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Fungrim entry: 268c9e

atan ⁣(x+y)=atan(x)+atan ⁣(y1+x(x+y))\operatorname{atan}\!\left(x + y\right) = \operatorname{atan}(x) + \operatorname{atan}\!\left(\frac{y}{1 + x \left(x + y\right)}\right)
Assumptions:xC  and  yC  and  x+y<1  and  x<1x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x + y\right| < 1 \;\mathbin{\operatorname{and}}\; \left|x\right| < 1
Alternative assumptions:xR  and  yR  and  x(x+y)>1x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \left(x + y\right) > -1
\operatorname{atan}\!\left(x + y\right) = \operatorname{atan}(x) + \operatorname{atan}\!\left(\frac{y}{1 + x \left(x + y\right)}\right)

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x + y\right| < 1 \;\mathbin{\operatorname{and}}\; \left|x\right| < 1

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \left(x + y\right) > -1
Fungrim symbol Notation Short description
Atanatan(z)\operatorname{atan}(z) Inverse tangent
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
RRR\mathbb{R} Real numbers
Source code for this entry:
    Formula(Equal(Atan(Add(x, y)), Add(Atan(x), Atan(Div(y, Add(1, Mul(x, Add(x, y)))))))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC), Less(Abs(Add(x, y)), 1), Less(Abs(x), 1)), And(Element(x, RR), Element(y, RR), Greater(Mul(x, Add(x, y)), -1))))

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2021-03-15 19:12:00.328586 UTC