PlanetMath: beta function

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beta function

(Definition)

The beta function is defined as

$\displaystyle B(p,q) = \int_0^1 x^{p-1} (1-x)^{q-1} dx$

. For other complex values of

and

, we can define

The beta function has the property

$\displaystyle B(p,q) = \frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$

and

for which the right-hand side is defined. Here $\Gamma$ is the gamma function.

Also,

$\displaystyle B(p,q) = B(q,p)$

and

$\displaystyle B({\textstyle\frac{1}{2},\frac{1}{2}}) = \pi.$

The beta function was first defined by L. Euler in 1730, and the name was given by J. Binet.

"beta function" is owned by yark. [ full author list (2) | owner history (1) ]

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Attachments:: evaluation of beta function using Laplace transform (Derivation) by rspuzio
alternate integral representation of beta function (Result) by rspuzio

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Cross-references: gamma function, complex numbers, analytic continuation, complex, real numbers
There are 5 references to this entry.

This is version 18 of beta function, born on 2003-02-09, modified 2007-04-28.
Object id is 4001, canonical name is BetaFunction.
Accessed 8168 times total.

Classification:

33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions)

Pending Errata and Addenda

None.[ View all 11 ]

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