Fungrim entry: 2090c3

{\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:

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\!\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
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Arg`	$\arg\!\left(z\right)$	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, 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

Powers