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

Lambert W-function