# Fungrim entry: b4165c

Symbol: BesselJ $J_{\nu}\!\left(z\right)$ Bessel function of the first kind
BesselJ(nu, z), rendered as $J_{\nu}\!\left(z\right)$, denotes the Bessel function of the first kind.
The input $\nu$ is called the order. The input $z$ is called the argument.
Called with three arguments, BesselJ(nu, z, r), rendered as $J'_{\nu}\!\left(z\right)$, $J''_{\nu}\!\left(z\right)$, $J'''_{\nu}\!\left(z\right)$ ( $1 \le r \le 3$ ), or $J^{(r)}_{\nu}\!\left(z\right)$, represents the order $r$ derivative of the Bessel function with respect to the argument $z$.
The following table lists conditions such that BesselJ(nu, z) or BesselJ(nu, z, r) is defined in Fungrim.
Domain Codomain
Numbers
$\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{R}$ $J_{\nu}\!\left(z\right) \in \mathbb{R}$
$\nu \in \mathbb{R} \;\mathbin{\operatorname{and}}\; z \in \left(0, \infty\right)$ $J_{\nu}\!\left(z\right) \in \mathbb{R}$
$\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$ $J_{\nu}\!\left(z\right) \in \mathbb{C}$
$\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}$ $J_{\nu}\!\left(z\right) \in \mathbb{C}$
$\nu \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$ $J_{\nu}\!\left(z\right) \in \mathbb{C}$
$\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{R} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$ $J^{(r)}_{\nu}\!\left(z\right) \in \mathbb{R}$
$\nu \in \mathbb{R} \;\mathbin{\operatorname{and}}\; z \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$ $J^{(r)}_{\nu}\!\left(z\right) \in \mathbb{R}$
$\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$ $J^{(r)}_{\nu}\!\left(z\right) \in \mathbb{C}$
$\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$ $J^{(r)}_{\nu}\!\left(z\right) \in \mathbb{C}$
Table data: $\left(P, Q\right)$ such that $\left(P\right) \;\implies\; \left(Q\right)$
Definitions:
Fungrim symbol Notation Short description
BesselJ$J_{\nu}\!\left(z\right)$ Bessel function of the first kind
ZZ$\mathbb{Z}$ Integers
RR$\mathbb{R}$ Real numbers
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("b4165c"),
SymbolDefinition(BesselJ, BesselJ(nu, z), "Bessel function of the first kind"),
Description(SourceForm(BesselJ(nu, z)), ", rendered as", BesselJ(nu, z), ", denotes the Bessel function of the first kind. "),
Description("The input", nu, "is called the order. The input", z, "is called the argument."),
Description("Called with three arguments, ", SourceForm(BesselJ(nu, z, r)), ", rendered as", BesselJ(nu, z, 1), ", ", BesselJ(nu, z, 2), ", ", BesselJ(nu, z, 3), " (", LessEqual(1, r, 3), "), or", BesselJ(nu, z, r), ", represents the order", r, "derivative of the Bessel function with respect to the argument", z, "."),
Description("The following table lists conditions such that", SourceForm(BesselJ(nu, z)), "or", SourceForm(BesselJ(nu, z, r)), "is defined in Fungrim."),
Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(And(Element(nu, ZZ), Element(z, RR)), Element(BesselJ(nu, z), RR)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity))), Element(BesselJ(nu, z), RR)), Tuple(And(Element(nu, ZZ), Element(z, CC)), Element(BesselJ(nu, z), CC)), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0)))), Element(BesselJ(nu, z), CC)), Tuple(And(Element(nu, ClosedOpenInterval(0, Infinity)), Element(z, CC)), Element(BesselJ(nu, z), CC)), Tuple(And(Element(nu, ZZ), Element(z, RR), Element(r, ZZGreaterEqual(0))), Element(BesselJ(nu, z, r), RR)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity)), Element(r, ZZGreaterEqual(0))), Element(BesselJ(nu, z, r), RR)), Tuple(And(Element(nu, ZZ), Element(z, CC), Element(r, ZZGreaterEqual(0))), Element(BesselJ(nu, z, r), CC)), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0))), Element(BesselJ(nu, z, r), CC)))))

## 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