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Fungrim entry: b4165c

Symbol: BesselJ Jν ⁣(z)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
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}
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
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 entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC