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 (2)
| Label | Occurrences |
|---|---|
| Lambert W function (distinct but historically related name) | 1 |
| Lambert W function (later named in his honor) canonical | 1 |
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 ⓘ CAS (Computer Algebra System) ⓘ
surface form:
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lambert W function (distinct but historically related name)