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

Sinc function