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Fungrim entry: f4fd7d

sinc ⁣(eiθ)=limxsinc ⁣(eiθx)={0,eiθ{1,1},otherwise\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:θR\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
Absz\left|z\right| Absolute value
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
Infinity\infty Positive infinity
RealLimitlimxaf(x)\lim_{x \to a} f(x) Limiting value, real variable
RRR\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

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2020-08-27 09:56:25.682319 UTC