Fungrim home page

# Fungrim entry: 1596d2

$\int_{0}^{\infty} \operatorname{sinc}^{n}\!\left(x\right) \, dx = \frac{\pi}{{2}^{n} \left(n - 1\right)!} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {\left(n - 2 k\right)}^{n - 1}$
Assumptions:$n \in \mathbb{Z}_{\ge 1}$
TeX:
\int_{0}^{\infty} \operatorname{sinc}^{n}\!\left(x\right) \, dx = \frac{\pi}{{2}^{n} \left(n - 1\right)!} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {\left(n - 2 k\right)}^{n - 1}

n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Integral$\int_{a}^{b} f(x) \, dx$ Integral
Pow${a}^{b}$ Power
Sinc$\operatorname{sinc}(z)$ Sinc function
Infinity$\infty$ Positive infinity
Pi$\pi$ The constant pi (3.14...)
Factorial$n !$ Factorial
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("1596d2"),
Formula(Equal(Integral(Pow(Sinc(x), n), For(x, 0, Infinity)), Mul(Div(Pi, Mul(Pow(2, n), Factorial(Sub(n, 1)))), Sum(Mul(Mul(Pow(-1, k), Binomial(n, k)), Pow(Sub(n, Mul(2, k)), Sub(n, 1))), For(k, 0, Floor(Div(n, 2))))))),
Variables(n),
Assumptions(Element(n, ZZGreaterEqual(1))))

## 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