Fungrim entry: f5887b

\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:

n \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
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`ZZ`	$\mathbb{Z}$	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

Sinc function