# Fungrim entry: 108daa

$\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(a x\right) \operatorname{sinc}\!\left(b x\right) \, dx = \frac{\pi}{2} \frac{\left|a + b\right| - \left|a - b\right|}{a b}$
Assumptions:$a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; b \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a \ne 0 \;\mathbin{\operatorname{and}}\; b \ne 0$
TeX:
\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(a x\right) \operatorname{sinc}\!\left(b x\right) \, dx = \frac{\pi}{2} \frac{\left|a + b\right| - \left|a - b\right|}{a b}

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; b \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a \ne 0 \;\mathbin{\operatorname{and}}\; b \ne 0
Definitions:
Fungrim symbol Notation Short description
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Sinc$\operatorname{sinc}(z)$ Sinc function
Infinity$\infty$ Positive infinity
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("108daa"),
Formula(Equal(Integral(Mul(Sinc(Mul(a, x)), Sinc(Mul(b, x))), For(x, Neg(Infinity), Infinity)), Mul(Div(Pi, 2), Div(Sub(Abs(Add(a, b)), Abs(Sub(a, b))), Mul(a, b))))),
Variables(a, b),
Assumptions(And(Element(a, RR), Element(b, RR), NotEqual(a, 0), NotEqual(b, 0))))

## 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