# Fungrim entry: 9ad254

$J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \frac{{e}^{-i z}}{\Gamma\!\left(\nu + 1\right)} \,{}_1F_1\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)$
Assumptions:$\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$
Alternative assumptions:$\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \nu \notin \{-1, -2, \ldots\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}$
TeX:
J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \frac{{e}^{-i z}}{\Gamma\!\left(\nu + 1\right)} \,{}_1F_1\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 i z\right)

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

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \nu \notin \{-1, -2, \ldots\} \;\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
Exp${e}^{z}$ Exponential function
ConstI$i$ Imaginary unit
Gamma$\Gamma(z)$ Gamma function
Hypergeometric1F1$\,{}_1F_1\!\left(a, b, z\right)$ Kummer confluent hypergeometric function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("9ad254"),
Formula(Equal(BesselJ(nu, z), Mul(Mul(Pow(Div(z, 2), nu), Div(Exp(Neg(Mul(ConstI, z))), Gamma(Add(nu, 1)))), Hypergeometric1F1(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(Mul(2, ConstI), z))))),
Variables(nu, z),
Assumptions(And(Element(nu, ZZGreaterEqual(0)), Element(z, CC)), And(Element(nu, CC), NotElement(nu, ZZLessEqual(-1)), 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