Mathematics · Calculus · Integral Calculus
Beta Function Calculator
Computes the Beta function B(x, y) using the relationship with the Gamma function for positive real inputs.
Calculator
Formula
B(x, y) is the Beta function evaluated at positive real numbers x and y. \Gamma denotes the Gamma function, which generalises the factorial: \Gamma(n) = (n-1)! for positive integers. The integral representation holds for x > 0 and y > 0. The relationship \Gamma(n+1) = n\,\Gamma(n) is used recursively in computation.
Source: Abramowitz & Stegun, Handbook of Mathematical Functions, §6.2 (NIST Digital Library of Mathematical Functions, dlmf.nist.gov/5.12)
How it works
The Beta function, sometimes called Euler's integral of the first kind, is defined for positive real numbers x and y by the integral B(x, y) = ∫₀¹ t^(x−1)(1−t)^(y−1) dt. This integral converges as long as both x and y are strictly positive. The function is symmetric — B(x, y) = B(y, x) — which follows directly from the substitution t → 1 − t in the integral.
The most computationally useful form uses the Gamma function: B(x, y) = Γ(x)Γ(y) / Γ(x + y). For positive integers, Γ(n) = (n − 1)!, so B(m, n) = (m−1)!(n−1)! / (m+n−1)!, linking the Beta function directly to binomial coefficients via C(m+n, m) = (m+n)! / (m! n!) = 1 / ((m+n) B(m+1, n+1)). This calculator uses the Lanczos approximation to compute ln Γ with high accuracy across a wide range of inputs, then exponentiates to obtain B(x, y).
In practice, the Beta function is indispensable in Bayesian statistics — the Beta distribution with parameters α and β has a normalisation constant B(α, β). It arises in the computation of Student's t-distribution, the F-distribution, and incomplete beta function integrals used for hypothesis testing. In physics it appears in string theory amplitudes, Feynman integrals, and solutions to the hypergeometric differential equation.
Worked example
Example 1 — Integer inputs: Compute B(3, 5).
Using the Gamma relation: B(3, 5) = Γ(3)Γ(5) / Γ(8) = 2! × 4! / 7! = 2 × 24 / 5040 = 48 / 5040 ≈ 0.00952381. Enter x = 3 and y = 5 to confirm.
Example 2 — Half-integer inputs: Compute B(1/2, 1/2).
We know Γ(1/2) = √π ≈ 1.7724539 and Γ(1) = 1, so B(0.5, 0.5) = Γ(0.5)² / Γ(1) = π / 1 = π ≈ 3.14159265. This is a well-known identity connecting the Beta function to π, verifiable by entering x = 0.5, y = 0.5.
Example 3 — Beta distribution normalisation: A Bayesian analyst uses a Beta(2, 3) prior. The normalisation constant is B(2, 3) = Γ(2)Γ(3) / Γ(5) = 1! × 2! / 4! = 2 / 24 ≈ 0.08333333. Enter x = 2, y = 3 to confirm.
Limitations & notes
This calculator requires both x and y to be strictly positive (x > 0, y > 0); the Beta function is not defined for zero or negative integer inputs because the Gamma function has poles there. For very large inputs (x or y greater than about 170 in direct computation), overflow in Γ(x) and Γ(y) individually can occur — this calculator mitigates that by working in log-space using ln B(x, y) = ln Γ(x) + ln Γ(y) − ln Γ(x + y) and only exponentiating at the end, but extreme inputs approaching the limits of IEEE 754 double precision (roughly x + y > 700) may still yield Infinity. The Lanczos approximation used here achieves about 15 significant digits of accuracy for positive real arguments; for complex-valued inputs or negative non-integer inputs, a more specialised library is required. Note also that the incomplete Beta function I_x(a, b) — used in CDF computations for the Beta and F distributions — is a separate object not computed here.
Frequently asked questions
What is the Beta function used for in statistics?
The Beta function is the normalisation constant of the Beta probability distribution, which models random variables constrained to [0, 1] such as proportions, probabilities, and rates. It also appears in the CDF of the F-distribution and Student's t-distribution, making it central to hypothesis testing and Bayesian inference.
How is the Beta function related to the Gamma function?
The fundamental identity is B(x, y) = Γ(x)Γ(y) / Γ(x + y). Because Γ(n) = (n−1)! for positive integers, this reduces the Beta function to a ratio of factorials in the integer case. This relation is used by virtually all numerical implementations, including this calculator.
Is the Beta function symmetric?
Yes — B(x, y) = B(y, x) for all valid inputs. This symmetry follows directly from the integral definition via the substitution t → 1 − t, and also from the symmetry of the Gamma-function formula in x and y.
What is the difference between the Beta function and the Beta distribution?
The Beta function B(α, β) is a deterministic mathematical function producing a single number. The Beta distribution Beta(α, β) is a probability distribution whose probability density function is f(x) = x^(α−1)(1−x)^(β−1) / B(α, β) for x ∈ [0, 1]. The Beta function appears as the normalising denominator ensuring the PDF integrates to 1.
Why does B(1/2, 1/2) equal π?
Because Γ(1/2) = √π, so B(1/2, 1/2) = Γ(1/2)² / Γ(1) = (√π)² / 1 = π. This striking identity can also be derived directly from the integral ∫₀¹ t^(−1/2)(1−t)^(−1/2) dt via the substitution t = sin²θ, which transforms it into ∫₀^(π/2) 2 dθ = π.
Last updated: 2025-01-15 · Formula verified against primary sources.