Fungrim symbol: BesselK

Symbol: BesselK —

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

— Modified Bessel function of the second kind

BesselK(nu, z), rendered as

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

, denotes the modified Bessel function of the second kind.

The input

\nu

is called the order. The input

z

is called the argument.

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

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

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

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

(

1 \le r \le 3

), or

K^{(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 BesselK(nu, z) or BesselK(nu, z, r) is defined in Fungrim.

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

Table data:

\left(P, Q\right)

such that

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

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second 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("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 symbol

The symbol BesselK appears in 5 topics:

Bessel functions (in 8 entries)
Specific values of Bessel functions (in 6 entries)
Recurrence relations for Bessel functions (in 5 entries)
Hypergeometric representations of Bessel functions (in 2 entries)
Euler's constant (in 1 entries)

Entries using this symbol

The symbol BesselK appears in 18 entries: