Fungrim symbol: BesselJ

Symbol: BesselJ —

J_{\nu}\!\left(z\right)

— Bessel function of the first kind

The following table lists all conditions such that BesselJ(nu, z) is defined in Fungrim.

Domain	Codomain
Numbers
$\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{R}$	$J_{\nu}\!\left(z\right) \in \mathbb{R}$
$\nu \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \left(0, \infty\right)$	$J_{\nu}\!\left(z\right) \in \mathbb{R}$
$\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}$	$J_{\nu}\!\left(z\right) \in \mathbb{C}$
$\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\}$	$J_{\nu}\!\left(z\right) \in \mathbb{C}$
$\nu \in \left[0, \infty\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C}$	$J_{\nu}\!\left(z\right) \in \mathbb{C}$

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

Source code for this entry:

Entry(ID("b4165c"),
    SymbolDefinition(BesselJ, BesselJ(nu, z), "Bessel function of the first kind"),
    Description("The following table lists all conditions such that", SourceForm(BesselJ(nu, z)), "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)))))

Topics using this symbol

The symbol BesselJ appears in 6 topics:

Bessel functions (in 20 entries)
Specific values of Bessel functions (in 6 entries)
Recurrence relations for Bessel functions (in 4 entries)
Hypergeometric representations of Bessel functions (in 4 entries)
Sine (in 1 entries)
Confluent hypergeometric functions (in 1 entries)

Entries using this symbol

The symbol BesselJ appears in 32 entries: