Lambert W function (later named in his honor)

E279121

The Lambert W function is a special multivalued function that solves equations where a variable appears both inside and outside an exponential, defined as the inverse of f(w) = w e^w and widely used in mathematics, physics, and engineering.

All labels observed (5)

How this entity was disambiguated

Statements (47)

Predicate Object
instanceOf multivalued function ⓘ
special function ⓘ
alsoKnownAs product logarithm ⓘ
appearsIn closed-form solutions of some delay equations ⓘ
closed-form solutions of some differential equations ⓘ
solutions of equations of the form a x e^{b x} = c ⓘ
branchPoint z = -1/e ⓘ
z = 0 ⓘ
classification transcendental function ⓘ
codomain complex numbers ⓘ
definesInverseOf f(w) = w e^w ⓘ
domain complex numbers ⓘ
growthOrder logarithmic for large arguments ⓘ
hasBranch lower branch W_{-1} ⓘ
principal branch W_0 ⓘ
hasBranchCut (-∞,-1/e] on principal branch ⓘ
hasSeriesExpansionAtZero W(z) = ∑_{n=1}^{∞} (-n)^{n-1} z^n / n! ⓘ
implementedIn MATLAB ⓘ
Maple ⓘ
Mathematica ⓘ
SciPy ⓘ
isAnalyticOn ℂ minus branch cuts ⓘ
isInverseOf w ↦ w e^w ⓘ
isMultivaluedOn [-1/e,0) ⓘ
isSingleValuedOn (-1/e,∞) ⓘ
namedAfter Johann Heinrich Lambert ⓘ
relatedTo exponential function ⓘ
logarithm ⓘ
tree function in combinatorics ⓘ
satisfiesDifferentialEquation W'(z) = W(z) / (z (1+W(z))) ⓘ
satisfiesEquation W(z) e^{W(z)} = z ⓘ
solvesEquationType x e^x = z ⓘ
usedIn algorithm analysis ⓘ
asymptotic analysis ⓘ
chemical kinetics ⓘ
combinatorics ⓘ
control theory ⓘ
delay differential equations ⓘ
electrical engineering ⓘ
number theory ⓘ
population dynamics ⓘ
quantum physics ⓘ
statistical mechanics ⓘ
transcendental equation solving ⓘ
valueAt W(-1/e) = -1 ⓘ
W(0) = 0 ⓘ
W(e) ≈ 1 ⓘ

How these facts were elicited

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Johann Heinrich Lambert → knownFor → Lambert W function (later named in his honor) ⓘ
Lambert series → relatedConcept → Lambert W function (distinct but historically related name) ⓘ
linked to: Lambert W function (later named in his honor)
product logarithm → alsoKnownAs → Lambert W function ⓘ
linked to: Lambert W function (later named in his honor)
product logarithm → alsoKnownAs → Lambert W ⓘ
linked to: Lambert W function (later named in his honor)
product logarithm → implementedIn → SciPy as scipy.special.lambertw ⓘ
linked to: Lambert W function (later named in his honor)