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Fungrim entry: ff93d0

Symbol: BesselK Kν ⁣(z)K_{\nu}\!\left(z\right) Modified Bessel function of the second kind
BesselK(nu, z), rendered as Kν ⁣(z)K_{\nu}\!\left(z\right), denotes the modified Bessel function of the second kind.
The input ν\nu is called the order. The input zz is called the argument.
Called with three arguments, BesselK(nu, z, r), rendered as Kν ⁣(z)K'_{\nu}\!\left(z\right), Kν ⁣(z)K''_{\nu}\!\left(z\right), Kν ⁣(z)K'''_{\nu}\!\left(z\right) ( 1r31 \le r \le 3 ), or Kν(r) ⁣(z)K^{(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 BesselK(nu, z) or BesselK(nu, z, r) is defined in Fungrim.
Domain Codomain
Numbers
νRandz(0,)\nu \in \mathbb{R} \,\mathbin{\operatorname{and}}\, z \in \left(0, \infty\right) Kν ⁣(z)RK_{\nu}\!\left(z\right) \in \mathbb{R}
νC{0}andzC\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} Kν ⁣(z)CK_{\nu}\!\left(z\right) \in \mathbb{C}
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} Kν(r) ⁣(z)RK^{(r)}_{\nu}\!\left(z\right) \in \mathbb{R}
νC{0}andzCandrZ0\nu \in \mathbb{C} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} Kν(r) ⁣(z)CK^{(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
BesselKKν ⁣(z)K_{\nu}\!\left(z\right) Modified Bessel function of the second kind
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("ff93d0"),
    SymbolDefinition(BesselK, BesselK(nu, z), "Modified Bessel function of the second kind"),
    Description(SourceForm(BesselK(nu, z)), ", rendered as", BesselK(nu, z), ", denotes the modified Bessel function of the second kind. "),
    Description("The input", nu, "is called the order. The input", z, "is called the argument."),
    Description("Called with three arguments, ", SourceForm(BesselK(nu, z, r)), ", rendered as", BesselK(nu, z, 1), ", ", BesselK(nu, z, 2), ", ", BesselK(nu, z, 3), " (", LessEqual(1, r, 3), "), or", BesselK(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(BesselK(nu, z)), "or", SourceForm(BesselK(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, RR), Element(z, OpenInterval(0, Infinity))), Element(BesselK(nu, z), RR)), Tuple(And(Element(nu, SetMinus(CC, Set(0))), Element(z, CC)), Element(BesselK(nu, z), CC)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity)), Element(r, ZZGreaterEqual(0))), Element(BesselK(nu, z, r), RR)), Tuple(And(Element(nu, SetMinus(CC, Set(0))), Element(z, CC), Element(r, ZZGreaterEqual(0))), Element(BesselK(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.

2019-10-05 13:11:19.856591 UTC