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

Sinc function