HomeCalculatorsMathSpecial FunctionsGamma Function Calculator
Math calculator

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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 15/15 declared partitions (erf, erfc, gamma, lgamma, bessel-j, bessel-y, bessel-i, bessel-k, lambert-w, lambert-wm1, beta, incomplete-beta, sinc, normalized-sinc, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Euler gamma function Γ(x), with an engine-linked Live graph of y = Γ(x).
Scope
Real argument
Verification
Engine tested · Specification checked · v1.10.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.10.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.special-transform-cvp
Verification revision
2026-09-22.special-transform-cvp · 1/1 property · digest ac95a7aec889
Legacy regression
58/58 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
16/16 golden · 8/8 CVP boundary · 10/10 invalid · 1/1 property · 4/4 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
13 legacy golden · 8 legacy boundary · legacy regression suite · 16/16 oracle-backed golden · 10/10 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.10.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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. The Live graph shows y = Γ(x); poles sit at 0, −1, −2, ….

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), with an engine-linked Live graph of y = 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, …
  • The Result Card includes an engine-linked Live graph of y = Γ(x) using the same Overview / Focus template as erf (Overview for x>0 is 0.2…max(5, x); negative x shows the poles). Same Lanczos Γ as REST/MCP. Integer n! stays on /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 6/6 · 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, Error Function Calculator, Incomplete Beta Calculator, Lambert W Calculator . Explore all Special Functions.

→ 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. › → Error Function Calculator erf(x) = (2/√π) ∫₀ˣ e^{−t²} dt. Same engine as Special Functions. Odd, erf(0)=0, erf(∞)→1. Not CAS. Runs locally. › → 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. › → Lambert W Calculator Principal real W_0(x) for x ≥ −1/e. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. ›

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.