Fungrim home page

Fungrim entry: a68e0e

sgn(k)Im ⁣(Wk ⁣(z))((2k2)π,(2k+1)π)\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:zC{0}  and  kZ{1,0,1}z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}
\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\}
Fungrim symbol Notation Short description
Signsgn(z)\operatorname{sgn}(z) Sign function
ImIm(z)\operatorname{Im}(z) Imaginary part
LambertWW ⁣(z)W\!\left(z\right) Lambert W-function
OpenInterval(a,b)\left(a, b\right) Open interval
Absz\left|z\right| Absolute value
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
Source code for this entry:
    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