# Documentation

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# beta

Beta function

## Syntax

```B = beta(Z,W) ```

## Description

`B = beta(Z,W)` computes the beta function for corresponding elements of arrays `Z` and `W`. The arrays must be real and nonnegative. They must be the same size, or either can be scalar.

## Examples

In this example, which uses integer arguments,

```beta(n,3) = (n-1)!*2!/(n+2)! = 2/(n*(n+1)*(n+2))```

is the ratio of fairly small integers, and the rational format is able to recover the exact result.

```format rat beta((0:10)',3) ans = 1/0 1/3 1/12 1/30 1/60 1/105 1/168 1/252 1/360 1/495 1/660 ```

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### Beta Function

The beta function is

`$B\left(z,w\right)={\int }_{0}^{1}{t}^{z-1}{\left(1-t\right)}^{w-1}dt=\frac{\Gamma \left(z\right)\Gamma \left(w\right)}{\Gamma \left(z+w\right)}$`

where Γ(z) is the gamma function.

## Algorithms

`beta(z,w) = exp(gammaln(z)+gammaln(w)-gammaln(z+w))`