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# Fungrim entry: b120b9

Symbol: Atan $\operatorname{atan}\!\left(z\right)$ Inverse tangent
The inverse tangent function $\operatorname{atan}\!\left(z\right)$ (denoted by Atan(z) in the Fungrim formula language) is a function of a single variable. The following table lists conditions such that Atan(z) is defined in Fungrim.
Domain Codomain
Numbers
$z \in \mathbb{R}$ $\operatorname{atan}\!\left(z\right) \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
$z \in \mathbb{C} \setminus \left\{-i, +i\right\}$ $\operatorname{atan}\!\left(z\right) \in \mathbb{C}$
Infinities
$z \in \left\{-\infty, \infty\right\}$ $\operatorname{atan}\!\left(z\right) \in \left\{-\frac{\pi}{2}, \frac{\pi}{2}\right\}$
$z \in \left\{-i, +i\right\}$ $\operatorname{atan}\!\left(z\right) \in \left\{-i \infty, +i \infty\right\}$
Formal power series
$z \in \mathbb{R}[[x]]$ $\operatorname{atan}\!\left(z\right) \in \mathbb{R}[[x]]$
$z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z \notin \left\{-i, +i\right\}$ $\operatorname{atan}\!\left(z\right) \in \mathbb{C}[[x]]$
Table data: $\left(P, Q\right)$ such that $\left(P\right) \implies \left(Q\right)$
Definitions:
Fungrim symbol Notation Short description
Atan$\operatorname{atan}\!\left(z\right)$ Inverse tangent
RR$\mathbb{R}$ Real numbers
OpenInterval$\left(a, b\right)$ Open interval
ConstPi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
ConstI$i$ Imaginary unit
Infinity$\infty$ Positive infinity
FormalPowerSeries$K[[x]]$ Formal power series
Source code for this entry:
Entry(ID("b120b9"),
SymbolDefinition(Atan, Atan(z), "Inverse tangent"),
Description("The inverse tangent function", Atan(z), "(denoted by", SourceForm(Atan(z)), "in the Fungrim formula language)", "is a function of a single variable.", "The following table lists conditions such that", SourceForm(Atan(z)), "is defined in Fungrim."),
Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(z, RR), Element(Atan(z), OpenInterval(Neg(Div(ConstPi, 2)), Div(ConstPi, 2)))), Tuple(Element(z, SetMinus(CC, Set(Neg(ConstI), Pos(ConstI)))), Element(Atan(z), CC)), TableSection("Infinities"), Tuple(Element(z, Set(Neg(Infinity), Infinity)), Element(Atan(z), Set(Neg(Div(ConstPi, 2)), Div(ConstPi, 2)))), Tuple(Element(z, Set(Neg(ConstI), Pos(ConstI))), Element(Atan(z), Set(Mul(Neg(ConstI), Infinity), Mul(Pos(ConstI), Infinity)))), TableSection("Formal power series"), Tuple(Element(z, FormalPowerSeries(RR, x)), Element(Atan(z), FormalPowerSeries(RR, x))), Tuple(And(Element(z, FormalPowerSeries(CC, x)), NotElement(SeriesCoefficient(z, x, 0), Set(Neg(ConstI), Pos(ConstI)))), Element(Atan(z), FormalPowerSeries(CC, x))))))

## Topics using this entry

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

2019-08-21 11:44:15.926409 UTC