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Fungrim entry: 500c0a

atan ⁣(z)=i2log ⁣(1iz1+iz)\operatorname{atan}\!\left(z\right) = \frac{i}{2} \log\!\left(\frac{1 - i z}{1 + i z}\right)
Assumptions:zCandzi[1,)z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, -z i \notin \left[1, \infty\right)
TeX:
\operatorname{atan}\!\left(z\right) = \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
Atanatan ⁣(z)\operatorname{atan}\!\left(z\right) Inverse tangent
ConstIii Imaginary unit
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
CCC\mathbb{C} Complex numbers
ClosedOpenInterval[a,b)\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)))))

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

2019-06-18 07:49:59.356594 UTC