About

The Euler Beta function B(x, y) appears in normalization constants for probability distributions (Beta, Dirichlet), combinatorial identities, and integral representations across physics and engineering. Errors in its evaluation propagate directly into posterior densities in Bayesian inference and into binomial coefficient generalizations for non-integer arguments. This calculator implements the Lanczos approximation with 9 coefficients at g = 7, yielding approximately 15 significant digits of accuracy across the positive real domain. It computes B(x, y), its natural logarithm ln B, the regularized incomplete beta I_x(a, b), and the reciprocal 1/B.

The log-beta path avoids overflow for large arguments where direct Gamma products exceed 10³⁰⁸. The incomplete beta uses Lentz’s continued fraction algorithm with up to 200 iterations. Note: this tool assumes real positive inputs. The Beta function is undefined at zero and negative integers where the Gamma function has poles.

Formulas

The Beta function is defined as a ratio of Gamma functions:

B(x, y) = Γ(x) ⋅ Γ(y)Γ(x + y)

For numerical stability with large arguments, the logarithmic form is used:

ln B(x, y) = ln Γ(x) + ln Γ(y) − ln Γ(x + y)

The Gamma function is computed via the Lanczos approximation:

Γ(z) ≈ √2π ⋅ (z + g + 0.5)^{z + 0.5} ⋅ e^{−(z + g + 0.5)} ⋅ A_g(z)

where g = 7 and A_g(z) is a truncated series of 9 precomputed coefficients.

The regularized incomplete beta function is defined as:

I_x(a, b) = B(x; a, b)B(a, b) = 1B(a, b) x∫0 t^a−1(1 − t)^b−1 dt

Evaluated via Lentz’s continued fraction method for x ∈ [0, 1], with the symmetry relation I_x(a, b) = 1 − I_1−x(b, a) used when x > (a + 1) / (a + b + 2).

Where B(x, y) = Beta function, Γ(z) = Gamma function, x, y = positive real parameters, g = Lanczos constant, I_x = regularized incomplete beta ratio, a, b = shape parameters, t = integration variable.

Reference Data

x	y	B(x, y)	ln B(x, y)	1/B(x, y)
1	1	1.000000	0.000000	1.000000
2	2	0.166667	−1.791759	6.000000
3	3	0.033333	−3.401197	30.000000
0.5	0.5	3.141593	1.144729	0.318310
1	2	0.500000	−0.693147	2.000000
1	3	0.333333	−1.098612	3.000000
5	5	0.001587	−6.445720	630.000000
10	10	2.385e−6	−12.945086	419244.000
0.5	1	2.000000	0.693147	0.500000
2	5	0.033333	−3.401197	30.000000
0.1	0.1	19.714464	2.981364	0.050724
3	7	0.003968	−5.530108	252.000000
4	4	0.004762	−5.347108	210.000000
1	0.5	2.000000	0.693147	0.500000
100	100	3.656e−61	−138.934	2.735e+60
0.25	0.75	4.442883	1.491273	0.225079
6	2	0.023810	−3.737670	42.000000
1.5	2.5	0.196350	−1.627313	5.092958
50	50	1.569e−30	−68.544	6.374e+29
0.01	0.01	199.714	5.297	0.005007

Frequently Asked Questions

The Gamma function has poles at zero and negative integers. Since B(x, y) = Γ(x)·Γ(y) / Γ(x+y), if x or y equals 0 or a negative integer, the result is undefined. For very large positive values (e.g., x = 200, y = 200), the direct Beta value underflows to 0 in IEEE 754 double-precision. In such cases, use the Log-Beta mode, which remains numerically stable up to arguments of several thousand.

The 9-coefficient Lanczos series with g = 7 provides approximately 15 significant digits of accuracy for all positive real arguments. Relative error stays below 2 × 10⁻¹⁵ for z > 0.5. For z near zero (e.g., z = 0.001), the reflection formula Γ(z)·Γ(1−z) = π/sin(πz) can supplement precision if needed. This calculator uses the log-gamma path internally, which avoids intermediate overflow.

For positive integers n and k, the binomial coefficient C(n, k) = 1 / ((n+1) · B(n−k+1, k+1)). The reciprocal Beta mode (1/B) directly generalizes this to non-integer arguments, which appears in fractional calculus and Pochhammer symbol identities. For example, 1/B(3, 3) = 30, which equals C(5,2) · 1.

The regularized incomplete beta I_x(a, b) is the CDF of the Beta distribution. Use it when computing p-values in statistical tests (e.g., Student's t-test, F-test), confidence intervals for proportions, or Bayesian posterior probabilities. The input x must lie in [0, 1], with a and b being the shape parameters (both > 0). For example, I₀.₅(2, 5) ≈ 0.8906 gives the probability that a Beta(2,5)-distributed variable is ≤ 0.5.

Lentz's algorithm evaluates the continued fraction representation of I_x(a, b) iteratively, with a maximum of 200 iterations and a tolerance of 1 × 10⁻¹⁰. The symmetry relation I_x(a, b) = 1 − I_{1−x}(b, a) is applied when x > (a+1)/(a+b+2) to ensure the continued fraction converges rapidly. If convergence fails within 200 iterations, the calculator reports a warning.

Yes. B(0.5, 0.5) = Γ(0.5)² / Γ(1) = (√π)² / 1 = π exactly. This identity connects the Beta function to the Wallis integral and the normalization of the arcsine distribution. It serves as a useful verification benchmark: if your calculator returns B(0.5, 0.5) ≈ 3.14159265, the Lanczos implementation is correct.