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Fungrim entry: a1e634

Wk ⁣(z)1z(1+23321ez+1)\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:zCandk{1,1}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
Absz\left|z\right| Absolute value
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
Sqrtz\sqrt{z} Principal square root
ConstEee The constant e (2.718...)
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("a1e634"),
    Formula(LessEqual(Abs(LambertW(k, z, 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)))))

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

2019-10-05 13:11:19.856591 UTC