Fungrim entry: 1232f7

\sqrt{r {e}^{i \theta}} = \sqrt{r} {e}^{i \theta / 2}

Assumptions:

r \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \left(-\pi, \pi\right]

TeX:

\sqrt{r {e}^{i \theta}} = \sqrt{r} {e}^{i \theta / 2}

r \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \left(-\pi, \pi\right]

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("1232f7"),
    Formula(Equal(Sqrt(Mul(r, Exp(Mul(ConstI, theta)))), Mul(Sqrt(r), Exp(Div(Mul(ConstI, theta), 2))))),
    Variables(r, theta),
    Assumptions(And(Element(r, ClosedOpenInterval(0, Infinity)), Element(theta, OpenClosedInterval(Neg(Pi), Pi)))))

Topics using this entry

Square roots