Fungrim entry: 8d486c

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

Assumptions:

\left(k = 0 \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{-{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\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{1}{\left(1 + W_{k}\!\left(z\right)\right) \exp\!\left(W_{k}\!\left(z\right)\right)}

\left(k = 0 \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{-{e}^{-1}\right\}\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\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
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("8d486c"),
    Formula(Equal(LambertW(k, z, 1), Div(1, Mul(Add(1, LambertW(k, z, 0)), Exp(LambertW(k, z, 0)))))),
    Variables(k, z),
    Assumptions(Or(And(Equal(k, 0), Element(z, SetMinus(CC, Set(Neg(Exp(-1)))))), And(Equal(k, -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