# Fungrim entry: ff93d0

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 entry

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

2021-03-15 19:12:00.328586 UTC