# 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$W\!\left(z\right)$ 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(z, k, 1), Div(1, Mul(Add(1, LambertW(z, k)), Exp(LambertW(z, k)))))),
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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC