Fungrim home page

Fungrim entry: 6da738

Symbol: LambertW Wk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
Called with two arguments LambertW(k, z) (rendered Wk ⁣(z)W_{k}\!\left(z\right) ) represents the kk -th branch of the Lambert W-function.
Called with three arguments LambertW(k, z, r) (rendered Wk(r) ⁣(z)W^{(r)}_{k}\!\left(z\right) ) represents the rr -th derivative of the kk -th branch of the Lambert W-function, with inherited branch cuts.
LambertW(k, z) is equivalent to LambertW(k, z, 0).
The following table lists conditions such that LambertW(k, z, r) is defined in Fungrim.
Domain Codomain
Numbers
kZandzC{0}k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} Wk ⁣(z)CW_{k}\!\left(z\right) \in \mathbb{C}
kZandzC{0,e1}andrZ0k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0, -{e}^{-1}\right\} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} Wk(r) ⁣(z)CW^{(r)}_{k}\!\left(z\right) \in \mathbb{C}
rZ0r \in \mathbb{Z}_{\ge 0} W0(r) ⁣(0)QW^{(r)}_{0}\!\left(0\right) \in \mathbb{Q}
Table data: (P,Q)\left(P, Q\right) such that (P)    (Q)\left(P\right) \implies \left(Q\right)
Definitions:
Fungrim symbol Notation Short description
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
Expez{e}^{z} Exponential function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
QQQ\mathbb{Q} Rational numbers
Source code for this entry:
Entry(ID("6da738"),
    SymbolDefinition(LambertW, LambertW(k, z), "Lambert W-function"),
    Description("Called with two arguments", SourceForm(LambertW(k, z)), "(rendered", LambertW(k, z), ") represents the", k, "-th branch", "of the Lambert W-function."),
    Description("Called with three arguments", SourceForm(LambertW(k, z, r)), "(rendered", LambertW(k, z, r), ") represents the", r, "-th derivative of the", k, "-th branch of the Lambert W-function, with inherited branch cuts."),
    Description(SourceForm(LambertW(k, z)), "is equivalent to", SourceForm(LambertW(k, z, 0)), "."),
    Description("The following table lists conditions such that", SourceForm(LambertW(k, z, r)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(And(Element(k, ZZ), Element(z, SetMinus(CC, Set(0)))), Element(LambertW(k, z), CC)), Tuple(And(Element(k, ZZ), Element(z, SetMinus(CC, Set(0, Neg(Exp(-1))))), Element(r, ZZGreaterEqual(0))), Element(LambertW(k, z, r), CC)), Tuple(Element(r, ZZGreaterEqual(0)), Element(LambertW(0, 0, r), QQ)))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-17 11:32:46.829430 UTC