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Fungrim entry: 636929

solution*w(,1][wew=x]=W1 ⁣(x)\mathop{\operatorname{solution*}\,}\limits_{w \in \left(-\infty, -1\right]} \left[w {e}^{w} = x\right] = W_{-1}\!\left(x\right)
Assumptions:x[1e,0)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
UniqueSolutionsolution*P(x)Q ⁣(x)\mathop{\operatorname{solution*}\,}\limits_{P\left(x\right)} Q\!\left(x\right) Unique solution
Expez{e}^{z} Exponential function
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
ConstEee The constant e (2.718...)
Source code for this entry:
Entry(ID("636929"),
    Formula(Equal(UniqueSolution(Brackets(Equal(Mul(w, Exp(w)), x)), w, Element(w, OpenClosedInterval(Neg(Infinity), -1))), LambertW(-1, x))),
    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.

2019-09-16 21:17:18.797188 UTC