Fungrim entry: 108daa

\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:

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; b \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a \ne 0 \;\mathbin{\operatorname{and}}\; b \ne 0

TeX:

\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

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Infinity`	$\infty$	Positive infinity
`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("108daa"),
    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))))

Topics using this entry

Sinc function