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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 any real numbers $ p,q > 0$. For other complex values of $ p$ and $ q$, we can define $ B(p,q)$ by analytic continuation.

The beta function has the property

$\displaystyle B(p,q) = \frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)} $
for all complex numbers $ p$ and $ q$ 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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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:
AMS MSC33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions)

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