Fungrim entry: 72b6ca

W'_{k}\!\left(z\right) = \frac{W_{k}\!\left(z\right)}{z \left(1 + W_{k}\!\left(z\right)\right)}

Assumptions:

\left(k \in \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0, -{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k \in \mathbb{Z} \setminus \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}\right)

TeX:

W'_{k}\!\left(z\right) = \frac{W_{k}\!\left(z\right)}{z \left(1 + W_{k}\!\left(z\right)\right)}

\left(k \in \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0, -{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k \in \mathbb{Z} \setminus \left\{0, 1\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`CC`	$\mathbb{C}$	Complex numbers
`Exp`	${e}^{z}$	Exponential function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("72b6ca"),
    Formula(Equal(LambertW(k, z, 1), Div(LambertW(k, z), Mul(z, Add(1, LambertW(k, z, 0)))))),
    Variables(k, z),
    Assumptions(Or(And(Element(k, Set(0, 1)), Element(z, SetMinus(CC, Set(0, Neg(Exp(-1)))))), And(Element(k, SetMinus(ZZ, Set(0, 1))), Element(z, SetMinus(CC, Set(0)))))))

Topics using this entry

Lambert W-function