Fungrim entry: d7136f

\mathop{\operatorname{solutions}\,}\limits_{w \in \mathbb{C}} \left[w {e}^{w} = z\right] = \left\{ W_{k}\!\left(z\right) : k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \left(z \ne 0 \,\mathbin{\operatorname{or}}\, k = 0\right) \right\}

Assumptions:

z \in \mathbb{C}

TeX:

\mathop{\operatorname{solutions}\,}\limits_{w \in \mathbb{C}} \left[w {e}^{w} = z\right] = \left\{ W_{k}\!\left(z\right) : k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \left(z \ne 0 \,\mathbin{\operatorname{or}}\, k = 0\right) \right\}

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d7136f"),
    Formula(Equal(Solutions(Brackets(Equal(Mul(w, Exp(w)), z)), w, Element(w, CC)), SetBuilder(LambertW(k, z), k, And(Element(k, ZZ), Or(Unequal(z, 0), Equal(k, 0)))))),
    Variables(z),
    Assumptions(Element(z, CC)))

Topics using this entry

Lambert W-function