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Fungrim entry: 72b6ca

Wk ⁣(z)=Wk ⁣(z)z(1+Wk ⁣(z))W'_{k}\!\left(z\right) = \frac{W_{k}\!\left(z\right)}{z \left(1 + W_{k}\!\left(z\right)\right)}
Assumptions:(k{0,1}  and  zC{0,e1})  or  (kZ{0,1}  and  zC{0})\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)
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)
Fungrim symbol Notation Short description
LambertWW ⁣(z)W\!\left(z\right) Lambert W-function
CCC\mathbb{C} Complex numbers
Expez{e}^{z} Exponential function
ZZZ\mathbb{Z} Integers
Source code for this entry:
    Formula(Equal(LambertW(z, k, 1), Div(LambertW(z, k), Mul(z, Add(1, LambertW(z, k)))))),
    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)))))))

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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