# Fungrim entry: 6da738

Symbol: LambertW $W\!\left(z\right)$ Lambert W-function
Called with one argument LambertW(z) (rendered $W\!\left(z\right)$ ) represents the principal branch of the Lambert W-function.
Called with two arguments LambertW(z, k) (rendered $W_{k}\!\left(z\right)$ ) represents the $k$ -th branch of the Lambert W-function.
Called with three arguments LambertW(z, k, r) (rendered $W^{(r)}_{k}\!\left(z\right)$ ) represents the $r$ -th derivative of the $k$ -th branch of the Lambert W-function, with inherited branch cuts.
LambertW(z, k) is equivalent to LambertW(z, k, 0).
The following table lists conditions such that LambertW(z, k, r) is defined in Fungrim.
Domain Codomain
Numbers
$k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}$ $W_{k}\!\left(z\right) \in \mathbb{C}$
$k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0, -{e}^{-1}\right\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$ $W^{(r)}_{k}\!\left(z\right) \in \mathbb{C}$
$r \in \mathbb{Z}_{\ge 0}$ $W^{(r)}_{0}\!\left(0\right) \in \mathbb{Q}$
Table data: $\left(P, Q\right)$ such that $\left(P\right) \;\implies\; \left(Q\right)$
Definitions:
Fungrim symbol Notation Short description
LambertW$W\!\left(z\right)$ Lambert W-function
ZZ$\mathbb{Z}$ Integers
CC$\mathbb{C}$ Complex numbers
Exp${e}^{z}$ Exponential function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
QQ$\mathbb{Q}$ Rational numbers
Source code for this entry:
Entry(ID("6da738"),
SymbolDefinition(LambertW, LambertW(z), "Lambert W-function"),
Description("Called with one argument", SourceForm(LambertW(z)), "(rendered", LambertW(z), ") represents the principal branch", "of the Lambert W-function."),
Description("Called with two arguments", SourceForm(LambertW(z, k)), "(rendered", LambertW(z, k), ") represents the", k, "-th branch", "of the Lambert W-function."),
Description("Called with three arguments", SourceForm(LambertW(z, k, r)), "(rendered", LambertW(z, k, r), ") represents the", r, "-th derivative of the", k, "-th branch of the Lambert W-function, with inherited branch cuts."),
Description(SourceForm(LambertW(z, k)), "is equivalent to", SourceForm(LambertW(z, k, 0)), "."),
Description("The following table lists conditions such that", SourceForm(LambertW(z, k, 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(z, k), CC)), Tuple(And(Element(k, ZZ), Element(z, SetMinus(CC, Set(0, Neg(Exp(-1))))), Element(r, ZZGreaterEqual(0))), Element(LambertW(z, k, 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.

2021-03-15 19:12:00.328586 UTC