# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC