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

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2021-03-15 19:12:00.328586 UTC