# 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

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