Beta Function Calculator
The beta function B(x, y) is a special function that pairs naturally with the gamma function and underlies the beta probability distribution. For positive x and y it can be written as the ratio gamma(x) times gamma(y) divided by gamma(x + y). This calculator evaluates each gamma factor with the Lanczos approximation and combines them, returning the beta value and its natural logarithm. The function is symmetric, so swapping x and y gives the same result. Inputs must be positive real numbers.
Beta function formula
B(x, y) = integral from 0 to 1 of t^(x-1) (1 - t)^(y-1) dt
B(x, y) = Gamma(x) * Gamma(y) / Gamma(x + y)
B(x, y) = B(y, x) (symmetry)
The calculator uses the gamma-ratio identity, computing each gamma value numerically and dividing, which is exact to high precision for the positive arguments allowed here.
About the beta function
- B(1, 1) = 1, the area under a flat density on the unit interval.
- The function is symmetric in its two arguments.
- It normalises the beta distribution, central to Bayesian inference.
- It generalises binomial coefficients to continuous arguments.
- It appears in the F and Student t distribution densities.
Beta function: frequently asked questions
What is the beta function?
The beta function B(x, y) is a special function defined for positive x and y by the integral from 0 to 1 of t^(x-1) (1 - t)^(y-1) dt. It is closely tied to the gamma function and to the binomial coefficients.
How is the beta function calculated?
The standard identity is B(x, y) = gamma(x) times gamma(y) divided by gamma(x + y). This calculator evaluates each gamma value with the Lanczos approximation, then combines them.
Is the beta function symmetric?
Yes. B(x, y) equals B(y, x) because swapping the arguments swaps the two gamma factors in the numerator, leaving the product unchanged.
How does beta relate to binomial coefficients?
For positive integers, 1 divided by ((m + n + 1) times B(m + 1, n + 1)) equals the binomial coefficient C(m + n, m). The beta function is therefore a continuous extension of binomial structure.
Where is the beta function used?
It defines the beta probability distribution used in Bayesian statistics, appears in the normalising constant of the Student t and F distributions, and arises throughout combinatorics and physics.
Official sources
- NIST Digital Library of Mathematical Functions: Beta Function.
- NIST Digital Library of Mathematical Functions: Gamma Function: Definitions.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.