflection Formula, Gamma Function, Integral, Math, Monday Math

This entry was posted on June 15, 2009 at 12:08 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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June 15, 2009 at 5:25 pm |

Hi I’m an amateur mathematician. I’ve been trying for about year to get the lowdown on the beta function.

I need to know why and how Euler derived the function, its significance, and how and when it is used.

Any clue will be appreciated.

Thanks, Martin Flaxman

June 17, 2009 at 9:42 am |

Here is a short biography of Euler, which mentions that he introduced the beta and gamma functions, as integrals, in 1729, and that they were named by later mathematicians. In particular:

One use of the beta function is as the normalization factor in the beta distribution. It also arises in the stochastic urn process known as a preferential attachment process.

One might also read about physicist Gabriele Veneziano, who, in applying the beta function to the scattering amplitudes of particles interacting via the strong force, led to the creation of string theory.

I hope these give you places to start