Beta Function Calculator
Compute B(x,y) = Γ(x)·Γ(y)/Γ(x+y) — Euler integral of the first kind with step-by-step solution
Key Formulas
Gamma: B(x,y) = Γ(x)·Γ(y) / Γ(x+y)
Symmetry: B(x,y) = B(y,x)
Step-by-step solution
What is the Beta Function?
The beta function B(x,y) — also written Β(α,β) using Greek letters — is a special mathematical function known as the Euler integral of the first kind. It is defined for all real numbers x > 0 and y > 0 by the integral:
The beta function is intimately connected to the gamma function Γ, which generalises the factorial to real and complex numbers. The connection is expressed as:
This relationship allows the beta function to be computed efficiently for any positive real arguments using the Lanczos approximation for the gamma function — exactly what this calculator does. The domain is strictly x > 0 and y > 0; the function is undefined (diverges) at zero and negative integers.
Beta Function Formula
The beta function satisfies several important identities:
Gamma relationship: B(x,y) = Γ(x)·Γ(y) / Γ(x+y)
Symmetry property: B(x,y) = B(y,x)
Recurrence: B(x+1,y) = x/(x+y) · B(x,y)
Special case: B(x,1) = 1/x
The symmetry property B(x,y) = B(y,x) follows immediately from the gamma formula because Γ(x)Γ(y) = Γ(y)Γ(x) and Γ(x+y) = Γ(y+x). The recurrence relation links B(x+1,y) to B(x,y), enabling efficient computation for integer arguments.
Special Values Table
| x | y | B(x,y) | Notes |
|---|---|---|---|
| 1 | 1 | 1 | B(1,1) = Γ(1)²/Γ(2) = 1 |
| 1/2 | 1/2 | π ≈ 3.14159 | Famous result via Γ(½)=√π |
| 2 | 2 | 1/6 ≈ 0.16667 | Γ(2)²/Γ(4) = 1/6 |
| 2 | 3 | 1/12 ≈ 0.08333 | Γ(2)Γ(3)/Γ(5) = 1/12 |
| 3 | 3 | 1/30 ≈ 0.03333 | Γ(3)²/Γ(6) = 1/30 |
| 1 | n | 1/n | General: B(1,n) = 1/n |
Worked Examples
Example 1 — B(2, 3)
Using the gamma relationship with Γ(n) = (n−1)! for positive integers:
Γ(3) = 2! = 2
Γ(2+3) = Γ(5) = 4! = 24
B(2,3) = Γ(2)·Γ(3) / Γ(5) = 1·2 / 24 = 2/24 = 1/12 ≈ 0.083333
This can also be verified via the integral: ∫₀¹ t(1-t)² dt = ∫₀¹ (t - 2t² + t³) dt = 1/2 − 2/3 + 1/4 = 1/12.
Example 2 — B(1/2, 1/2) = π
One of the most celebrated identities in mathematical analysis:
Γ(1/2 + 1/2) = Γ(1) = 1
B(1/2, 1/2) = Γ(1/2)·Γ(1/2) / Γ(1) = √π · √π / 1 = π
This result connects the beta function to the circle constant π via the integral ∫₀¹ t^(-1/2)(1-t)^(-1/2) dt = π.
Example 3 — B(1, n) = 1/n
For any positive integer (or real) n:
This generalises the harmonic series: B(1,1)=1, B(1,2)=1/2, B(1,3)=1/3, and so on. For non-integer n the same formula holds exactly.
Applications of the Beta Function
- Probability — Beta distribution: The beta distribution PDF f(x;α,β) = x^(α-1)(1-x)^(β-1) / B(α,β) models random variables confined to [0,1], such as proportions, probabilities, and rates.
- Bayesian statistics: The beta distribution is the conjugate prior for the Bernoulli and binomial likelihood, making it the workhorse of Bayesian A/B testing and click-through-rate modelling.
- Combinatorics: The binomial coefficient C(n,k) satisfies 1/((n+1)·B(k+1, n-k+1)) = C(n,k), linking combinatorics directly to the beta function.
- Physics — Scattering amplitudes: The Veneziano amplitude in early string theory is expressed as a sum of beta functions B(−α(s), −α(t)), a historically important result.
- Integration: Many definite integrals involving powers of trigonometric functions reduce to beta function values via the substitution t = sin²θ.
- Signal processing: The beta distribution models noise and channel error probabilities in certain communication systems.