Bessel functions

Symbol: BesselJ —

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

— Bessel function of the first kind

BesselJ(nu, z), rendered as

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

, denotes the Bessel function of the first kind.

The input

\nu

is called the order. The input

z

is called the argument.

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

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

,

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

,

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

(

1 \le r \le 3

), or

J^{(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 BesselJ(nu, z) or BesselJ(nu, z, r) is defined in Fungrim.

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

Table data:

\left(P, Q\right)

such that

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

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	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
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("b4165c"),
    SymbolDefinition(BesselJ, BesselJ(nu, z), "Bessel function of the first kind"),
    Description(SourceForm(BesselJ(nu, z)), ", rendered as", BesselJ(nu, z), ", denotes the 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(BesselJ(nu, z, r)), ", rendered as", BesselJ(nu, z, 1), ", ", BesselJ(nu, z, 2), ", ", BesselJ(nu, z, 3), " (", LessEqual(1, r, 3), "), or", BesselJ(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(BesselJ(nu, z)), "or", SourceForm(BesselJ(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(BesselJ(nu, z), RR)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity))), Element(BesselJ(nu, z), RR)), Tuple(And(Element(nu, ZZ), Element(z, CC)), Element(BesselJ(nu, z), CC)), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0)))), Element(BesselJ(nu, z), CC)), Tuple(And(Element(nu, ClosedOpenInterval(0, Infinity)), Element(z, CC)), Element(BesselJ(nu, z), CC)), Tuple(And(Element(nu, ZZ), Element(z, RR), Element(r, ZZGreaterEqual(0))), Element(BesselJ(nu, z, r), RR)), Tuple(And(Element(nu, RR), Element(z, OpenInterval(0, Infinity)), Element(r, ZZGreaterEqual(0))), Element(BesselJ(nu, z, r), RR)), Tuple(And(Element(nu, ZZ), Element(z, CC), Element(r, ZZGreaterEqual(0))), Element(BesselJ(nu, z, r), CC)), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))), Element(r, ZZGreaterEqual(0))), Element(BesselJ(nu, z, r), CC)))))

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)))))

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)))))

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)))))

ad9caa

{z}^{2} J''_{\nu}\!\left(z\right) + z J'_{\nu}\!\left(z\right) + \left({z}^{2} - {\nu}^{2}\right) J_{\nu}\!\left(z\right) = 0

Assumptions:

\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

{z}^{2} J''_{\nu}\!\left(z\right) + z J'_{\nu}\!\left(z\right) + \left({z}^{2} - {\nu}^{2}\right) J_{\nu}\!\left(z\right) = 0

\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ad9caa"),
    Formula(Equal(Add(Add(Mul(Pow(z, 2), BesselJ(nu, z, 2)), Mul(z, BesselJ(nu, z, 1))), Mul(Sub(Pow(z, 2), Pow(nu, 2)), BesselJ(nu, z))), 0)),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZ), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

62f23c

{z}^{2} Y''_{\nu}\!\left(z\right) + z Y'_{\nu}\!\left(z\right) + \left({z}^{2} - {\nu}^{2}\right) Y_{\nu}\!\left(z\right) = 0

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

{z}^{2} Y''_{\nu}\!\left(z\right) + z Y'_{\nu}\!\left(z\right) + \left({z}^{2} - {\nu}^{2}\right) Y_{\nu}\!\left(z\right) = 0

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("62f23c"),
    Formula(Equal(Add(Add(Mul(Pow(z, 2), BesselY(nu, z, 2)), Mul(z, BesselY(nu, z, 1))), Mul(Sub(Pow(z, 2), Pow(nu, 2)), BesselY(nu, z))), 0)),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

95e561

{z}^{2} I''_{\nu}\!\left(z\right) + z I'_{\nu}\!\left(z\right) - \left({z}^{2} + {\nu}^{2}\right) I_{\nu}\!\left(z\right) = 0

Assumptions:

