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 \lt u \lt 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 \lt u \lt 2 k + 1 \,\mathbin{\operatorname{and}}\, t \gt v\right)\right)\; \text{ where } t = x \operatorname{sinc}\!\left(y\right),\,v = -\cos\!\left(y\right),\,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 \lt u \lt 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 \lt u \lt 2 k + 1 \,\mathbin{\operatorname{and}}\, t \gt 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`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers
`ConstPi`	$\pi$	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(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))))

Topics using this entry

Lambert W-function