Fungrim entry: 99c077

J_{n}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(n t - z \sin(t)\right) \, dt

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

J_{n}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(n t - z \sin(t)\right) \, dt

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("99c077"),
    Formula(Equal(BesselJ(n, z), Mul(Div(1, Pi), Integral(Cos(Sub(Mul(n, t), Mul(z, Sin(t)))), For(t, 0, Pi))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, CC))))

Topics using this entry

Bessel functions

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