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Fungrim entry: 21d9a0

{W1 ⁣(z):zC{0}}=(,1]{x+yi:xRandyRand((0<u<2andtv)or(1u2)or(1<u<3andt>v))   where t=xsinc ⁣(y),v=cos ⁣(y),u=yπ}\left\{ W_{-1}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left(-\infty, -1\right] \cup \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(0 < u < 2 \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(1 \le u \le 2\right) \,\mathbin{\operatorname{or}}\, \left(1 < u < 3 \,\mathbin{\operatorname{and}}\, t > v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}
TeX:
\left\{ W_{-1}\!\left(z\right) : z \in \mathbb{C} \setminus \left\{0\right\} \right\} = \left(-\infty, -1\right] \cup \left\{ x + y i : x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left(\left(0 < u < 2 \,\mathbin{\operatorname{and}}\, t \le v\right) \,\mathbin{\operatorname{or}}\, \left(1 \le u \le 2\right) \,\mathbin{\operatorname{or}}\, \left(1 < u < 3 \,\mathbin{\operatorname{and}}\, t > v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}
Definitions:
Fungrim symbol Notation Short description
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ConstIii Imaginary unit
RRR\mathbb{R} Real numbers
ConstPiπ\pi The constant pi (3.14...)
Source code for this entry:
Entry(ID("21d9a0"),
    Formula(Equal(SetBuilder(LambertW(-1, z), z, Element(z, SetMinus(CC, Set(0)))), Union(OpenClosedInterval(Neg(Infinity), -1), SetBuilder(Add(x, Mul(y, ConstI)), Tuple(x, y), Where(And(Element(x, RR), Element(y, RR), Or(And(Less(0, u, 2), LessEqual(t, v)), Parentheses(LessEqual(1, u, 2)), And(Less(1, u, 3), Greater(t, v)))), Equal(t, Mul(x, Sinc(y))), Equal(v, Neg(Cos(y))), Equal(u, Neg(Div(y, ConstPi)))))))))

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

2019-09-16 21:17:18.797188 UTC