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Fungrim entry: bf3e29

{Wk ⁣(z):zC{0}}={x+yi:xRandyRand((2k2<u<2kandtv)or(2k1u2k)or(2k1<u<2k+1andt>v))   where t=xsinc ⁣(y),v=cos ⁣(y),u=yπ}\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}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}
Assumptions:kZ2k \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}\!\left(y\right),\,v = -\cos\!\left(y\right),\,u = -\frac{y}{\pi} \right\}

k \in \mathbb{Z}_{\ge 2}
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
ConstIii Imaginary unit
RRR\mathbb{R} Real numbers
ConstPiπ\pi The constant pi (3.14...)
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("bf3e29"),
    Formula(Equal(SetBuilder(LambertW(Neg(k), z), z, Element(z, SetMinus(CC, Set(0)))), SetBuilder(Add(x, Mul(y, ConstI)), 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, ConstPi))))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))))

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

2019-08-17 11:32:46.829430 UTC