# Fungrim entry: a1e634

$\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{23}{32} \frac{1}{\sqrt{\left|e z + 1\right|}}\right)\right|$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \left\{-1, 1\right\}$
TeX:
\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{23}{32} \frac{1}{\sqrt{\left|e z + 1\right|}}\right)\right|

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \left\{-1, 1\right\}
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
LambertW$W\!\left(z\right)$ Lambert W-function
Sqrt$\sqrt{z}$ Principal square root
ConstE$e$ The constant e (2.718...)
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("a1e634"),
Formula(LessEqual(Abs(LambertW(z, k, 1)), Abs(Mul(Div(1, Abs(z)), Add(1, Mul(Div(23, 32), Div(1, Sqrt(Abs(Add(Mul(ConstE, z), 1)))))))))),
Variables(k, z),
Assumptions(And(Element(z, CC), Element(k, Set(-1, 1)))))

## 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