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Fungrim entry: d5917b

{Wk ⁣(z):zC{0}}={x+yi:xR  and  yR  and  ((2k2<u<2k  and  t<v)  or  (2k1u2k)  or  (2k1<u<2k+1  and  tv))   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 < 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 \ge v\right)\right)\; \text{ where } t = x \operatorname{sinc}(y),\;v = -\cos(y),\;u = \frac{y}{\pi} \right\}
Assumptions:kZ1k \in \mathbb{Z}_{\ge 1}
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 < 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 \ge v\right)\right)\; \text{ where } t = x \operatorname{sinc}(y),\;v = -\cos(y),\;u = \frac{y}{\pi} \right\}

k \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
CCC\mathbb{C} Complex numbers
ConstIii Imaginary unit
RRR\mathbb{R} Real numbers
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Coscos(z)\cos(z) Cosine
Piπ\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("d5917b"),
    Formula(Equal(Set(LambertW(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)), Less(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)), GreaterEqual(t, v)))), Equal(t, Mul(x, Sinc(y))), Equal(v, Neg(Cos(y))), Equal(u, Div(y, Pi)))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(1))))

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

2020-08-27 09:56:25.682319 UTC