\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

{z}^{2} I''_{\nu}\!\left(z\right) + z I'_{\nu}\!\left(z\right) - \left({z}^{2} + {\nu}^{2}\right) I_{\nu}\!\left(z\right) = 0

\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("95e561"),
    Formula(Equal(Sub(Add(Mul(Pow(z, 2), BesselI(nu, z, 2)), Mul(z, BesselI(nu, z, 1))), Mul(Add(Pow(z, 2), Pow(nu, 2)), BesselI(nu, z))), 0)),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZ), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

fd9add

{z}^{2} K''_{\nu}\!\left(z\right) + z K'_{\nu}\!\left(z\right) - \left({z}^{2} + {\nu}^{2}\right) K_{\nu}\!\left(z\right) = 0

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

{z}^{2} K''_{\nu}\!\left(z\right) + z K'_{\nu}\!\left(z\right) - \left({z}^{2} + {\nu}^{2}\right) K_{\nu}\!\left(z\right) = 0

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("fd9add"),
    Formula(Equal(Sub(Add(Mul(Pow(z, 2), BesselK(nu, z, 2)), Mul(z, BesselK(nu, z, 1))), Mul(Add(Pow(z, 2), Pow(nu, 2)), BesselK(nu, z))), 0)),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

d56914

J_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(J_{\nu - 1}\!\left(z\right) + J_{\nu + 1}\!\left(z\right)\right)

Assumptions:

\nu \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(J_{\nu - 1}\!\left(z\right) + J_{\nu + 1}\!\left(z\right)\right)

\nu \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d56914"),
    Formula(Equal(BesselJ(nu, z), Mul(Div(z, Mul(2, nu)), Add(BesselJ(Sub(nu, 1), z), BesselJ(Add(nu, 1), z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(ZZ, Set(0))), Element(z, CC)), And(Element(nu, SetMinus(CC, Set(0))), Element(z, SetMinus(CC, Set(0))))))

b6d600

Y_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(Y_{\nu - 1}\!\left(z\right) + Y_{\nu + 1}\!\left(z\right)\right)

Assumptions:

\nu \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(Y_{\nu - 1}\!\left(z\right) + Y_{\nu + 1}\!\left(z\right)\right)

\nu \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("b6d600"),
    Formula(Equal(BesselY(nu, z), Mul(Div(z, Mul(2, nu)), Add(BesselY(Sub(nu, 1), z), BesselY(Add(nu, 1), z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(ZZ, Set(0))), Element(z, CC)), And(Element(nu, SetMinus(CC, Set(0))), Element(z, SetMinus(CC, Set(0))))))

4fb391

I_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(I_{\nu - 1}\!\left(z\right) - I_{\nu + 1}\!\left(z\right)\right)

Assumptions:

\nu \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

I_{\nu}\!\left(z\right) = \frac{z}{2 \nu} \left(I_{\nu - 1}\!\left(z\right) - I_{\nu + 1}\!\left(z\right)\right)

\nu \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4fb391"),
    Formula(Equal(BesselI(nu, z), Mul(Div(z, Mul(2, nu)), Sub(BesselI(Sub(nu, 1), z), BesselI(Add(nu, 1), z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(ZZ, Set(0))), Element(z, CC)), And(Element(nu, SetMinus(CC, Set(0))), Element(z, SetMinus(CC, Set(0))))))

9d98f8

K_{\nu}\!\left(z\right) = -\frac{z}{2 \nu} \left(K_{\nu - 1}\!\left(z\right) - K_{\nu + 1}\!\left(z\right)\right)

Assumptions:

\nu \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{\nu}\!\left(z\right) = -\frac{z}{2 \nu} \left(K_{\nu - 1}\!\left(z\right) - K_{\nu + 1}\!\left(z\right)\right)

