# Fungrim entry: 073e1a

$\operatorname{atan}\!\left(\frac{1}{z}\right) = -\frac{\pi}{2} - \operatorname{atan}(z)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}(z) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) \in \left(-\infty, -1\right) \cup \left(0, 1\right)\right)\right)$
TeX:
\operatorname{atan}\!\left(\frac{1}{z}\right) = -\frac{\pi}{2} - \operatorname{atan}(z)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}(z) < 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) \in \left(-\infty, -1\right) \cup \left(0, 1\right)\right)\right)
Definitions:
Fungrim symbol Notation Short description
Atan$\operatorname{atan}(z)$ Inverse tangent
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Im$\operatorname{Im}(z)$ Imaginary part
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("073e1a"),
Formula(Equal(Atan(Div(1, z)), Sub(Neg(Div(Pi, 2)), Atan(z)))),
Variables(z),
Assumptions(And(Element(z, CC), Or(Less(Re(z), 0), And(Equal(Re(z), 0), Element(Im(z), Union(OpenInterval(Neg(Infinity), -1), OpenInterval(0, 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