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Math calculator

Lambert W Calculator

Principal real W_0(x) for x ≥ −1/e. Same engine as Special Functions. Not incomplete beta. 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
Principal real Lambert W_0(x).
Scope
Principal real branch only
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Embedded snapshot: unpublished · Build schema 1.10.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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.

DefinitionW(x) e^{W(x)} = x
Branch pointW_0(−1/e) = −1
iSame engine as /special-functions. Not a second engine. Not /omega. Incomplete beta and normalized sinc later on this engine.

How to use

1

Enter x

Finite real x ≥ −1/e. Default x=1 (Ω constant).

2

Read W_0(x)

W_0(1) ≈ 0.56714329041. W_0(e)=1.

Example calculations

Common configurations with formula and result.

ϟ

Omega constant

x = 1

W_0(1) = Ω
0.56714329041
ϟ

W_0(e)

x = e

W_0(e) = 1
1

Lambert W calculator specification

Version 1.10.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The principal real Lambert W function W_0(x) solves W e^W = x for x ≥ −1/e. W_0(0)=0, W_0(e)=1, W_0(−1/e)=−1. This page is mode=lambert_w on math.special. Not incomplete beta. Not CAS.
What it calculates
Principal real Lambert W_0(x).
Inputs
  • x
Outputs
  • value
  • branch
Formula
W(x) e^{W(x)} = x
Assumptions
  • Principal real branch only
  • x ≥ −1/e
  • Not incomplete beta (W_{-1} is on /lambert-w-minus-1)
  • Not incomplete beta / sinc
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • non-finite x → INVALID_NUMBER
  • x < −1/e → INVALID_INPUT
  • unknown mode (incomplete beta / CAS) → INVALID_MODE
Example
x=1 → 0.56714329041
Validation cases

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

  • x=1 → 0.56714329041
  • x=e → 1
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=lambert_w on math.special, the same engine as /calc/math/special-functions.

Does this compute W_{-1}?

W_{-1} is on /calc/math/lambert-w-minus-1 for −1/e ≤ x < 0. This page is the principal real branch W_0. Not CAS.

Where does this run?

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