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Fungrim entry: b4165c

Symbol: BesselJ Jν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
BesselJ(nu, z), rendered as Jν ⁣(z)J_{\nu}\!\left(z\right), denotes the Bessel function of the first kind.
The input ν\nu is called the order. The input zz is called the argument.
Called with three arguments, BesselJ(nu, z, r), rendered as Jν ⁣(z)J'_{\nu}\!\left(z\right), Jν ⁣(z)J''_{\nu}\!\left(z\right), Jν ⁣(z)J'''_{\nu}\!\left(z\right) ( 1r31 \le r \le 3 ), or Jν(r) ⁣(z)J^{(r)}_{\nu}\!\left(z\right), represents the order rr derivative of the Bessel function with respect to the argument zz.
The following table lists conditions such that BesselJ(nu, z) or BesselJ(nu, z, r) is defined in Fungrim.
Domain Codomain
Numbers
νZandzR\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{R} Jν ⁣(z)RJ_{\nu}\!\left(z\right) \in \mathbb{R}
νRandz(0,)\nu \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \left(0, \infty\right) Jν ⁣(z)RJ_{\nu}\!\left(z\right) \in \mathbb{R}
νZandzC\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} Jν ⁣(z)CJ_{\nu}\!\left(z\right) \in \mathbb{C}
νCandzC{0}\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} Jν ⁣(z)CJ_{\nu}\!\left(z\right) \in \mathbb{C}
ν[0,)andzC\nu \in \left[0, \infty\right) \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} Jν ⁣(z)CJ_{\nu}\!\left(z\right) \in \mathbb{C}
νZandzRandrZ0\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{R} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} Jν(r) ⁣(z)RJ^{(r)}_{\nu}\!\left(z\right) \in \mathbb{R}
νRandz(0,)andrZ0\nu \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \left(0, \infty\right) \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} Jν(r) ⁣(z)RJ^{(r)}_{\nu}\!\left(z\right) \in \mathbb{R}
νZandzCandrZ0\nu \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} Jν(r) ⁣(z)CJ^{(r)}_{\nu}\!\left(z\right) \in \mathbb{C}
νCandzC{0}andrZ0\nu \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} Jν(r) ⁣(z)CJ^{(r)}_{\nu}\!\left(z\right) \in \mathbb{C}
Table data: (P,Q)\left(P, Q\right) such that (P)    (Q)\left(P\right) \implies \left(Q\right)
Definitions:
Fungrim symbol Notation Short description
BesselJJν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
ZZZ\mathbb{Z} Integers
RRR\mathbb{R} Real numbers
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
ZZGreaterEqualZn\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.

2019-09-19 20:12:49.583742 UTC