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Fungrim entry: 6d936e

Wk ⁣(z)=Wk ⁣(z)W_{k}\!\left(\overline{z}\right) = \overline{W_{-k}\!\left(z\right)}
Assumptions:kZandzCand((k=0andz(,e1))or(k0andz(,0]))k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k = 0 \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, -{e}^{-1}\right)\right) \,\mathbin{\operatorname{or}}\, \left(k \ne 0 \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, 0\right]\right)\right)
TeX:
W_{k}\!\left(\overline{z}\right) = \overline{W_{-k}\!\left(z\right)}

k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k = 0 \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, -{e}^{-1}\right)\right) \,\mathbin{\operatorname{or}}\, \left(k \ne 0 \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, 0\right]\right)\right)
Definitions:
Fungrim symbol Notation Short description
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
Conjugatez\overline{z} Complex conjugate
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Source code for this entry:
Entry(ID("6d936e"),
    Formula(Equal(LambertW(k, Conjugate(z)), Conjugate(LambertW(Neg(k), z)))),
    Variables(k, z),
    Assumptions(And(Element(k, ZZ), Element(z, CC), Or(And(Equal(k, 0), NotElement(z, OpenInterval(Neg(Infinity), Neg(Exp(-1))))), And(Unequal(k, 0), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)))))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-19 15:10:20.037976 UTC