product logarithm
E944313
The product logarithm is a special function, commonly denoted as the Lambert W function, that serves as the inverse of f(w) = w e^w and is widely used in solving equations involving exponentials and products.
All labels observed (1)
| Label | Occurrences |
|---|---|
| product logarithm canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11739768 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: product logarithm Context triple: [Lambert W function, alsoKnownAs, product logarithm]
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A.
Plug One
Plug One is the stage name of Posdnuos, a founding member and primary lyricist of the influential hip hop group De La Soul.
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B.
The Product G&B
The Product G&B is an R&B and hip hop duo best known for their collaboration with Wyclef Jean and their appearance on the hit single "Maria Maria" by Santana.
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C.
Symbolyc One
Symbolyc One is an American hip-hop record producer best known for his work with artists like Kanye West, Jay-Z, and Beyoncé.
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D.
Preki
Preki is a former Yugoslav-American professional soccer player and coach best known for his standout MLS career and successful transition into coaching.
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E.
Precisely
Precisely is a data integrity and location intelligence company known for providing software and services that help organizations ensure accurate, consistent, and contextualized data for analytics and decision-making.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: product logarithm Target entity description: The product logarithm is a special function, commonly denoted as the Lambert W function, that serves as the inverse of f(w) = w e^w and is widely used in solving equations involving exponentials and products.
-
A.
Plug One
Plug One is the stage name of Posdnuos, a founding member and primary lyricist of the influential hip hop group De La Soul.
-
B.
The Product G&B
The Product G&B is an R&B and hip hop duo best known for their collaboration with Wyclef Jean and their appearance on the hit single "Maria Maria" by Santana.
-
C.
Symbolyc One
Symbolyc One is an American hip-hop record producer best known for his work with artists like Kanye West, Jay-Z, and Beyoncé.
-
D.
Preki
Preki is a former Yugoslav-American professional soccer player and coach best known for his standout MLS career and successful transition into coaching.
-
E.
Precisely
Precisely is a data integrity and location intelligence company known for providing software and services that help organizations ensure accurate, consistent, and contextualized data for analytics and decision-making.
- F. None of above. chosen
Statements (53)
| Predicate | Object |
|---|---|
| instanceOf |
complex function
ⓘ
special function ⓘ transcendental function ⓘ |
| alsoKnownAs |
Lambert W
ⓘ
Lambert W function ⓘ Omega function ⓘ |
| asymptoticExpansion | W(z) ~ ln z - ln ln z as |z| → ∞ ⓘ |
| branchIndexNotation | W_k(z) ⓘ |
| codomain | complex numbers ⓘ |
| definesInverseOf | f(w) = w e^w ⓘ |
| definingEquation | W(z) e^{W(z)} = z ⓘ |
| definition | W(z) is the multivalued inverse of the function w ↦ w e^w ⓘ |
| derivativeFormula | W'(z) = W(z) / (z (1 + W(z))) for z ≠ 0, -1/e ⓘ |
| domain | complex numbers ⓘ |
| growthOrder | logarithmic for large |z| ⓘ |
| hasBranch |
lower branch W_{-1}
ⓘ
principal branch W_0 ⓘ |
| hasBranchCut | (-∞,-1/e] on the real axis ⓘ |
| hasBranchPoint | z = -1/e ⓘ |
| hasInfinitelyManyBranches | true ⓘ |
| implementedIn |
MATLAB as lambertw(z)
ⓘ
Maple as LambertW(z) ⓘ Mathematica as ProductLog[z] ⓘ SciPy as scipy.special.lambertw ⓘ |
| isMultivalued | true ⓘ |
| namedAfter | Johann Heinrich Lambert ⓘ |
| principalBranchDomain | {z ∈ ℂ : z ≥ -1/e on ℝ} ⓘ |
| principalBranchRange | {w ∈ ℂ : w ≥ -1 on ℝ} ⓘ |
| realBranches |
W_0 on [-1/e,∞)
ⓘ
W_{-1} on [-1/e,0) ⓘ |
| realBranchesInterval | [-1/e,0) ⓘ |
| relatedFunction |
exponential function
ⓘ
logarithm ⓘ |
| relatedTo | tree function in combinatorics ⓘ |
| seriesExpansionAtZero |
W(z) = ∑_{n=1}^{∞} [(-n)^{n-1} / n!] z^n
ⓘ
radius of convergence e^{-1} ⓘ |
| symbol |
W
ⓘ
W(x) ⓘ W(z) ⓘ |
| usedFor |
solving equations of the form x e^x = a
ⓘ
solving equations where the unknown appears in both base and exponent ⓘ |
| usedInField |
algorithm analysis
ⓘ
asymptotic analysis ⓘ combinatorics ⓘ control theory ⓘ delay differential equations ⓘ number theory ⓘ physics ⓘ quantum statistics ⓘ |
| valueAtMinusLog2Over2 | W(-\ln 2 / 2) = -\ln 2 ⓘ |
| valueAtMinusOneOverE | W(-1/e) = -1 ⓘ |
| valueAtOne | W(1) ≈ 0.567143290409783872999968 ⓘ |
| valueAtZero | W(0) = 0 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: product logarithm Description of subject: The product logarithm is a special function, commonly denoted as the Lambert W function, that serves as the inverse of f(w) = w e^w and is widely used in solving equations involving exponentials and products.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Lambert W function (later named in his honor)