Fungrim entry: 38dc04

\int_{0}^{\infty} {e}^{-a x} \operatorname{sinc}(x) \, dx = \operatorname{acot}(a)

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

TeX:

\int_{0}^{\infty} {e}^{-a x} \operatorname{sinc}(x) \, dx = \operatorname{acot}(a)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("38dc04"),
    Formula(Equal(Integral(Mul(Exp(Neg(Mul(a, x))), Sinc(x)), For(x, 0, Infinity)), Acot(a))),
    Variables(a),
    Assumptions(And(Element(a, CC), Greater(Re(a), 0))))

Topics using this entry

Sinc function

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC