Fungrim entry: 8ac81d

Symbol: BesselI —

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

— Modified Bessel function of the first kind

BesselI(nu, z), rendered as

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

, denotes the modified Bessel function of the first kind.

The input

\nu

is called the order. The input

z

is called the argument.

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

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

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

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

(

1 \le r \le 3

), or

I^{(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 BesselI(nu, z) or BesselI(nu, z, r) is defined in Fungrim.

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

Table data:

\left(P, Q\right)

such that

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

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`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("8ac81d"),
    SymbolDefinition(BesselI, BesselI(nu, z), "Modified Bessel function of the first kind"),
    Description(SourceForm(BesselI(nu, z)), ", rendered as", BesselI(nu, z), ", denotes the modified Bessel function of the first kind. "),
    Description("The input", nu, "is called the order. The input", z, "is called the argument."),
    Description("Called with three arguments, ", SourceForm(BesselI(nu, z, r)), ", rendered as", BesselI(nu, z, 1), ", ", BesselI(nu, z, 2), ", ", BesselI(nu, z, 3), " (", LessEqual(1, r, 3), "), or", BesselI(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(BesselI(nu, z)), "or", SourceForm(BesselI(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, ZZ), Element(z, RR)), Element(BesselI(nu, z), RR)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity))), Element(BesselI(nu, z), RR)), Tuple(And(Element(nu, ZZ), Element(z, CC)), Element(BesselI(nu, z), CC)), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0)))), Element(BesselI(nu, z), CC)), Tuple(And(Element(nu, ZZ), Element(z, RR), Element(r, ZZGreaterEqual(0))), Element(BesselI(nu, z, r), RR)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity)), Element(r, ZZGreaterEqual(0))), Element(BesselI(nu, z, r), RR)), Tuple(And(Element(nu, ZZ), Element(z, CC), Element(r, ZZGreaterEqual(0))), Element(BesselI(nu, z, r), CC)), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0))), Element(BesselI(nu, z, r), CC)))))

Topics using this entry

Bessel functions