# Fungrim entry: 2a69ce

$\int_{-\infty}^{\infty} {e}^{i a x} \operatorname{sinc}(x) \, dx = \int_{-\infty}^{\infty} \cos\!\left(a x\right) \operatorname{sinc}(x) \, dx = \begin{cases} \pi, & \left|a\right| < 1\\\frac{\pi}{2}, & \left|a\right| = 1\\0, & \left|a\right| > 1\\ \end{cases}$
Assumptions:$a \in \mathbb{R}$
TeX:
\int_{-\infty}^{\infty} {e}^{i a x} \operatorname{sinc}(x) \, dx = \int_{-\infty}^{\infty} \cos\!\left(a x\right) \operatorname{sinc}(x) \, dx = \begin{cases} \pi, & \left|a\right| < 1\\\frac{\pi}{2}, & \left|a\right| = 1\\0, & \left|a\right| > 1\\ \end{cases}

a \in \mathbb{R}
Definitions:
Fungrim symbol Notation Short description
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Exp${e}^{z}$ Exponential function
ConstI$i$ Imaginary unit
Sinc$\operatorname{sinc}(z)$ Sinc function
Infinity$\infty$ Positive infinity
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
Abs$\left|z\right|$ Absolute value
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("2a69ce"),
Formula(Equal(Integral(Mul(Exp(Mul(Mul(ConstI, a), x)), Sinc(x)), For(x, Neg(Infinity), Infinity)), Integral(Mul(Cos(Mul(a, x)), Sinc(x)), For(x, Neg(Infinity), Infinity)), Cases(Tuple(Pi, Less(Abs(a), 1)), Tuple(Div(Pi, 2), Equal(Abs(a), 1)), Tuple(0, Greater(Abs(a), 1))))),
Variables(a),
Assumptions(Element(a, RR)))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC