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Fungrim entry: 2090c3

(xy)a=xayaexp ⁣(2πiaπarg(x)arg(y)2π){\left(x y\right)}^{a} = {x}^{a} {y}^{a} \exp\!\left(2 \pi i a \left\lfloor \frac{\pi - \arg(x) - \arg(y)}{2 \pi} \right\rfloor\right)
Assumptions:xC{0}  and  yC{0}  and  aCx \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}
{\left(x y\right)}^{a} = {x}^{a} {y}^{a} \exp\!\left(2 \pi i a \left\lfloor \frac{\pi - \arg(x) - \arg(y)}{2 \pi} \right\rfloor\right)

x \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}
Fungrim symbol Notation Short description
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Argarg(z)\arg(z) Complex argument
CCC\mathbb{C} Complex numbers
Source code for this entry:
    Formula(Equal(Pow(Mul(x, y), a), Mul(Mul(Pow(x, a), Pow(y, a)), Exp(Mul(Mul(Mul(Mul(2, Pi), ConstI), a), Floor(Div(Sub(Sub(Pi, Arg(x)), Arg(y)), Mul(2, Pi)))))))),
    Variables(x, y, a),
    Assumptions(And(Element(x, SetMinus(CC, Set(0))), Element(y, SetMinus(CC, Set(0))), Element(a, CC))))

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2021-03-15 19:12:00.328586 UTC