Fungrim entry: fa30c7

\left|\operatorname{atan}\!\left(x + y\right) - \operatorname{atan}\!\left(x\right)\right| \le \frac{\left|y\right|}{1 + {\left(\max\!\left(0, \left|x\right| - \left|y\right|\right)\right)}^{2}}

Assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R}

TeX:

\left|\operatorname{atan}\!\left(x + y\right) - \operatorname{atan}\!\left(x\right)\right| \le \frac{\left|y\right|}{1 + {\left(\max\!\left(0, \left|x\right| - \left|y\right|\right)\right)}^{2}}

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Atan`	$\operatorname{atan}\!\left(z\right)$	Inverse tangent
`Pow`	${a}^{b}$	Power
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("fa30c7"),
    Formula(LessEqual(Abs(Sub(Atan(Add(x, y)), Atan(x))), Div(Abs(y), Add(1, Pow(Max(0, Sub(Abs(x), Abs(y))), 2))))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR))))

Topics using this entry

Inverse tangent