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Fungrim entry: 1596d2

0sincn ⁣(x)dx=π2n(n1)!k=0n/2(1)k(nk)(n2k)n1\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:nZ1n \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
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Infinity\infty Positive infinity
Piπ\pi The constant pi (3.14...)
Factorialn!n ! Factorial
Sumnf(n)\sum_{n} f(n) Sum
Binomial(nk){n \choose k} Binomial coefficient
ZZGreaterEqualZn\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))))

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2020-08-27 09:56:25.682319 UTC