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Fungrim entry: 2d3356

{W0 ⁣(z):zCR}={x+yi:y(π,π){0}andx(ycot(y),)}\left\{ W_{0}\!\left(z\right) : z \in \mathbb{C} \setminus \mathbb{R} \right\} = \left\{ x + y i : y \in \left(-\pi, \pi\right) \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \in \left(-y \cot(y), \infty\right) \right\}
TeX:
\left\{ W_{0}\!\left(z\right) : z \in \mathbb{C} \setminus \mathbb{R} \right\} = \left\{ x + y i : y \in \left(-\pi, \pi\right) \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, x \in \left(-y \cot(y), \infty\right) \right\}
Definitions:
Fungrim symbol Notation Short description
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
CCC\mathbb{C} Complex numbers
RRR\mathbb{R} Real numbers
ConstIii Imaginary unit
OpenInterval(a,b)\left(a, b\right) Open interval
ConstPiπ\pi The constant pi (3.14...)
Infinity\infty Positive infinity
Source code for this entry:
Entry(ID("2d3356"),
    Formula(Equal(Set(LambertW(0, z), ForElement(z, SetMinus(CC, RR))), Set(Add(x, Mul(y, ConstI)), For(Tuple(x, y)), And(Element(y, SetMinus(OpenInterval(Neg(ConstPi), ConstPi), Set(0))), Element(x, OpenInterval(Mul(Neg(y), Cot(y)), Infinity)))))))

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

2019-10-05 13:11:19.856591 UTC