HomeCalculatorsMathNumerical CalculusLambert W₋₁ Calculator
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Lambert W₋₁ Calculator

Real W_{-1}(x) for −1/e ≤ x < 0. 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
Secondary real Lambert W_{-1}(x) for −1/e ≤ x < 0.
Scope
Secondary 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Embedded snapshot: unpublished · Build schema 1.10.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain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_{-1}(x) e^{W_{-1}(x)} = x
Domain−1/e ≤ x < 0
iSame engine as /special-functions. Not a second engine. Not /wm1. Incomplete beta and normalized sinc later on this engine.

How to use

1

Enter x

Finite real x with −1/e ≤ x < 0. Default x=−0.1.

2

Read W_{-1}(x)

W_{-1}(−0.1) ≈ −3.57715206396. W_{-1}(−2/e²)=−2. x=0 is not real.

Example calculations

Common configurations with formula and result.

ϟ

W_{-1}(−0.1)

x = −0.1

W_{-1}(−0.1)
−3.57715206396
ϟ

W_{-1}(−2/e²)

x = −2/e²

W_{-1}(−2/e²) = −2
−2

Lambert W₋₁ calculator specification

Version 1.10.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The secondary real Lambert W function W_{-1}(x) solves W e^W = x for −1/e ≤ x < 0. W_{-1}(−1/e)=−1, W_{-1}(−2/e²)=−2, and W_{-1}(x)→−∞ as x→0⁻. This page is mode=lambert_wm1 on math.special. Not incomplete beta. Not CAS.
What it calculates
Secondary real Lambert W_{-1}(x) for −1/e ≤ x < 0.
Inputs
  • x
Outputs
  • value
  • branch
Formula
W_{-1}(x) e^{W_{-1}(x)} = x
Assumptions
  • Secondary real branch only
  • −1/e ≤ x < 0
  • Not incomplete beta / normalized sinc
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • non-finite x → INVALID_NUMBER
  • x < −1/e or x ≥ 0 → INVALID_INPUT
  • unknown mode (incomplete beta / CAS) → INVALID_MODE
Example
x=−0.1 → −3.57715206396
Validation cases

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

  • x=−0.1 → −3.57715206396
  • x=−2/e² → −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=lambert_wm1 on math.special, the same engine as /calc/math/special-functions.

Does this compute incomplete beta?

No. This seed is the secondary real branch W_{-1} for −1/e ≤ x < 0. Incomplete beta is on /calc/math/incomplete-beta. Not CAS.

Where does this run?

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