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In mathematics, the **beta function** (occasionally written as *Beta function*), also called the Euler integral of the first kind, is a special function defined by

for Re(*x*), Re(*y*) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.

## Properties Edit

The beta function is symmetric, meaning that

It has many other forms, including:

where is the gamma function and (*x*)_{n} is the falling factorial; i.e., . The second identity shows in particular .

Like the gamma function for integers describes factorials, the beta function can define a binomial coefficient after adjusting indices:

The beta function was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano.

## Derivatives Edit

The derivatives follow:

where is the digamma function.

## Integrals Edit

The Nörlund-Rice integral is a contour integral involving the beta function.

## Incomplete beta functionEdit

The **incomplete beta function** is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. The situation is analogous to the incomplete gamma function being a generalization of the gamma function.

The incomplete beta function is defined as

For *x* = 1, the incomplete beta function coincides with the complete beta function.

The **regularized incomplete beta function** (or **regularized beta function** for short) is defined in terms of the incomplete beta function and the complete beta function:

### PropertiesEdit

*(Many other properties could be listed here.)*

## ReferencesEdit

- Milton Abramowitz and Irene A. Stegun, eds.
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.*New York: Dover, 1972.*(See §6.2, 6.6, and 26.5)* - W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling.
*Numerical Recipes in C*. Cambridge, UK: Cambridge University Press, 1992. Second edition.*(See section 6.4)*