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Fungrim entry: 8ac81d

Symbol: BesselI Iν ⁣(z)I_{\nu}\!\left(z\right) Modified Bessel function of the first kind
BesselI(nu, z), rendered as Iν ⁣(z)I_{\nu}\!\left(z\right), denotes the modified 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, BesselI(nu, z, r), rendered as Iν ⁣(z)I'_{\nu}\!\left(z\right), Iν ⁣(z)I''_{\nu}\!\left(z\right), Iν ⁣(z)I'''_{\nu}\!\left(z\right) ( 1r31 \le r \le 3 ), or Iν(r) ⁣(z)I^{(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 BesselI(nu, z) or BesselI(nu, z, r) is defined in Fungrim.
Domain Codomain
Numbers
νZ  and  zR\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{R} Iν ⁣(z)RI_{\nu}\!\left(z\right) \in \mathbb{R}
νR  and  z(0,)\nu \in \mathbb{R} \;\mathbin{\operatorname{and}}\; z \in \left(0, \infty\right) Iν ⁣(z)RI_{\nu}\!\left(z\right) \in \mathbb{R}
νZ  and  zC\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} Iν ⁣(z)CI_{\nu}\!\left(z\right) \in \mathbb{C}
νC  and  zC{0}\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} Iν ⁣(z)CI_{\nu}\!\left(z\right) \in \mathbb{C}
νZ  and  zR  and  rZ0\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{R} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} Iν(r) ⁣(z)RI^{(r)}_{\nu}\!\left(z\right) \in \mathbb{R}
νR  and  z(0,)  and  rZ0\nu \in \mathbb{R} \;\mathbin{\operatorname{and}}\; z \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} Iν(r) ⁣(z)RI^{(r)}_{\nu}\!\left(z\right) \in \mathbb{R}
νZ  and  zC  and  rZ0\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} Iν(r) ⁣(z)CI^{(r)}_{\nu}\!\left(z\right) \in \mathbb{C}
νC  and  zC{0}  and  rZ0\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} Iν(r) ⁣(z)CI^{(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
BesselIIν ⁣(z)I_{\nu}\!\left(z\right) Modified 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
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("8ac81d"),
    SymbolDefinition(BesselI, BesselI(nu, z), "Modified Bessel function of the first kind"),
    Description(SourceForm(BesselI(nu, z)), ", rendered as", BesselI(nu, z), ", denotes the modified 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(BesselI(nu, z, r)), ", rendered as", BesselI(nu, z, 1), ", ", BesselI(nu, z, 2), ", ", BesselI(nu, z, 3), " (", LessEqual(1, r, 3), "), or", BesselI(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(BesselI(nu, z)), "or", SourceForm(BesselI(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(BesselI(nu, z), RR)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity))), Element(BesselI(nu, z), RR)), Tuple(And(Element(nu, ZZ), Element(z, CC)), Element(BesselI(nu, z), CC)), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0)))), Element(BesselI(nu, z), CC)), Tuple(And(Element(nu, ZZ), Element(z, RR), Element(r, ZZGreaterEqual(0))), Element(BesselI(nu, z, r), RR)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity)), Element(r, ZZGreaterEqual(0))), Element(BesselI(nu, z, r), RR)), Tuple(And(Element(nu, ZZ), Element(z, CC), Element(r, ZZGreaterEqual(0))), Element(BesselI(nu, z, r), CC)), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0))), Element(BesselI(nu, z, r), CC)))))

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

2020-04-08 16:14:44.404316 UTC