Beta function
beta( returns the
beta function of
x,y)x and y.
Compute the beta function for these numbers. Because these numbers are not symbolic objects, you get floating-point results:
[beta(1, 5), beta(3, sqrt(2)), beta(pi, exp(1)), beta(0, 1)]
ans =
0.2000 0.1716 0.0379 InfCompute the beta function for the numbers converted to symbolic objects:
[beta(sym(1), 5), beta(3, sym(2)), beta(sym(4), sym(4))]
ans = [ 1/5, 1/12, 1/140]
If one or both parameters are complex numbers, convert these numbers to symbolic objects:
[beta(sym(i), 3/2), beta(sym(i), i), beta(sym(i + 2), 1 - i)]
ans = [ (pi^(1/2)*gamma(1i))/(2*gamma(3/2 + 1i)), gamma(1i)^2/gamma(2i),... (pi*(1/2 + 1i/2))/sinh(pi)]
Compute the beta function for negative parameters. If one or both arguments are negative numbers, convert these numbers to symbolic objects:
[beta(sym(-3), 2), beta(sym(-1/3), 2), beta(sym(-3), 4), beta(sym(-3), -2)]
ans = [ 1/6, -9/2, Inf, Inf]
Call beta for the matrix A and
the value 1. The result is a matrix of the beta functions
beta(A(i,j),1):
A = sym([1 2; 3 4]); beta(A,1)
ans = [ 1, 1/2] [ 1/3, 1/4]
Differentiate the beta function, then substitute the variable
t with the value 2/3 and approximate the result using vpa:
syms t u = diff(beta(t^2 + 1, t)) vpa(subs(u, t, 2/3), 10)
u = beta(t, t^2 + 1)*(psi(t) + 2*t*psi(t^2 + 1) -... psi(t^2 + t + 1)*(2*t + 1)) ans = -2.836889094
Expand these beta functions:
syms x y expand(beta(x, y)) expand(beta(x + 1, y - 1))
ans = (gamma(x)*gamma(y))/gamma(x + y) ans = -(x*gamma(x)*gamma(y))/(gamma(x + y) - y*gamma(x + y))
The beta function is uniquely defined for positive numbers and complex numbers with positive real parts. It is approximated for other numbers.
Calling beta for numbers that are not symbolic objects invokes the
MATLAB®
beta function. This function accepts real
arguments only. If you want to compute the beta function for complex numbers, use
sym to convert the numbers to symbolic
objects, and then call beta for those symbolic objects.
If one or both parameters are negative numbers, convert these numbers to symbolic
objects using sym, and then call beta
for those symbolic objects.
If the beta function has a singularity, beta returns the positive
infinity Inf.
beta(sym(0),0), beta(0,sym(0)), and
beta(sym(0),sym(0)) return NaN.
beta(x,y) = beta(y,x) and beta(x,A) =
beta(A,x).
At least one input argument must be a scalar or both arguments must be vectors or
matrices of the same size. If one input argument is a scalar and the other one is a vector
or a matrix, beta(x,y) expands the scalar into a vector or matrix of
the same size as the other argument with all elements equal to that scalar.
[1] Zelen, M. and N. C. Severo. “Probability Functions.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.