Fungrim entry: a68e0e

\operatorname{Im}\!\left(W_{k}\!\left(z\right)\right) \in \left(\operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| - 2\right) \pi, \operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| + 1\right) \pi\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}

TeX:

\operatorname{Im}\!\left(W_{k}\!\left(z\right)\right) \in \left(\operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| - 2\right) \pi, \operatorname{sgn}\!\left(k\right) \left(2 \left|k\right| + 1\right) \pi\right)

z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \setminus \left\{-1, 0, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Sign`	$\operatorname{sgn}\!\left(z\right)$	Sign function
`Abs`	$\left\|z\right\|$	Absolute value
`ConstPi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("a68e0e"),
    Formula(Element(Im(LambertW(k, z)), OpenInterval(Mul(Mul(Sign(k), Sub(Mul(2, Abs(k)), 2)), ConstPi), Mul(Mul(Sign(k), Add(Mul(2, Abs(k)), 1)), ConstPi)))),
    Variables(z, k),
    Assumptions(And(Element(z, SetMinus(CC, Set(0))), Element(k, SetMinus(ZZ, Set(-1, 0, 1))))))

Topics using this entry

Lambert W-function