Fungrim entry: f4fd7d

\left|\operatorname{sinc}\!\left({e}^{i \theta} \infty\right)\right| = \lim_{x \to \infty} \left|\operatorname{sinc}\!\left({e}^{i \theta} x\right)\right| = \begin{cases} 0, & {e}^{i \theta} \in \left\{-1, 1\right\}\\\infty, & \text{otherwise}\\ \end{cases}

Assumptions:

\theta \in \mathbb{R}

TeX:

\left|\operatorname{sinc}\!\left({e}^{i \theta} \infty\right)\right| = \lim_{x \to \infty} \left|\operatorname{sinc}\!\left({e}^{i \theta} x\right)\right| = \begin{cases} 0, & {e}^{i \theta} \in \left\{-1, 1\right\}\\\infty, & \text{otherwise}\\ \end{cases}

\theta \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`RealLimit`	$\lim_{x \to a} f(x)$	Limiting value, real variable
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("f4fd7d"),
    Formula(Equal(Abs(Sinc(Mul(Exp(Mul(ConstI, theta)), Infinity))), RealLimit(Abs(Sinc(Mul(Exp(Mul(ConstI, theta)), x))), For(x, Infinity)), Cases(Tuple(0, Element(Exp(Mul(ConstI, theta)), Set(-1, 1))), Tuple(Infinity, Otherwise)))),
    Variables(theta),
    Assumptions(Element(theta, RR)))

Topics using this entry

Sinc function