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Fungrim entry: 073e1a

atan ⁣(1z)=π2atan(z)\operatorname{atan}\!\left(\frac{1}{z}\right) = -\frac{\pi}{2} - \operatorname{atan}(z)
Assumptions:zCand(Re(z)<0or(Re(z)=0andIm(z)(,1)(0,1)))z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\operatorname{Re}(z) < 0 \,\mathbin{\operatorname{or}}\, \left(\operatorname{Re}(z) = 0 \,\mathbin{\operatorname{and}}\, \operatorname{Im}(z) \in \left(-\infty, -1\right) \cup \left(0, 1\right)\right)\right)
TeX:
\operatorname{atan}\!\left(\frac{1}{z}\right) = -\frac{\pi}{2} - \operatorname{atan}(z)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(\operatorname{Re}(z) < 0 \,\mathbin{\operatorname{or}}\, \left(\operatorname{Re}(z) = 0 \,\mathbin{\operatorname{and}}\, \operatorname{Im}(z) \in \left(-\infty, -1\right) \cup \left(0, 1\right)\right)\right)
Definitions:
Fungrim symbol Notation Short description
Atanatan(z)\operatorname{atan}(z) Inverse tangent
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ImIm(z)\operatorname{Im}(z) Imaginary part
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
Source code for this entry:
Entry(ID("073e1a"),
    Formula(Equal(Atan(Div(1, z)), Sub(Neg(Div(Pi, 2)), Atan(z)))),
    Variables(z),
    Assumptions(And(Element(z, CC), Or(Less(Re(z), 0), And(Equal(Re(z), 0), Element(Im(z), Union(OpenInterval(Neg(Infinity), -1), OpenInterval(0, 1))))))))

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2020-01-31 18:09:28.494564 UTC