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Fungrim entry: af8328

0k=0nsinc ⁣(x2k+1)dx={π2,n{0,1,,6}467807924713440738696537864469467807924720320453655260875000π2,n=7\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{0,1,,7}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
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Productnf(n)\prod_{n} f(n) Product
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Infinity\infty Positive infinity
Piπ\pi The constant pi (3.14...)
Range{a,a+1,,b}\{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))))

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2020-08-27 09:56:25.682319 UTC