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# Fungrim entry: bfc13f

$\operatorname{atan}\!\left(\frac{1}{z}\right) = \frac{\pi}{2} \operatorname{csgn}\!\left(\frac{1}{z}\right) - \operatorname{atan}(z)$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; i z \notin \left\{0\right\} \cup \left(-\infty, -1\right] \cup \left[1, \infty\right)$
TeX:
\operatorname{atan}\!\left(\frac{1}{z}\right) = \frac{\pi}{2} \operatorname{csgn}\!\left(\frac{1}{z}\right) - \operatorname{atan}(z)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; i z \notin \left\{0\right\} \cup \left(-\infty, -1\right] \cup \left[1, \infty\right)
Definitions:
Fungrim symbol Notation Short description
Atan$\operatorname{atan}(z)$ Inverse tangent
Pi$\pi$ The constant pi (3.14...)
Csgn$\operatorname{csgn}(z)$ Real-valued sign function for complex numbers
CC$\mathbb{C}$ Complex numbers
ConstI$i$ Imaginary unit
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Source code for this entry:
Entry(ID("bfc13f"),
Formula(Equal(Atan(Div(1, z)), Sub(Mul(Div(Pi, 2), Csgn(Div(1, z))), Atan(z)))),
Variables(z),
Assumptions(And(Element(z, CC), NotElement(Mul(ConstI, z), Union(Set(0), OpenClosedInterval(Neg(Infinity), -1), ClosedOpenInterval(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