Fungrim entry: 40baa9

\mathcal{E}_{\rho} = \left\{ \frac{\rho {e}^{i \theta} + {\rho}^{-1} {e}^{-i \theta}}{2} : \theta \in \left[0, 2 \pi\right) \right\}

Assumptions:

\rho \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \rho \gt 1

TeX:

\mathcal{E}_{\rho} = \left\{ \frac{\rho {e}^{i \theta} + {\rho}^{-1} {e}^{-i \theta}}{2} : \theta \in \left[0, 2 \pi\right) \right\}

\rho \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \rho \gt 1

Definitions:

Fungrim symbol	Notation	Short description
`BernsteinEllipse`	$\mathcal{E}_{\rho}$	Bernstein ellipse with foci -1,+1 and semi-axis sum rho
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`ConstPi`	$\pi$	The constant pi (3.14...)
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("40baa9"),
    Formula(Equal(BernsteinEllipse(rho), SetBuilder(Div(Add(Mul(rho, Exp(Mul(ConstI, theta))), Mul(Pow(rho, -1), Exp(Neg(Mul(ConstI, theta))))), 2), theta, Element(theta, ClosedOpenInterval(0, Mul(2, ConstPi)))))),
    Variables(rho),
    Assumptions(And(Element(rho, RR), Greater(rho, 1))))

Topics using this entry

Complex plane