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Fungrim entry: 108daa

sinc ⁣(ax)sinc ⁣(bx)dx=π2a+babab\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(a x\right) \operatorname{sinc}\!\left(b x\right) \, dx = \frac{\pi}{2} \frac{\left|a + b\right| - \left|a - b\right|}{a b}
Assumptions:aR  and  bR  and  a0  and  b0a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; b \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a \ne 0 \;\mathbin{\operatorname{and}}\; b \ne 0
\int_{-\infty}^{\infty} \operatorname{sinc}\!\left(a x\right) \operatorname{sinc}\!\left(b x\right) \, dx = \frac{\pi}{2} \frac{\left|a + b\right| - \left|a - b\right|}{a b}

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; b \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a \ne 0 \;\mathbin{\operatorname{and}}\; b \ne 0
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Infinity\infty Positive infinity
Piπ\pi The constant pi (3.14...)
Absz\left|z\right| Absolute value
RRR\mathbb{R} Real numbers
Source code for this entry:
    Formula(Equal(Integral(Mul(Sinc(Mul(a, x)), Sinc(Mul(b, x))), For(x, Neg(Infinity), Infinity)), Mul(Div(Pi, 2), Div(Sub(Abs(Add(a, b)), Abs(Sub(a, b))), Mul(a, b))))),
    Variables(a, b),
    Assumptions(And(Element(a, RR), Element(b, RR), NotEqual(a, 0), NotEqual(b, 0))))

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2021-03-15 19:12:00.328586 UTC