Fungrim home page

Fungrim entry: f5887b

sinc ⁣(x+πn)sinc ⁣(x+πm)dx={π,n=m0,nm\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(x + \pi n\right) \operatorname{sinc}\!\left(x + \pi m\right) \, dx = \begin{cases} \pi, & n = m\\0, & n \ne m\\ \end{cases}
Assumptions:nZ  and  mZn \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}
TeX:
\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(x + \pi n\right) \operatorname{sinc}\!\left(x + \pi m\right) \, dx = \begin{cases} \pi, & n = m\\0, & n \ne m\\ \end{cases}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Piπ\pi The constant pi (3.14...)
Infinity\infty Positive infinity
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("f5887b"),
    Formula(Equal(Integral(Mul(Sinc(Add(x, Mul(Pi, n))), Sinc(Add(x, Mul(Pi, m)))), For(x, Neg(Infinity), Infinity)), Cases(Tuple(Pi, Equal(n, m)), Tuple(0, NotEqual(n, m))))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZ), Element(m, ZZ))))

Topics using this entry

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

2020-08-27 09:56:25.682319 UTC