# beta

beta function (Euler integral of the first kind)

## Calling Sequence

    z = beta(x,y)

## Arguments

x, y

  2 positive real scalars, vectors or matricesof equal sizes.

z

  a real or a matrix of the same size than `x` with `z(i,j) = beta(x(i,j),y(i,j))`.

## Description

Computes the complete beta function :

For small `x` and `y` (`x+y≤2` elementwise), the algorithm uses the expression in function of the gamma function, else it applies the exponential function onto the result of the `betaln` function provided with the DCDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameter (see cdfbet for more information about DCDFLIB).

## Examples

    // example 1 :
    beta(5,2) - beta(2,5)   // symmetry (must be exactly 0)
    beta(0.5,0.5)           // exact value is pi

    // example 2 : an error study based on the relation  B(1,x) = 1/x
    // (computing 1/x must lead to only a relative error of eps_m, so
    //  it may be used near as a reference to evaluate the error in B(1,x))
    x = logspace(-8,8,20000)';
    e = beta(ones(x),x) - (1)./x;
    er = abs(e) .* x;
    ind = find(er ~= 0);
    eps = ones(x(ind))*number_properties("eps");
    clf()
    plot2d(x(ind),[er(ind) eps 2*eps],style=[1 2 3],logflag="ll",leg="er@eps_m@2 eps_m")
    xtitle("approximate relative error in computing beta(1,x)")
    show_window()

x = logspace(-8,8,20000)'; e = beta(ones(x),x) - (1)./x; er = abs(e) .* x; ind = find(er ~= 0); eps = ones(x(ind))*number_properties("eps"); plot2d(x(ind),[er(ind) eps 2*eps],style=[1 2 3],logflag="ll",leg="er@eps_m@2 eps_m") xtitle("approximate relative error in computing beta(1,x)")

    // example 3 : plotting the beta function
    t = linspace(0.2,10,60);
    X = t'*ones(t); Y = ones(t')*t;
    Z = beta(X,Y);
    clf()
    plot3d(t, t, Z, flag=[2 4 4], leg="x@y@z", alpha=75, theta=30)
    xtitle("The beta function on [0.2,10]x[0.2,10]")
    show_window()

t = linspace(0.2,10,60); X = t'*ones(t); Y = ones(t')*t; Z = beta(X,Y); clf() plot3d(t, t, Z, flag=[2 4 4], leg="x@y@z", alpha=75, theta=30) xtitle("The beta function on [0.2,10]x[0.2,10]")

## See Also

- gamma
- cdfbet