Fungrim entry: 45f05f

\frac{1}{\operatorname{sinc}\!\left(\frac{\pi}{z}\right)} = \int_{0}^{\infty} \frac{1}{{x}^{z} + 1} \, dx

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 1

TeX:

\frac{1}{\operatorname{sinc}\!\left(\frac{\pi}{z}\right)} = \int_{0}^{\infty} \frac{1}{{x}^{z} + 1} \, dx

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 1

Definitions:

Fungrim symbol	Notation	Short description
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("45f05f"),
    Formula(Equal(Div(1, Sinc(Div(Pi, z))), Integral(Div(1, Add(Pow(x, z), 1)), For(x, 0, Infinity)))),
    Variables(z),
    Assumptions(And(Element(z, CC), Greater(Re(z), 1))))

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