Fungrim entry: 268c9e

\operatorname{atan}\!\left(x + y\right) = \operatorname{atan}\!\left(x\right) + \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| \lt 1 \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1

Alternative assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x \left(x + y\right) \gt -1

TeX:

\operatorname{atan}\!\left(x + y\right) = \operatorname{atan}\!\left(x\right) + \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| \lt 1 \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x \left(x + y\right) \gt -1

Definitions:

Fungrim symbol	Notation	Short description
`Atan`	$\operatorname{atan}\!\left(z\right)$	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"),
    Formula(Equal(Atan(Add(x, y)), Add(Atan(x), Atan(Div(y, Add(1, Mul(x, Add(x, y)))))))),
    Variables(x, y),
    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))))

Topics using this entry

Inverse tangent