Fungrim entry: af8328

\int_{0}^{\infty} \prod_{k=0}^{n} \operatorname{sinc}\!\left(\frac{x}{2 k + 1}\right) \, dx = \begin{cases} \frac{\pi}{2}, & n \in \{0, 1, \ldots, 6\}\\\frac{467807924713440738696537864469}{467807924720320453655260875000} \frac{\pi}{2}, & n = 7\\ \end{cases}

Assumptions:

n \in \{0, 1, \ldots, 7\}

TeX:

\int_{0}^{\infty} \prod_{k=0}^{n} \operatorname{sinc}\!\left(\frac{x}{2 k + 1}\right) \, dx = \begin{cases} \frac{\pi}{2}, & n \in \{0, 1, \ldots, 6\}\\\frac{467807924713440738696537864469}{467807924720320453655260875000} \frac{\pi}{2}, & n = 7\\ \end{cases}

n \in \{0, 1, \ldots, 7\}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Product`	$\prod_{n} f(n)$	Product
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Infinity`	$\infty$	Positive infinity
`Pi`	$\pi$	The constant pi (3.14...)
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("af8328"),
    Formula(Equal(Integral(Product(Sinc(Div(x, Add(Mul(2, k), 1))), For(k, 0, n)), For(x, 0, Infinity)), Cases(Tuple(Div(Pi, 2), Element(n, Range(0, 6))), Tuple(Mul(Div(467807924713440738696537864469, 467807924720320453655260875000), Div(Pi, 2)), Equal(n, 7))))),
    Variables(n),
    Assumptions(Element(n, Range(0, 7))))

Topics using this entry

Sinc function