Beta Function B(a,b) Calculator

Compute the Beta function B(a,b)=∫_0^1 t^(a-1)(1-t)^(b-1)dt both by Gauss-Legendre numerical integration and the gamma form Γ(a)Γ(b)/Γ(a+b), with lnB and the agreement error.

Input

Enter positive reals a and b to integrate the Beta function B(a,b)=∫_0^1 t^(a-1)(1-t)^(b-1)dt with 64-point Gauss quadrature and compare it to the gamma form.

Parameter a

Positive real number

Parameter b

Positive real number

Result

Beta function B(2, 3) (numerical integral)

0.0833333333

Approximated with 64-point Gauss-Legendre quadrature

Gamma form Γ(a)Γ(b)/Γ(a+b)

0.0833333333

lnB(a,b)

-2.4849066498

Difference from integral

2.220446e-16

The gamma form B(a,b)=Γ(a)Γ(b)/Γ(a+b) is computed with the Lanczos approximation. When a or b is below 1 the integrand diverges at an endpoint, so the numerical integral carries more error.

How it works

The Beta function is defined as B(a,b)=∫_0^1 t^(a-1)(1-t)^(b-1)dt; enter positive real numbers for a and b.
The primary value comes from 64-point Gauss-Legendre quadrature, with the standard interval [-1,1] mapped linearly to [0,1].
The gamma form is B(a,b)=Γ(a)Γ(b)/Γ(a+b), where the gamma function is evaluated with the Lanczos approximation.
lnB(a,b)=lnΓ(a)+lnΓ(b)-lnΓ(a+b) is useful when you want to avoid overflow for large arguments.
When a or b is below 1 the integrand diverges at an endpoint, so the numerical integral carries slightly more error than the gamma form.

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