Modified Bessel function of the second kind: Differential equations (formula 03.04.13.0016)

BesselK

$Mathematica Notation$

Bessel-Type Functions

BesselK[nu,z]

Differential equations

Ordinary linear differential equations and wronskians

Involving related functions

http://functions.wolfram.com/03.04.13.0016.01

Input Form

Nest[z D[#1, z] & , w[z], 4] - 2 (\[Nu]^2 + \[Mu]^2) Nest[z D[#1, z] & , w[z], 2] + (\[Nu]^2 - \[Mu]^2)^2 w[z] - 4 z^2 (2 w[z] + 3 z Derivative[1][w][z] + Nest[z D[#1, z] & , w[z], 2]) == 0 /; w[z] == Subscript[c, 1] BesselI[\[Nu], z] BesselI[\[Mu], z] + Subscript[c, 2] BesselI[\[Nu], z] BesselK[\[Mu], z] + Subscript[c, 3] BesselK[\[Nu], z] BesselI[\[Mu], z] + Subscript[c, 4] BesselK[\[Nu], z] BesselK[\[Mu], z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Nest", "[", RowBox[List[RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "#1"]]]], "&"]], ",", RowBox[List["w", "[", "z", "]"]], ",", "4"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Nu]", "2"], "+", SuperscriptBox["\[Mu]", "2"]]], ")"]], " ", RowBox[List["Nest", "[", RowBox[List[RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "#1"]]]], "&"]], ",", RowBox[List["w", "[", "z", "]"]], ",", "2"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[Nu]", "2"], "-", SuperscriptBox["\[Mu]", "2"]]], ")"]], "2"], " ", RowBox[List["w", "[", "z", "]"]]]], "-", RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["w", "[", "z", "]"]]]], "+", RowBox[List["3", " ", "z", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List["Nest", "[", RowBox[List[RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "#1"]]]], "&"]], ",", RowBox[List["w", "[", "z", "]"]], ",", "2"]], "]"]]]], ")"]]]]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "3"], RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "4"], RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </munderover> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </munderover> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </munderover> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 4 </mn> </msub> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <cn type='integer'> 4 </cn> </uplimit> <apply> <times /> <ci> z </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <cn type='integer'> 2 </cn> </uplimit> <apply> <times /> <ci> z </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <cn type='integer'> 2 </cn> </uplimit> <apply> <times /> <ci> z </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> BesselI </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <ci> BesselI </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> BesselI </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <ci> BesselK </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> BesselI </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <ci> BesselK </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 4 </cn> </apply> <apply> <ci> BesselK </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <ci> BesselK </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Nest", "[", RowBox[List[RowBox[List[RowBox[List["z_", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], "#1"]]]], "&"]], ",", RowBox[List["w", "[", "z_", "]"]], ",", "4"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Nu]_", "2"], "+", SuperscriptBox["\[Mu]_", "2"]]], ")"]], " ", RowBox[List["Nest", "[", RowBox[List[RowBox[List[RowBox[List["z_", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], "#1"]]]], "&"]], ",", RowBox[List["w", "[", "z_", "]"]], ",", "2"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["\[Nu]_", "2"], "-", SuperscriptBox["\[Mu]_", "2"]]], ")"]], "2"], " ", RowBox[List["w", "[", "z_", "]"]]]], "-", RowBox[List["4", " ", SuperscriptBox["z_", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["w", "[", "z_", "]"]]]], "+", RowBox[List["3", " ", "z_", " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List["Nest", "[", RowBox[List[RowBox[List[RowBox[List["z_", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], "#1"]]]], "&"]], ",", RowBox[List["w", "[", "z_", "]"]], ",", "2"]], "]"]]]], ")"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "3"], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "4"], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]]]]]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29