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Fungrim entry: 5bb42e

Symbol: BesselY Yν ⁣(z)Y_{\nu}\!\left(z\right) Bessel function of the second kind
BesselJ(nu, z), rendered as Yν ⁣(z)Y_{\nu}\!\left(z\right), denotes the 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, BesselY(nu, z, r), rendered as Yν ⁣(z)Y'_{\nu}\!\left(z\right), Yν ⁣(z)Y''_{\nu}\!\left(z\right), Yν ⁣(z)Y'''_{\nu}\!\left(z\right) ( 1r31 \le r \le 3 ), or Yν(r) ⁣(z)Y^{(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 BesselY(nu, z) or BesselY(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) Yν ⁣(z)RY_{\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} Yν ⁣(z)CY_{\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} Yν(r) ⁣(z)RY^{(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} Yν(r) ⁣(z)CY^{(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
BesselYYν ⁣(z)Y_{\nu}\!\left(z\right) Bessel function of the second kind
BesselJJν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first 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("5bb42e"),
    SymbolDefinition(BesselY, BesselY(nu, z), "Bessel function of the second kind"),
    Description(SourceForm(BesselJ(nu, z)), ", rendered as", BesselY(nu, z), ", denotes the 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(BesselY(nu, z, r)), ", rendered as", BesselY(nu, z, 1), ", ", BesselY(nu, z, 2), ", ", BesselY(nu, z, 3), " (", LessEqual(1, r, 3), "), or", BesselY(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(BesselY(nu, z)), "or", SourceForm(BesselY(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(BesselY(nu, z), RR)), Tuple(And(Element(nu, SetMinus(CC, Set(0))), Element(z, CC)), Element(BesselY(nu, z), CC)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity)), Element(r, ZZGreaterEqual(0))), Element(BesselY(nu, z, r), RR)), Tuple(And(Element(nu, SetMinus(CC, Set(0))), Element(z, CC), Element(r, ZZGreaterEqual(0))), Element(BesselY(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.

2019-09-15 11:00:55.020619 UTC