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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:zCand((k{1,1}andRe ⁣(z)0)or(k=1andIm ⁣(z)<0)or(k=1andIm ⁣(z)0))z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\left(k \in \left\{1, -1\right\} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) < 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \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}\!\left(z\right) \ge 0\right) \,\mathbin{\operatorname{or}}\, \left(k = -1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) < 0\right) \,\mathbin{\operatorname{or}}\, \left(k = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(z\right) \ge 0\right)\right)
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
Source code for this entry:
Entry(ID("214b1c"),
    Formula(LessEqual(Abs(LambertW(k, z, 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.

2019-09-22 15:43:45.410764 UTC