# Fungrim entry: 268c9e

$\operatorname{atan}\!\left(x + y\right) = \operatorname{atan}(x) + \operatorname{atan}\!\left(\frac{y}{1 + x \left(x + y\right)}\right)$
Assumptions:$x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, y \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x + y\right| < 1 \,\mathbin{\operatorname{and}}\, \left|x\right| < 1$
Alternative assumptions:$x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x \left(x + y\right) > -1$
TeX:
\operatorname{atan}\!\left(x + y\right) = \operatorname{atan}(x) + \operatorname{atan}\!\left(\frac{y}{1 + x \left(x + y\right)}\right)

x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, y \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x + y\right| < 1 \,\mathbin{\operatorname{and}}\, \left|x\right| < 1

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x \left(x + y\right) > -1
Definitions:
Fungrim symbol Notation Short description
Atan$\operatorname{atan}(z)$ Inverse tangent
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("268c9e"),
Assumptions(And(Element(x, CC), Element(y, CC), Less(Abs(Add(x, y)), 1), Less(Abs(x), 1)), And(Element(x, RR), Element(y, RR), Greater(Mul(x, Add(x, y)), -1))))