Fungrim home page

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\!\left(x\right) - \arg\!\left(y\right)}{2 \pi} \right\rfloor\right)
Assumptions:xC{0}andyC{0}andaCx \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}
TeX:
{\left(x y\right)}^{a} = {x}^{a} {y}^{a} \exp\!\left(2 \pi i a \left\lfloor \frac{\pi - \arg\!\left(x\right) - \arg\!\left(y\right)}{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}
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
Argarg ⁣(z)\arg\!\left(z\right) Complex argument
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("2090c3"),
    Formula(Equal(Pow(Mul(x, y), a), Mul(Mul(Pow(x, a), Pow(y, a)), Exp(Mul(Mul(Mul(Mul(2, ConstPi), ConstI), a), Floor(Div(Sub(Sub(ConstPi, Arg(x)), Arg(y)), Mul(2, ConstPi)))))))),
    Variables(x, y, a),
    Assumptions(And(Element(x, SetMinus(CC, Set(0))), Element(y, SetMinus(CC, Set(0))), Element(a, CC))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-21 11:44:15.926409 UTC