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Fungrim entry: 503d4d

atan(x)atan(y)=atan ⁣(xy1+xy)\operatorname{atan}(x) - \operatorname{atan}(y) = \operatorname{atan}\!\left(\frac{x - y}{1 + x y}\right)
Assumptions:xC  and  yC  and  x<1  and  y<1x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1 \;\mathbin{\operatorname{and}}\; \left|y\right| < 1
Alternative assumptions:xR  and  yR  and  xy>1x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x y > -1
\operatorname{atan}(x) - \operatorname{atan}(y) = \operatorname{atan}\!\left(\frac{x - y}{1 + x y}\right)

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

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x y > -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(Sub(Atan(x), Atan(y)), Atan(Div(Sub(x, y), Add(1, Mul(x, y)))))),
    Variables(x, y),
    Assumptions(And(Element(x, CC), Element(y, CC), Less(Abs(x), 1), Less(Abs(y), 1)), And(Element(x, RR), Element(y, RR), Greater(Mul(x, y), -1))))

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2021-03-15 19:12:00.328586 UTC