Fungrim entry: 500c0a

\operatorname{atan}(z) = \frac{i}{2} \log\!\left(\frac{1 - i z}{1 + i z}\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; -z i \notin \left[1, \infty\right)

TeX:

\operatorname{atan}(z) = \frac{i}{2} \log\!\left(\frac{1 - i z}{1 + i z}\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; -z i \notin \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`ConstI`	$i$	Imaginary unit
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("500c0a"),
    Formula(Equal(Atan(z), Mul(Div(ConstI, 2), Log(Div(Sub(1, Mul(ConstI, z)), Add(1, Mul(ConstI, z))))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(Mul(Neg(z), ConstI), ClosedOpenInterval(1, Infinity)))))

Topics using this entry

Inverse tangent

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