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`	$W_{k}\!\left(z\right)$	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(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)))))

Topics using this entry

Lambert W-function