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Concept
Beta function
Summary
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
for complex number inputs
such that .
The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital beta.
The beta function is symmetric, meaning that
for all inputs and .
A key property of the beta function is its close relationship to the gamma function:
A proof is given below in .
The beta function is also closely related to binomial coefficients. When m (or n, by symmetry) is a positive integer, it follows from the definition of the gamma function Γ that
A simple derivation of the relation can be found in Emil Artin's book The Gamma Function, page 18–19.
To derive this relation, write the product of two factorials as
Changing variables by u = st and v = s(1 − t), because u + v = s and u / (u+v) = t, we have that the limits of integrations for s are 0 to ∞ and the limits of integration for t are 0 to 1. Thus produces
Dividing both sides by gives the desired result.
The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking
one has:
We have
where denotes the polygamma function.
Stirling's approximation gives the asymptotic formula
for large x and large y.
If on the other hand x is large and y is fixed, then
The integral defining the beta function may be rewritten in a variety of ways, including the following:
where in the second-to-last identity n is any positive real number. One may move from the first integral to the second one by substituting .
Official source
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