\nu \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9d98f8"),
    Formula(Equal(BesselK(nu, z), Neg(Mul(Div(z, Mul(2, nu)), Sub(BesselK(Sub(nu, 1), z), BesselK(Add(nu, 1), z)))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(ZZ, Set(0))), Element(z, CC)), And(Element(nu, SetMinus(CC, Set(0))), Element(z, SetMinus(CC, Set(0))))))

5aceb9

J'_{\nu}\!\left(z\right) = \frac{J_{\nu - 1}\!\left(z\right) - J_{\nu + 1}\!\left(z\right)}{2}

Assumptions:

\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J'_{\nu}\!\left(z\right) = \frac{J_{\nu - 1}\!\left(z\right) - J_{\nu + 1}\!\left(z\right)}{2}

\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("5aceb9"),
    Formula(Equal(BesselJ(nu, z, 1), Div(Sub(BesselJ(Sub(nu, 1), z), BesselJ(Add(nu, 1), z)), 2))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZ), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

40aeb6

Y'_{\nu}\!\left(z\right) = \frac{Y_{\nu - 1}\!\left(z\right) - Y_{\nu + 1}\!\left(z\right)}{2}

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y'_{\nu}\!\left(z\right) = \frac{Y_{\nu - 1}\!\left(z\right) - Y_{\nu + 1}\!\left(z\right)}{2}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("40aeb6"),
    Formula(Equal(BesselY(nu, z, 1), Div(Sub(BesselY(Sub(nu, 1), z), BesselY(Add(nu, 1), z)), 2))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

58d91f

I'_{\nu}\!\left(z\right) = \frac{I_{\nu - 1}\!\left(z\right) + I_{\nu + 1}\!\left(z\right)}{2}

Assumptions:

\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

I'_{\nu}\!\left(z\right) = \frac{I_{\nu - 1}\!\left(z\right) + I_{\nu + 1}\!\left(z\right)}{2}

\nu \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("58d91f"),
    Formula(Equal(BesselI(nu, z, 1), Div(Add(BesselI(Sub(nu, 1), z), BesselI(Add(nu, 1), z)), 2))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZ), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

a0ff0b

K'_{\nu}\!\left(z\right) = -\frac{K_{\nu - 1}\!\left(z\right) + K_{\nu + 1}\!\left(z\right)}{2}

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K'_{\nu}\!\left(z\right) = -\frac{K_{\nu - 1}\!\left(z\right) + K_{\nu + 1}\!\left(z\right)}{2}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a0ff0b"),
    Formula(Equal(BesselK(nu, z, 1), Neg(Div(Add(BesselK(Sub(nu, 1), z), BesselK(Add(nu, 1), z)), 2)))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

ecd36f

J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right)

Assumptions:

\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, -\frac{{z}^{2}}{4}\right)

\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("ecd36f"),
    Formula(Equal(BesselJ(nu, z), Mul(Pow(Div(z, 2), nu), Hypergeometric0F1Regularized(Add(nu, 1), Neg(Div(Pow(z, 2), 4)))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZGreaterEqual(0)), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

81eec6

I_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, \frac{{z}^{2}}{4}\right)

Assumptions:

\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Alternative assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

I_{\nu}\!\left(z\right) = {\left(\frac{z}{2}\right)}^{\nu} \,{}_0{\textbf F}_1\!\left(\nu + 1, \frac{{z}^{2}}{4}\right)

\nu \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("81eec6"),
    Formula(Equal(BesselI(nu, z), Mul(Pow(Div(z, 2), nu), Hypergeometric0F1Regularized(Add(nu, 1), Div(Pow(z, 2), 4))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ZZGreaterEqual(0)), Element(z, CC)), And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

7efe21

K_{\nu}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{-1 / 2} {e}^{-z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 z\right)

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{\nu}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{-1 / 2} {e}^{-z} U^{*}\!\left(\nu + \frac{1}{2}, 2 \nu + 1, 2 z\right)

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("7efe21"),
    Formula(Equal(BesselK(nu, z), Mul(Mul(Pow(Div(Mul(2, z), Pi), Neg(Div(1, 2))), Exp(Neg(z))), HypergeometricUStar(Add(nu, Div(1, 2)), Add(Mul(2, nu), 1), Mul(2, z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

