# Fungrim entry: e3d274

$\operatorname{atan}\!\left(\frac{1}{z}\right) = \frac{\pi}{2} - \operatorname{atan}(z)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}(z) > 0 \;\mathbin{\operatorname{or}}\; \left(\operatorname{Re}(z) = 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) \in \left(-1, 0\right) \cup \left(1, \infty\right)\right)\right)$
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(-1, 0\right) \cup \left(1, \infty\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("e3d274"),
Formula(Equal(Atan(Div(1, z)), Sub(Div(Pi, 2), Atan(z)))),
Variables(z),
Assumptions(And(Element(z, CC), Or(Greater(Re(z), 0), And(Equal(Re(z), 0), Element(Im(z), Union(OpenInterval(-1, 0), OpenInterval(1, Infinity))))))))

## 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