Fungrim entry: 78fca3

$\int_{0}^{\infty} {e}^{-a {x}^{2}} \operatorname{sinc}(x) \, dx = \frac{\pi}{2} \operatorname{erf}\!\left(\frac{1}{2 \sqrt{a}}\right)$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0$
TeX:
\int_{0}^{\infty} {e}^{-a {x}^{2}} \operatorname{sinc}(x) \, dx = \frac{\pi}{2} \operatorname{erf}\!\left(\frac{1}{2 \sqrt{a}}\right)

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
Pow${a}^{b}$ Power
Sinc$\operatorname{sinc}(z)$ Sinc function
Infinity$\infty$ Positive infinity
Pi$\pi$ The constant pi (3.14...)
Erf$\operatorname{erf}(z)$ Error function
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("78fca3"),
Formula(Equal(Integral(Mul(Exp(Neg(Mul(a, Pow(x, 2)))), Sinc(x)), For(x, 0, Infinity)), Mul(Div(Pi, 2), Erf(Div(1, Mul(2, Sqrt(a))))))),
Variables(a),
Assumptions(And(Element(a, CC), Greater(Re(a), 0))))

Topics using this entry

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