LambertW

The Lambert $W$ function (Wolfram's ProductLog), the inverse of $w\,e^w$. compute-engine's native LambertW already covers the principal ($k=0$) and lower-real ($k=-1$) branches numerically; extended here with exact values at algebraically nice points and every other integer branch.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaLambert W function↗MathWorldLambertW-Function↗WikidataQ429331↗nLabLambert W-function↗DLMF4.13.E1↗FungrimLambertW✓↗Fungrim entryWolfram LanguageProductLog✓↗

LambertW(z)the principal branch $W_0(z)$ — native, extended with exact values.

SageMathlambert_w(z)mpmathlambertw(z)

LambertW(z, k)the $k$-th branch $W_k(z)$, by Halley's iteration from the standard log-log seed (Corless et al. 1996). $k=0$ and $k=-1$ stay on compute-engine's own native handler.

SageMathlambert_w(k, z)mpmathlambertw(z, k)

Typed signatures

Type	Package	Applies
(z: complex | infinity, branch: integer?) -> number	analyticreplaces compute-engine	always
(z: complex | infinity, branch: integer?) -> number	compute-engine	always

Domain: Special functions

Details

• Differs from Wolfram: the branch is the second argument, $\mathrm{LambertW}(z, k)$, in compute-engine's order. Wolfram's ProductLog[k, z] and Sage's lambert_w(k, z) take $(k, z)$; mpmath's lambertw(z, k) takes $(z, k)$, like ours. Keep this order in mind when pasting a call across systems: the same branch $W_{-1}(-0.2) \approx -2.5426$ is LambertW(-0.2, -1) here, ProductLog[-1, -0.2] in Wolfram, lambert_w(-1, -0.2) in Sage and lambertw(-0.2, -1) in mpmath. Swapped, ProductLog[-0.2, -1] is an error in Wolfram (the branch must be an integer), and LambertW(-1, -0.2) declines here. The Wolfram transpiler reverses the arguments at the boundary.
• Checked directly: LambertW(-0.14, -1) is the $k=-1$ branch. The Sage form is documented prior art, not run here.
• Exact values are recognized structurally at $z=0$, $z=e$ (giving $w=1$), $z=-1/e$ (giving $w=-1$), $z = n\,e^n$ for small integer $n$, and $z=-\ln(k)/k$ for small integer $k \ge 2$ (giving $w=-\ln k$, since $e^{-\ln k}=1/k$) — even under plain evaluate(), not just N().
• Every other branch is solved by Halley's method in the complex plane; checked against wolframscript's N[ProductLog[k, z], 16] at several points (real and complex $z$, several $k$) to full double precision.

Plot

ComplexPlot

real slice

Examples

4 more kept as data, checked against the oracles

Basic #

live ↗

In$\operatorname{W}(0)$Out$0$

$1 \cdot e^1 = e$

live ↗

In$\operatorname{W}(e)$Out$1$

Scope #

The branch point $-1/e$

live ↗

In$\operatorname{W}(-\frac{1}{e})$Out$-1$

$2e^2 = w e^w$ at $w = 2$

live ↗

In$\operatorname{W}(2e^2)$Out$2$

$-\ln 2 \cdot e^{-\ln 2} = -\tfrac{\ln 2}{2}$

live ↗

In$\operatorname{W}(-\frac{\ln(2)}{2})$Out$-\ln(2)$

A branch other than 0 or -1, checked against N[ProductLog[-3, -0.14], 16]

live ↗

In$N(\operatorname{W}_{-3}(-0.14))$Out$-4.645\,279\,174\,278\,859-13.812\,745\,325\,218\,817i$

Known value · Wolfram 15.0.0 ProductLog

The branch $k = 1$, written in compute-engine's $(z, k)$ order — Wolfram's ProductLog[1, 1]

live ↗

In$N(\operatorname{W}_{1}(1))$Out$-1.533\,913\,319\,793\,574\,4+4.375\,185\,153\,061\,898i$

The real branch $k = -1$ on $-1/e < z < 0$, written $(z, k)$ as here; Wolfram's ProductLog[-1, -0.2] and mpmath's lambertw(-0.2, -1) give the same value

live ↗

In$N(\operatorname{W}_{-1}(-0.2))$Out$-2.542\,641\,357\,773\,526\,424\,29$

About

Examples	12
Checked against	Wolfram Language 12 agreeSageMath 12 agreempmath 12 agree
Crosswalk	WikipediaMathWorldWikidatanLabDLMFFungrimFungrim entry ×20Wolfram LanguageSageMath ×2mpmath ×2

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/lambert-w.ts

Halley's iteration in the complex plane for k ∉ {0, -1}; structural pattern matching for the exact cases.

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mpmath takes the branch second: lambertw(z, k).

mappedsagepackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

See also: Exp, Ln