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Fungrim entry: 2a69ce

eiaxsinc(x)dx=cos ⁣(ax)sinc(x)dx={π,a<1π2,a=10,a>1\int_{-\infty}^{\infty} {e}^{i a x} \operatorname{sinc}(x) \, dx = \int_{-\infty}^{\infty} \cos\!\left(a x\right) \operatorname{sinc}(x) \, dx = \begin{cases} \pi, & \left|a\right| < 1\\\frac{\pi}{2}, & \left|a\right| = 1\\0, & \left|a\right| > 1\\ \end{cases}
Assumptions:aRa \in \mathbb{R}
TeX:
\int_{-\infty}^{\infty} {e}^{i a x} \operatorname{sinc}(x) \, dx = \int_{-\infty}^{\infty} \cos\!\left(a x\right) \operatorname{sinc}(x) \, dx = \begin{cases} \pi, & \left|a\right| < 1\\\frac{\pi}{2}, & \left|a\right| = 1\\0, & \left|a\right| > 1\\ \end{cases}

a \in \mathbb{R}
Definitions:
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Infinity\infty Positive infinity
Coscos(z)\cos(z) Cosine
Piπ\pi The constant pi (3.14...)
Absz\left|z\right| Absolute value
RRR\mathbb{R} Real numbers
Source code for this entry:
Entry(ID("2a69ce"),
    Formula(Equal(Integral(Mul(Exp(Mul(Mul(ConstI, a), x)), Sinc(x)), For(x, Neg(Infinity), Infinity)), Integral(Mul(Cos(Mul(a, x)), Sinc(x)), For(x, Neg(Infinity), Infinity)), Cases(Tuple(Pi, Less(Abs(a), 1)), Tuple(Div(Pi, 2), Equal(Abs(a), 1)), Tuple(0, Greater(Abs(a), 1))))),
    Variables(a),
    Assumptions(Element(a, RR)))

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2020-08-27 09:56:25.682319 UTC