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Fungrim entry: df52fc

Re ⁣(atan ⁣(x+yi))=12atan2 ⁣(2x,1x2y2)\operatorname{Re}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{2} \operatorname{atan2}\!\left(2 x, 1 - {x}^{2} - {y}^{2}\right)
Assumptions:xR  and  yR  and  not(x=0  and  y(,1]{1})x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \operatorname{not} \left(x = 0 \;\mathbin{\operatorname{and}}\; y \in \left(-\infty, -1\right] \cup \left\{1\right\}\right)
\operatorname{Re}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{2} \operatorname{atan2}\!\left(2 x, 1 - {x}^{2} - {y}^{2}\right)

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\;  \operatorname{not} \left(x = 0 \;\mathbin{\operatorname{and}}\; y \in \left(-\infty, -1\right] \cup \left\{1\right\}\right)
Fungrim symbol Notation Short description
ReRe(z)\operatorname{Re}(z) Real part
Atanatan(z)\operatorname{atan}(z) Inverse tangent
ConstIii Imaginary unit
Atan2atan2 ⁣(y,x)\operatorname{atan2}\!\left(y, x\right) Two-argument inverse tangent
Powab{a}^{b} Power
RRR\mathbb{R} Real numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
Source code for this entry:
    Formula(Equal(Re(Atan(Add(x, Mul(y, ConstI)))), Mul(Div(1, 2), Atan2(Mul(2, x), Sub(Sub(1, Pow(x, 2)), Pow(y, 2)))))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR), Not(And(Equal(x, 0), Element(y, Union(OpenClosedInterval(Neg(Infinity), -1), Set(1))))))))

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2021-03-15 19:12:00.328586 UTC