# Fungrim entry: 6d936e

$W_{k}\!\left(\overline{z}\right) = \overline{W_{-k}\!\left(z\right)}$
Assumptions:$k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(k = 0 \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -{e}^{-1}\right)\right) \;\mathbin{\operatorname{or}}\; \left(k \ne 0 \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right]\right)\right)$
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
LambertW$W\!\left(z\right)$ Lambert W-function
Conjugate$\overline{z}$ Complex conjugate
ZZ$\mathbb{Z}$ Integers
CC$\mathbb{C}$ Complex numbers
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Source code for this entry:
Entry(ID("6d936e"),
Formula(Equal(LambertW(Conjugate(z), k), Conjugate(LambertW(z, Neg(k))))),
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(NotEqual(k, 0), NotElement(z, OpenClosedInterval(Neg(Infinity), 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