What is the Beta function?

The Beta function B(x, y) is defined by the integral ∫₀¹ tˣ⁻¹(1−t)ʸ⁻¹ dt, and is closely related to combinations and probability distributions like the Beta distribution.

How is the Beta function related to the Gamma function?

B(x, y) = Γ(x)Γ(y) / Γ(x + y) — this identity lets the Beta function be computed directly from three Gamma function evaluations rather than numerical integration.

Why must x and y be positive?

The defining integral for B(x, y) only converges for x > 0 and y > 0; outside that domain the Beta function isn't real-valued in the same simple sense, so this calculator restricts to positive inputs.

Is B(x, y) symmetric?

Yes — B(x, y) = B(y, x), which follows directly from the Gamma function formula since Γ(x)Γ(y) = Γ(y)Γ(x).

B(1, 1) = Γ(1)Γ(1) / Γ(2) = 1×1/1 = 1, matching the integral ∫₀¹ 1 dt = 1.

How accurate is this calculator?

It uses the Lanczos approximation for the Gamma function (the same method used throughout this site), which is accurate to about 15 significant digits for typical inputs — more than sufficient for virtually all practical uses.

Where is the Beta function used in real life?

It's central to Bayesian statistics (the Beta distribution is the conjugate prior for binomial proportions), combinatorics, and evaluating certain definite integrals in physics and engineering.

How is this related to the Binomial Coefficient Calculator?

For positive integers, B(m, n) relates directly to binomial coefficients: 1/B(m,n) = (m+n−1) × C(m+n−2, m−1), connecting the continuous Beta function back to discrete combinatorics.

Beta Function Calculator — Compute B(x,y) via the Gamma Function

Our Beta Function Calculator computes B(x, y) = Γ(x)Γ(y) / Γ(x + y) for any positive x and y, using the same Lanczos approximation for the Gamma function that powers the site's Gamma Function Calculator, showing each intermediate Gamma value along the way.

Quick Answer

B(x, y) = Γ(x)Γ(y) / Γ(x + y). Enter positive x and y below to instantly compute the Beta function via the Gamma function relationship.

Enter x and y, then click Calculate.

How to Use the Beta Function Calculator — B(x,y) Online

1
Enter positive values for x and y.
2
Click 'Calculate' to compute B(x, y) via the Gamma function relationship.
3
Review the intermediate Γ(x), Γ(y), and Γ(x+y) values used.

Why Use Beta Function Calculator — B(x,y) Online?

The Beta function is defined by the integral B(x, y) = ∫₀¹ tˣ⁻¹(1−t)ʸ⁻¹ dt, but it's far easier to compute using its closed-form relationship to the Gamma function, B(x, y) = Γ(x)Γ(y) / Γ(x + y) — the same identity that makes the Beta function a natural companion to the site's Gamma Function Calculator. This relationship means every Beta function value can be computed exactly the way this calculator does it: by evaluating three Gamma function values via the Lanczos approximation and combining them.

Frequently Asked Questions

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