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Fungrim entry: 214b1c

Wk ⁣(z)1z(1+14+z2)\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{1}{4 + {\left|z\right|}^{2}}\right)\right|
Assumptions:zC  and  ((k{1,1}  and  Re(z)0)  or  (k=1  and  Im(z)<0)  or  (k=1  and  Im(z)0))z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(k \in \left\{1, -1\right\} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0\right) \;\mathbin{\operatorname{or}}\; \left(k = -1 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) < 0\right) \;\mathbin{\operatorname{or}}\; \left(k = 1 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) \ge 0\right)\right)
TeX:
\left|W'_{k}\!\left(z\right)\right| \le \left|\frac{1}{\left|z\right|} \left(1 + \frac{1}{4 + {\left|z\right|}^{2}}\right)\right|

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\left(k \in \left\{1, -1\right\} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) \ge 0\right) \;\mathbin{\operatorname{or}}\; \left(k = -1 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) < 0\right) \;\mathbin{\operatorname{or}}\; \left(k = 1 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) \ge 0\right)\right)
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
LambertWW ⁣(z)W\!\left(z\right) Lambert W-function
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ImIm(z)\operatorname{Im}(z) Imaginary part
Source code for this entry:
Entry(ID("214b1c"),
    Formula(LessEqual(Abs(LambertW(z, k, 1)), Abs(Mul(Div(1, Abs(z)), Add(1, Div(1, Add(4, Pow(Abs(z), 2)))))))),
    Variables(k, z),
    Assumptions(And(Element(z, CC), Or(And(Element(k, Set(1, -1)), GreaterEqual(Re(z), 0)), And(Equal(k, -1), Less(Im(z), 0)), And(Equal(k, 1), GreaterEqual(Im(z), 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