Fungrim entry: 636929

\mathop{\operatorname{solution*}\,}\limits_{w \in \left(-\infty, -1\right]} \left[w {e}^{w} = x\right] = W_{-1}\!\left(x\right)

Assumptions:

x \in \left[-\frac{1}{e}, 0\right)

TeX:

\mathop{\operatorname{solution*}\,}\limits_{w \in \left(-\infty, -1\right]} \left[w {e}^{w} = x\right] = W_{-1}\!\left(x\right)

x \in \left[-\frac{1}{e}, 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`UniqueSolution`	$\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)$	Unique solution
`Exp`	${e}^{z}$	Exponential function
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`LambertW`	$W\!\left(z\right)$	Lambert W-function
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("636929"),
    Formula(Equal(UniqueSolution(Brackets(Equal(Mul(w, Exp(w)), x)), ForElement(w, OpenClosedInterval(Neg(Infinity), -1))), LambertW(x, -1))),
    Variables(x),
    Assumptions(Element(x, ClosedOpenInterval(Neg(Div(1, ConstE)), 0))))

Topics using this entry

Lambert W-function