# beta, betainc, betaln

## Purpose

Beta function.

## Synopsis

    y = beta(z,w)
    y = betainc(x,a,b)
    y = betaln(z,w)

## Description

`y = beta(z,w)` is the beta function

or equivalently `gamma(z)`*`gamma(w)./gamma(z+w)`. If both `z` and `w` are vectors or matrices, they must be the same size.

`y = betainc(x,z,w)` is the incomplete beta function

`y = betaln(z,w)` is the natural logarithm of the beta function, `log(beta(z,w)),` computed without computing `beta(z,w)`. Since the `beta` function can range over very large or very small values, its logarithm is sometimes more useful. If both `z` and `w` are vectors or matrices, they must be the same size.

## Examples

    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

In this case, with 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.

For `x = 510`, `betaln(x,x) = -708.8616`, which, on a computer with IEEE arithmetic, is slightly less than `log(realmin)`. Here `beta(x,x)` would underflow (or be denormal).

## Algorithm

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

(c) Copyright 1994 by The MathWorks, Inc.