# Fungrim entry: ecd36f

$J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right)$
Assumptions:$\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$
Alternative assumptions:$\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}$
TeX:
J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right)

\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}
Definitions:
Fungrim symbol Notation Short description
BesselJ$J_{\nu}\!\left(z\right)$ Bessel function of the first kind
Pow${a}^{b}$ Power
Hypergeometric0F1Regularized$\,{}_0{\textbf F}_1\!\left(a, z\right)$ Regularized confluent hypergeometric limit function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("ecd36f"),
Formula(Equal(BesselJ(nu, z), Mul(Pow(Div(z, 2), nu), Hypergeometric0F1Regularized(Add(nu, 1), Neg(Div(Pow(z, 2), 4)))))),
Variables(nu, z),
Assumptions(And(Element(nu, ZZGreaterEqual(0)), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(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