Fungrim entry: df52fc

\operatorname{Re}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{2} \operatorname{atan2}\!\left(2 x, 1 - {x}^{2} - {y}^{2}\right)

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \operatorname{not} \left(x = 0 \;\mathbin{\operatorname{and}}\; y \in \left(-\infty, -1\right] \cup \left\{1\right\}\right)

TeX:

\operatorname{Re}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{2} \operatorname{atan2}\!\left(2 x, 1 - {x}^{2} - {y}^{2}\right)

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\;  \operatorname{not} \left(x = 0 \;\mathbin{\operatorname{and}}\; y \in \left(-\infty, -1\right] \cup \left\{1\right\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Re`	$\operatorname{Re}(z)$	Real part
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`ConstI`	$i$	Imaginary unit
`Atan2`	$\operatorname{atan2}\!\left(y, x\right)$	Two-argument inverse tangent
`Pow`	${a}^{b}$	Power
`RR`	$\mathbb{R}$	Real numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("df52fc"),
    Formula(Equal(Re(Atan(Add(x, Mul(y, ConstI)))), Mul(Div(1, 2), Atan2(Mul(2, x), Sub(Sub(1, Pow(x, 2)), Pow(y, 2)))))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR), Not(And(Equal(x, 0), Element(y, Union(OpenClosedInterval(Neg(Infinity), -1), Set(1))))))))

Topics using this entry

Inverse tangent