54bce2

J_{-n}\!\left(z\right) = {\left(-1\right)}^{n} J_{n}\!\left(z\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

J_{-n}\!\left(z\right) = {\left(-1\right)}^{n} J_{n}\!\left(z\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("54bce2"),
    Formula(Equal(BesselJ(Neg(n), z), Mul(Pow(-1, n), BesselJ(n, z)))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, CC))))

afbd22

I_{-n}\!\left(z\right) = I_{n}\!\left(z\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

I_{-n}\!\left(z\right) = I_{n}\!\left(z\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("afbd22"),
    Formula(Equal(BesselI(Neg(n), z), BesselI(n, z))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, CC))))

86bc7d

I_{\nu}\!\left(z\right) = {z}^{\nu} {\left(i z\right)}^{-\nu} J_{\nu}\!\left(i z\right)

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(z \ne 0 \;\mathbin{\operatorname{or}}\; \nu = 0\right)

TeX:

I_{\nu}\!\left(z\right) = {z}^{\nu} {\left(i z\right)}^{-\nu} J_{\nu}\!\left(i z\right)

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(z \ne 0 \;\mathbin{\operatorname{or}}\; \nu = 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("86bc7d"),
    Formula(Equal(BesselI(nu, z), Mul(Mul(Pow(z, nu), Pow(Mul(ConstI, z), Neg(nu))), BesselJ(nu, Mul(ConstI, z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, CC), Or(NotEqual(z, 0), Equal(nu, 0)))))

15bbb1

I_{n}\!\left(z\right) = {i}^{-n} J_{n}\!\left(i z\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

I_{n}\!\left(z\right) = {i}^{-n} J_{n}\!\left(i z\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("15bbb1"),
    Formula(Equal(BesselI(n, z), Mul(Pow(ConstI, Neg(n)), BesselJ(n, Mul(ConstI, z))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, CC))))

2a4195

Y_{\nu}\!\left(z\right) = \frac{\cos\!\left(\pi \nu\right) J_{\nu}\!\left(z\right) - J_{-\nu}\!\left(z\right)}{\sin\!\left(\pi \nu\right)}

Assumptions:

\nu \in \mathbb{C} \setminus \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y_{\nu}\!\left(z\right) = \frac{\cos\!\left(\pi \nu\right) J_{\nu}\!\left(z\right) - J_{-\nu}\!\left(z\right)}{\sin\!\left(\pi \nu\right)}

\nu \in \mathbb{C} \setminus \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Sin`	$\sin(z)$	Sine
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("2a4195"),
    Formula(Equal(BesselY(nu, z), Div(Sub(Mul(Cos(Mul(Pi, nu)), BesselJ(nu, z)), BesselJ(Neg(nu), z)), Sin(Mul(Pi, nu))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, SetMinus(CC, ZZ)), Element(z, SetMinus(CC, Set(0))))))

d5b7e8

Y_{n}\!\left(z\right) = -\frac{2}{\pi} \left({i}^{n} K_{n}\!\left(i z\right) + \left(\log\!\left(i z\right) - \log(z)\right) J_{n}\!\left(z\right)\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

Y_{n}\!\left(z\right) = -\frac{2}{\pi} \left({i}^{n} K_{n}\!\left(i z\right) + \left(\log\!\left(i z\right) - \log(z)\right) J_{n}\!\left(z\right)\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Log`	$\log(z)$	Natural logarithm
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d5b7e8"),
    Formula(Equal(BesselY(n, z), Mul(Neg(Div(2, Pi)), Add(Mul(Pow(ConstI, n), BesselK(n, Mul(ConstI, z))), Mul(Sub(Log(Mul(ConstI, z)), Log(z)), BesselJ(n, z)))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, SetMinus(CC, Set(0))))))

