# Fungrim entry: 2090c3

${\left(x y\right)}^{a} = {x}^{a} {y}^{a} \exp\!\left(2 \pi i a \left\lfloor \frac{\pi - \arg(x) - \arg(y)}{2 \pi} \right\rfloor\right)$
Assumptions:$x \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}$
TeX:
{\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}
Definitions:
Fungrim symbol Notation Short description
Pow${a}^{b}$ Power
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
Arg$\arg(z)$ Complex argument
CC$\mathbb{C}$ 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, 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))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC