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Gamma Function Calculator

Γ(x) = ∫₀^∞ t^{x−1} e^{−t} dt except poles at 0, −1, −2, …. Same engine as Special Functions. Not factorial. 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
Euler gamma function Γ(x).
Scope
Real argument
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Embedded snapshot: unpublished · Build schema 1.10.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

IntegralΓ(x) = ∫₀^∞ t^{x−1} e^{−t} dt
Half-integerΓ(1/2) = √π
iSame engine as /special-functions. log-gamma is a hub mode. Not a second engine. Not /calc/electric/gamma-z.

How to use

1

Enter x

Finite real, not a non-positive integer. Default x=6.

2

Read Γ(x)

Γ(6)=120. Γ(1/2)=√π.

Example calculations

Common configurations with formula and result.

ϟ

Gamma of 6

x = 6

Γ(6) = 5!
120
ϟ

Half-integer

x = 0.5

Γ(1/2) = √π
√π

Gamma Function calculator specification

Version 1.10.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The Euler gamma function Γ(x)=∫₀^∞ t^{x−1} e^{−t} dt extends factorial: Γ(n)=(n−1)! for positive integers n. Poles at 0, −1, −2, …. This page is mode=gamma on math.special. Integer n! stays on /factorial. Not CAS.
What it calculates
Euler gamma function Γ(x).
Inputs
  • x
Outputs
  • value
  • gamma
Formula
Γ(x) = ∫₀^∞ t^{x−1} e^{−t} dt
Assumptions
  • Real argument
  • Poles at 0, −1, −2, …
  • Not factorial
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • non-finite x → INVALID_NUMBER
  • x = 0, −1, −2, … → INVALID_INPUT
  • unknown mode → INVALID_MODE
Example
x=6 → 120
Validation cases

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

  • x=6 → 120
  • x=0.5 → √π
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=gamma on math.special, the same engine as /calc/math/special-functions.

Is Γ(6) the factorial calculator?

No. Γ(n)=(n−1)! for positive integers, on a different model. Integer n! stays on /calc/math/factorial.

Where does this run?

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