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.
Statements (53)
| Predicate | Object |
|---|---|
| instanceOf |
complex function
ⓘ
special function ⓘ transcendental function ⓘ |
| alsoKnownAs |
Lambert W
NERFINISHED
ⓘ
Lambert W function NERFINISHED ⓘ 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 NERFINISHED ⓘ |
| isMultivalued | true ⓘ |
| namedAfter | Johann Heinrich Lambert NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Lambert W function