121b21

J_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

J_{1 / 2}\!\left(z\right) = {\left(\frac{2 z}{\pi}\right)}^{1 / 2} \frac{\sin(z)}{z}

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Sin`	$\sin(z)$	Sine
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("121b21"),
    Formula(Equal(BesselJ(Div(1, 2), z), Mul(Pow(Div(Mul(2, z), Pi), Div(1, 2)), Div(Sin(z), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

d1f5c5

K_{1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

K_{1 / 2}\!\left(z\right) = {\left(\frac{\pi z}{2}\right)}^{1 / 2} \frac{{e}^{-z}}{z}

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d1f5c5"),
    Formula(Equal(BesselK(Div(1, 2), z), Mul(Pow(Div(Mul(Pi, z), 2), Div(1, 2)), Div(Exp(Neg(z)), z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

24d383

Symbol: HankelH1 —

H^{(1)}_{\nu}\!\left(z\right)

— Hankel function of the first kind

The following table lists all conditions such that HankelH1(nu, z) is defined in Fungrim.

Domain	Codomain
Numbers
$\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}$	$H^{(1)}_{\nu}\!\left(z\right) \in \mathbb{C}$

Table data:

\left(P, Q\right)

such that

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

Definitions:

Fungrim symbol	Notation	Short description
`HankelH1`	$H^{(1)}_{\nu}\!\left(z\right)$	Hankel function of the first kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("24d383"),
    SymbolDefinition(HankelH1, HankelH1(nu, z), "Hankel function of the first kind"),
    Description("The following table lists all conditions such that", SourceForm(HankelH1(nu, z)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0)))), Element(HankelH1(nu, z), CC)))))

d154dd

Symbol: HankelH2 —

H^{(2)}_{\nu}\!\left(z\right)

— Hankel function of the second kind

The following table lists all conditions such that HankelH2(nu, z) is defined in Fungrim.

Domain	Codomain
Numbers
$\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}$	$H^{(2)}_{\nu}\!\left(z\right) \in \mathbb{C}$

Table data:

\left(P, Q\right)

such that

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

Definitions:

Fungrim symbol	Notation	Short description
`HankelH2`	$H^{(2)}_{\nu}\!\left(z\right)$	Hankel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d154dd"),
    SymbolDefinition(HankelH2, HankelH2(nu, z), "Hankel function of the second kind"),
    Description("The following table lists all conditions such that", SourceForm(HankelH2(nu, z)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0)))), Element(HankelH2(nu, z), CC)))))

6a6a09

H^{(1)}_{\nu}\!\left(z\right) = J_{\nu}\!\left(z\right) + i Y_{\nu}\!\left(z\right)

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

H^{(1)}_{\nu}\!\left(z\right) = J_{\nu}\!\left(z\right) + i Y_{\nu}\!\left(z\right)

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`HankelH1`	$H^{(1)}_{\nu}\!\left(z\right)$	Hankel function of the first kind
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ConstI`	$i$	Imaginary unit
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("6a6a09"),
    Formula(Equal(HankelH1(nu, z), Add(BesselJ(nu, z), Mul(ConstI, BesselY(nu, z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

1dce21

H^{(2)}_{\nu}\!\left(z\right) = J_{\nu}\!\left(z\right) - i Y_{\nu}\!\left(z\right)

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

H^{(2)}_{\nu}\!\left(z\right) = J_{\nu}\!\left(z\right) - i Y_{\nu}\!\left(z\right)

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`HankelH2`	$H^{(2)}_{\nu}\!\left(z\right)$	Hankel function of the second kind
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ConstI`	$i$	Imaginary unit
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("1dce21"),
    Formula(Equal(HankelH2(nu, z), Sub(BesselJ(nu, z), Mul(ConstI, BesselY(nu, z))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, SetMinus(CC, Set(0))))))

99c077

J_{n}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(n t - z \sin(t)\right) \, dt

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

J_{n}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(n t - z \sin(t)\right) \, dt

