Fungrim entry: 214b1c

\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:

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) \lt 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) \lt 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
`Abs`	$\left\|z\right\|$	Absolute value
`LambertW`	$W_{k}\!\left(z\right)$	Lambert W-function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part
`Im`	$\operatorname{Im}\!\left(z\right)$	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))))))

Topics using this entry

Lambert W-function