# Fungrim entry: bf3e29

$\left\{ W_{-k}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(2 k - 2 < u < 2 k \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \le u \le 2 k\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 < u < 2 k + 1 \,\mathbin{\operatorname{and}}\, t > v\right)\right)\; \text{ where } t = x \operatorname{sinc}(y),\;v = -\cos(y),\;u = -\frac{y}{\pi} \right\}$
Assumptions:$k \in \mathbb{Z}_{\ge 2}$
TeX:
\left\{ W_{-k}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(2 k - 2 < u < 2 k \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 \le u \le 2 k\right) \,\mathbin{\operatorname{or}}\, \left(2 k - 1 < u < 2 k + 1 \,\mathbin{\operatorname{and}}\, t > v\right)\right)\; \text{ where } t = x \operatorname{sinc}(y),\;v = -\cos(y),\;u = -\frac{y}{\pi} \right\}

k \in \mathbb{Z}_{\ge 2}
Definitions:
Fungrim symbol Notation Short description
LambertW$W_{k}\!\left(z\right)$ Lambert W-function
CC$\mathbb{C}$ Complex numbers
ConstI$i$ Imaginary unit
RR$\mathbb{R}$ Real numbers
Sinc$\operatorname{sinc}(z)$ Sinc function
Pi$\pi$ The constant pi (3.14...)
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("bf3e29"),
Formula(Equal(Set(LambertW(Neg(k), z), ForElement(z, SetMinus(CC, Set(0)))), Set(Add(x, Mul(y, ConstI)), For(Tuple(x, y)), Where(And(Element(x, RR), Element(y, RR), Or(And(Less(Sub(Mul(2, k), 2), u, Mul(2, k)), LessEqual(t, v)), Parentheses(LessEqual(Sub(Mul(2, k), 1), u, Mul(2, k))), And(Less(Sub(Mul(2, k), 1), u, Add(Mul(2, k), 1)), Greater(t, v)))), Equal(t, Mul(x, Sinc(y))), Equal(v, Neg(Cos(y))), Equal(u, Neg(Div(y, Pi))))))),
Variables(k),
Assumptions(Element(k, ZZGreaterEqual(2))))

## Topics using this entry

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

2020-01-31 18:09:28.494564 UTC