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Incomplete Beta Calculator

Incomplete beta B_x(a, b) = ∫₀ˣ t^{a−1}(1−t)^{b−1} dt for a>0, b>0, 0≤x≤1. Same engine as Special Functions. Not normalized sinc. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 58/58 tests · Production surface contract 4/4 · v1.10.0
Input interpretation
Enter values to calculate.
Result
—
Model
Incomplete beta B_x(a, b) for a>0, b>0, 0≤x≤1.
Scope
Incomplete Euler beta
Verification
Engine tested · 58/58 tests · Production surface contract 4/4 · Specification checked · v1.10.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
13 golden · 8 boundary · 37 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.10.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

DefinitionB_x(a, b) = ∫₀ˣ t^{a−1} (1−t)^{b−1} dt
Complete at x=1B_1(a, b) = B(a, b)
iSame engine as /special-functions. Not a second engine. Not /ibeta. Normalized sinc is on /normalized-sinc.

How to use

1

Enter x, a, and b

Finite reals with a>0, b>0, and 0≤x≤1. Defaults x=0.5, a=2, b=3.

2

Read B_x(a, b)

B_{0.5}(2, 3) ≈ 0.0572916666667. B_1(2, 3)=1/12. Regularized I_x is B_x/B.

Example calculations

Common configurations with formula and result.

ϟ

B_{0.5}(2, 3)

x = 0.5, a = 2, b = 3

B_{0.5}(2, 3)
0.0572916666667
ϟ

B_{0.5}(1/2, 1/2)

x = 0.5, a = 0.5, b = 0.5

B_{1/2}(1/2, 1/2) = π/2
π/2

Incomplete Beta calculator specification

Version 1.10.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The incomplete beta B_x(a, b) = ∫₀ˣ t^{a−1}(1−t)^{b−1} dt for a>0, b>0, and 0≤x≤1. B_0=0, B_1(a,b)=B(a,b), B_x(1,1)=x, B_{1/2}(1/2,1/2)=π/2. Regularized I_x(a,b)=B_x/B is returned as regularized. This page is mode=incomplete_beta on math.special. Not normalized sinc. Not CAS.
What it calculates
Incomplete beta B_x(a, b) for a>0, b>0, 0≤x≤1.
Inputs
  • x
  • a
  • b
Outputs
  • value
  • regularized
  • x
  • a
  • b
Formula
B_x(a, b) = ∫₀ˣ t^{a−1} (1−t)^{b−1} dt
Assumptions
  • Incomplete Euler beta
  • a>0, b>0, 0≤x≤1
  • Not normalized sinc
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • non-finite x, a, or b → INVALID_NUMBER
  • a ≤ 0 or b ≤ 0 → INVALID_INPUT
  • x < 0 or x > 1 → INVALID_INPUT
  • unknown mode (normalized sinc / CAS) → INVALID_MODE
Example
x=0.5 a=2 b=3 → 0.0572916666667
Validation cases

2 published on this page · 58/58 tests · Production surface contract 4/4 · View evidence

  • x=0.5 a=2 b=3 → 0.0572916666667
  • x=0.5 a=0.5 b=0.5 → π/2
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.10.0

Other calculators in this family: Bessel I Calculator, Bessel J Calculator, Bessel K Calculator, Bessel Y Calculator, Beta Function Calculator, Brent Method Calculator, Definite Integral Calculator, DFT Calculator . Explore all Numerical Calculus.

→ Bessel I Calculator I_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel J Calculator J_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel K Calculator K_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel Y Calculator Y_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Beta Function Calculator Complete Euler B(x, y) = Γ(x)Γ(y)/Γ(x+y) for x>0, y>0. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. › → Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → DFT Calculator Forward DFT X[k]=Σ x[n] e^{−2πi kn/N}, N≤64, unnormalized. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not inverse, not Laplace. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second special-functions engine?

No. It is mode=incomplete_beta on math.special, the same engine as /calc/math/special-functions.

Is this normalized sinc?

No. This seed is the incomplete beta B_x(a,b) for a>0, b>0, 0≤x≤1. Complete Euler B is on /beta-function. Normalized sinc is on /normalized-sinc. Not CAS.

Where does this run?

Locally in the browser by default. REST and MCP call the same special-functions engine.