Fungrim entry: 6d936e

W_{k}\!\left(\overline{z}\right) = \overline{W_{-k}\!\left(z\right)}

Assumptions:

k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(k = 0 \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -{e}^{-1}\right)\right) \;\mathbin{\operatorname{or}}\; \left(k \ne 0 \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right]\right)\right)

TeX:

W_{k}\!\left(\overline{z}\right) = \overline{W_{-k}\!\left(z\right)}

k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(k = 0 \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, -{e}^{-1}\right)\right) \;\mathbin{\operatorname{or}}\; \left(k \ne 0 \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right]\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W\!\left(z\right)$	Lambert W-function
`Conjugate`	$\overline{z}$	Complex conjugate
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`Exp`	${e}^{z}$	Exponential function
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval

Source code for this entry:

Entry(ID("6d936e"),
    Formula(Equal(LambertW(Conjugate(z), k), Conjugate(LambertW(z, Neg(k))))),
    Variables(k, z),
    Assumptions(And(Element(k, ZZ), Element(z, CC), Or(And(Equal(k, 0), NotElement(z, OpenInterval(Neg(Infinity), Neg(Exp(-1))))), And(NotEqual(k, 0), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)))))))

Topics using this entry

Lambert W-function