Fungrim entry: 5bb42e

Symbol: BesselY —

Y_{\nu}\!\left(z\right)

— Bessel function of the second kind

BesselJ(nu, z), rendered as

Y_{\nu}\!\left(z\right)

, denotes the Bessel function of the second kind.

The input

\nu

is called the order. The input

z

is called the argument.

Called with three arguments, BesselY(nu, z, r), rendered as

Y'_{\nu}\!\left(z\right)

Y''_{\nu}\!\left(z\right)

Y'''_{\nu}\!\left(z\right)

(

1 \le r \le 3

), or

Y^{(r)}_{\nu}\!\left(z\right)

, represents the order

r

derivative of the Bessel function with respect to the argument

z

The following table lists conditions such that BesselY(nu, z) or BesselY(nu, z, r) is defined in Fungrim.

Domain	Codomain
Numbers
$\nu \in \mathbb{R} \;\mathbin{\operatorname{and}}\; z \in \left(0, \infty\right)$	$Y_{\nu}\!\left(z\right) \in \mathbb{R}$
$\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$	$Y_{\nu}\!\left(z\right) \in \mathbb{C}$
$\nu \in \mathbb{R} \;\mathbin{\operatorname{and}}\; z \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$	$Y^{(r)}_{\nu}\!\left(z\right) \in \mathbb{R}$
$\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$	$Y^{(r)}_{\nu}\!\left(z\right) \in \mathbb{C}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \;\implies\; \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`RR`	$\mathbb{R}$	Real numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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)))))

Topics using this entry

Bessel functions