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Fungrim entry: 78fca3

0eax2sinc(x)dx=π2erf ⁣(12a)\int_{0}^{\infty} {e}^{-a {x}^{2}} \operatorname{sinc}(x) \, dx = \frac{\pi}{2} \operatorname{erf}\!\left(\frac{1}{2 \sqrt{a}}\right)
Assumptions:aC  and  Re(a)>0a \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
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Expez{e}^{z} Exponential function
Powab{a}^{b} Power
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Infinity\infty Positive infinity
Piπ\pi The constant pi (3.14...)
Erferf(z)\operatorname{erf}(z) Error function
Sqrtz\sqrt{z} Principal square root
CCC\mathbb{C} Complex numbers
ReRe(z)\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.

2020-08-27 09:56:25.682319 UTC