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 < 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}(y),\;v = -\cos(y),\;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 < 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}(y),\;v = -\cos(y),\;u = -\frac{y}{\pi} \right\}

k \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	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(Set(LambertW(z, Neg(k)), 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)), 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, Pi))))))),
    Variables(k),
    Assumptions(Element(k, ZZGreaterEqual(2))))

Topics using this entry

Lambert W-function