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("99c077"),
    Formula(Equal(BesselJ(n, z), Mul(Div(1, Pi), Integral(Cos(Sub(Mul(n, t), Mul(z, Sin(t)))), For(t, 0, Pi))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, CC))))

cac83e

J_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(\nu t - z \sin(t)\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \sinh(t) - \nu t\right) \, dt

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

TeX:

J_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \cos\!\left(\nu t - z \sin(t)\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \sinh(t) - \nu t\right) \, dt

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

Definitions:

Fungrim symbol	Notation	Short description
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("cac83e"),
    Formula(Equal(BesselJ(nu, z), Sub(Mul(Div(1, Pi), Integral(Cos(Sub(Mul(nu, t), Mul(z, Sin(t)))), For(t, 0, Pi))), Mul(Div(Sin(Mul(Pi, nu)), Pi), Integral(Exp(Sub(Neg(Mul(z, Sinh(t))), Mul(nu, t))), For(t, 0, Infinity)))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, CC), Greater(Re(z), 0))))

7ae3ed

I_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \exp\!\left(z \cos(t)\right) \cos\!\left(\nu t\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \cosh(t) - \nu t\right) \, dt

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

TeX:

I_{\nu}\!\left(z\right) = \frac{1}{\pi} \int_{0}^{\pi} \exp\!\left(z \cos(t)\right) \cos\!\left(\nu t\right) \, dt - \frac{\sin\!\left(\pi \nu\right)}{\pi} \int_{0}^{\infty} \exp\!\left(-z \cosh(t) - \nu t\right) \, dt

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

Definitions:

Fungrim symbol	Notation	Short description
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Cos`	$\cos(z)$	Cosine
`Sin`	$\sin(z)$	Sine
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("7ae3ed"),
    Formula(Equal(BesselI(nu, z), Sub(Mul(Div(1, Pi), Integral(Mul(Exp(Mul(z, Cos(t))), Cos(Mul(nu, t))), For(t, 0, Pi))), Mul(Div(Sin(Mul(Pi, nu)), Pi), Integral(Exp(Sub(Neg(Mul(z, Cosh(t))), Mul(nu, t))), For(t, 0, Infinity)))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, CC), Greater(Re(z), 0))))

c29d6f

K_{\nu}\!\left(z\right) = \int_{0}^{\infty} \exp\!\left(-z \cosh(t)\right) \cosh\!\left(\nu t\right) \, dt

Assumptions:

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

TeX:

K_{\nu}\!\left(z\right) = \int_{0}^{\infty} \exp\!\left(-z \cosh(t)\right) \cosh\!\left(\nu t\right) \, dt

\nu \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0

Definitions:

Fungrim symbol	Notation	Short description
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("c29d6f"),
    Formula(Equal(BesselK(nu, z), Integral(Mul(Exp(Neg(Mul(z, Cosh(t)))), Cosh(Mul(nu, t))), For(t, 0, Infinity)))),
    Variables(nu, z),
    Assumptions(And(Element(nu, CC), Element(z, CC), Greater(Re(z), 0))))

3bffa9

\left|J_{\nu}\!\left(x\right)\right| \le 1

Assumptions:

\nu \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}

TeX:

\left|J_{\nu}\!\left(x\right)\right| \le 1

\nu \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("3bffa9"),
    Formula(LessEqual(Abs(BesselJ(nu, x)), 1)),
    Variables(nu, x),
    Assumptions(And(Element(nu, ClosedOpenInterval(0, Infinity)), Element(x, RR))))

e7b5be

\left|J_{\nu}\!\left(x\right)\right| \le 0.6749 {\nu}^{-1 / 3}

Assumptions:

\nu \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; x \in \left[0, \infty\right)

References:

L. Landau. Monotonicity and bounds on Bessel functions. Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory. Vol. 4. Southwest Texas State Univ. San Marcos, TX, 2000. http://emis.ams.org/journals/EJDE/conf-proc/04/l1/landau.pdf

