# Fungrim entry: a68e0e

$\operatorname{sgn}(k) \operatorname{Im}\!\left(W_{k}\!\left(z\right)\right) \in \left(\left(2 \left|k\right| - 2\right) \pi, \left(2 \left|k\right| + 1\right) \pi\right)$
Assumptions:$z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}$
TeX:
\operatorname{sgn}(k) \operatorname{Im}\!\left(W_{k}\!\left(z\right)\right) \in \left(\left(2 \left|k\right| - 2\right) \pi, \left(2 \left|k\right| + 1\right) \pi\right)

z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}
Definitions:
Fungrim symbol Notation Short description
Sign$\operatorname{sgn}(z)$ Sign function
Im$\operatorname{Im}(z)$ Imaginary part
LambertW$W\!\left(z\right)$ Lambert W-function
OpenInterval$\left(a, b\right)$ Open interval
Abs$\left|z\right|$ Absolute value
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("a68e0e"),
Formula(Element(Mul(Sign(k), Im(LambertW(z, k))), OpenInterval(Mul(Sub(Mul(2, Abs(k)), 2), Pi), Mul(Add(Mul(2, Abs(k)), 1), Pi)))),
Variables(z, k),
Assumptions(And(Element(z, SetMinus(CC, Set(0))), Element(k, SetMinus(ZZ, Set(-1, 0, 1))))))

## 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