Fungrim entry: 2a69ce

\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:

a \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
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Infinity`	$\infty$	Positive infinity
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Abs`	$\left\|z\right\|$	Absolute value
`RR`	$\mathbb{R}$	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)))

Topics using this entry

Sinc function