TeX:

\left|J_{\nu}\!\left(x\right)\right| \le 0.6749 {\nu}^{-1 / 3}

\nu \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; x \in \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval

Source code for this entry:

Entry(ID("e7b5be"),
    Formula(LessEqual(Abs(BesselJ(nu, x)), Mul(Decimal("0.6749"), Pow(nu, Neg(Div(1, 3)))))),
    Variables(nu, x),
    Assumptions(And(Element(nu, OpenInterval(0, Infinity)), Element(x, ClosedOpenInterval(0, Infinity)))),
    References("L. Landau. Monotonicity and bounds on Bessel functions. Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory. Vol. 4. Southwest Texas State Univ. San Marcos, TX, 2000. http://emis.ams.org/journals/EJDE/conf-proc/04/l1/landau.pdf"))

7f3485

\left|J_{\nu}\!\left(x\right)\right| \le 0.7858 {x}^{-1 / 3}

Assumptions:

\nu \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; x \in \left(0, \infty\right)

References:

L. Landau. Monotonicity and bounds on Bessel functions. Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory. Vol. 4. Southwest Texas State Univ. San Marcos, TX, 2000. http://emis.ams.org/journals/EJDE/conf-proc/04/l1/landau.pdf

TeX:

\left|J_{\nu}\!\left(x\right)\right| \le 0.7858 {x}^{-1 / 3}

\nu \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; x \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Pow`	${a}^{b}$	Power
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("7f3485"),
    Formula(LessEqual(Abs(BesselJ(nu, x)), Mul(Decimal("0.7858"), Pow(x, Neg(Div(1, 3)))))),
    Variables(nu, x),
    Assumptions(And(Element(nu, ClosedOpenInterval(0, Infinity)), Element(x, OpenInterval(0, Infinity)))),
    References("L. Landau. Monotonicity and bounds on Bessel functions. Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory. Vol. 4. Southwest Texas State Univ. San Marcos, TX, 2000. http://emis.ams.org/journals/EJDE/conf-proc/04/l1/landau.pdf"))

0836b4

\left|J_{n}\!\left(z\right)\right| \le {e}^{\left|\operatorname{Im}(z)\right|}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\left|J_{n}\!\left(z\right)\right| \le {e}^{\left|\operatorname{Im}(z)\right|}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Exp`	${e}^{z}$	Exponential function
`Im`	$\operatorname{Im}(z)$	Imaginary part
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0836b4"),
    Formula(LessEqual(Abs(BesselJ(n, z)), Exp(Abs(Im(z))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZ), Element(z, CC))))

cc4572

\left|J_{\nu}\!\left(z\right)\right| \le \frac{1}{\Gamma\!\left(\nu + 1\right)} {\left|\frac{z}{2}\right|}^{\nu} {e}^{\left|\operatorname{Im}(z)\right|}

Assumptions:

\nu \in \left[-\frac{1}{2}, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

\left|J_{\nu}\!\left(z\right)\right| \le \frac{1}{\Gamma\!\left(\nu + 1\right)} {\left|\frac{z}{2}\right|}^{\nu} {e}^{\left|\operatorname{Im}(z)\right|}

\nu \in \left[-\frac{1}{2}, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`Gamma`	$\Gamma(z)$	Gamma function
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Im`	$\operatorname{Im}(z)$	Imaginary part
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("cc4572"),
    Formula(LessEqual(Abs(BesselJ(nu, z)), Mul(Mul(Div(1, Gamma(Add(nu, 1))), Pow(Abs(Div(z, 2)), nu)), Exp(Abs(Im(z)))))),
    Variables(nu, z),
    Assumptions(And(Element(nu, ClosedOpenInterval(Neg(Div(1, 2)), Infinity)), Element(z, SetMinus(CC, Set(0))))))

Differential equations

Derivatives and recurrence relations

Hypergeometric representations

Connection formulas

Specific values

Hankel functions

Integral representations

Bounds and inequalities