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Fungrim entry: bfc13f

atan ⁣(1z)=π2csgn ⁣(1z)atan(z)\operatorname{atan}\!\left(\frac{1}{z}\right) = \frac{\pi}{2} \operatorname{csgn}\!\left(\frac{1}{z}\right) - \operatorname{atan}(z)
Assumptions:zCandiz{0}(,1][1,)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
Atanatan(z)\operatorname{atan}(z) Inverse tangent
Piπ\pi The constant pi (3.14...)
Csgncsgn(z)\operatorname{csgn}(z) Real-valued sign function for complex numbers
CCC\mathbb{C} Complex numbers
ConstIii Imaginary unit
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ClosedOpenInterval[a,b)\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))))))

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2020-01-31 18:09:28.494564 UTC