beta

Beta function

Syntax

B = beta(Z,W)

Description

B = beta(Z,W) returns the beta function evaluated at the elements of Z and W. Both Z and W must be real and nonnegative.

Examples

Compute Beta Function for Integer Arguments

Compute the beta function for integer arguments $$w=3$$ and $$z=1,...,10$$. Based on the definition, the beta function can be calculated as

$\mathit{B}\left(\mathit{z},3\right)=\frac{\Gamma \left(\mathit{z}\right)\Gamma \left(3\right)}{\Gamma \left(\mathit{z}+3\right)}=\frac{\left(\mathit{z}-1\right)!2!}{\left(\mathit{z}+2\right)!}=\frac{2}{\mathit{z}\left(\mathit{z}+1\right)\left(\mathit{z}+2\right)}$.

Set the output format to rational to show the results as ratios of integers.

format rat
B = beta((1:10)',3)

B = 
       1/3     
       1/12    
       1/30    
       1/60    
       1/105   
       1/168   
       1/252   
       1/360   
       1/495   
       1/660   

Plot Beta Function

Calculate the beta function for $$z$$ = 0.05, 0.1, 0.2, and 1 within the interval $$0\le w\le 10$$. Loop over values of $$z$$, evaluate the function at each one, and assign each result to a row of B.

Z = [0.05 0.1 0.2 1];
W = 0:0.05:10;
B = zeros(4,201);
for i = 1:4
    B(i,:) = beta(Z(i),W);
end

Plot all of the beta functions in the same figure.

plot(W,B)
grid on
legend('$z = 0.05$','$z = 0.1$','$z = 0.2$','$z = 1$','interpreter','latex')
title('Beta function for $z = 0.05, 0.1, 0.2$, and $1$','interpreter','latex')
xlabel('$w$','interpreter','latex')
ylabel('$B(z,w)$','interpreter','latex')

Input Arguments

Z — Input array

scalar | vector | matrix | multidimensional array

Input array, specified as a scalar, vector, matrix, or multidimensional array. The elements of Z must be real and nonnegative. Z and W must be the same size, or else one of them must be a scalar.

Data Types: single | double

W — Input array

scalar | vector | matrix | multidimensional array

Input array, specified as a scalar, vector, matrix, or multidimensional array. The elements of W must be real and nonnegative. Z and W must be the same size, or else one of them must be a scalar.

• If Z or W is equal to 0, the beta function returns Inf.

• If Z and W are both 0, the beta function returns NaN.

Data Types: single | double

More About

Beta Function

The beta function is defined by

$$B(z,w)={\displaystyle {\int }_0^1{t}^{z-1}{(1-t)}^{w-1}}dt=\frac{\Gamma (z)\Gamma (w)}{\Gamma (z+w)}.$$

The $$\Gamma \left(z\right)$$ term is the gamma function

$$\Gamma \left(z\right)={\displaystyle {\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt}.$$

References

[1] Olver, F. W. J., A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds., Chapter 5.12 Beta Function, NIST Digital Library of Mathematical Functions, Release 1.0.22, Mar. 15, 2018.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function fully supports tall arrays. For more information, see Tall Arrays.

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).