Modified Bessel function of the second kind: Representations through equivalent functions (formula 03.04.27.0008)

BesselK

$Mathematica Notation$

Bessel-Type Functions

BesselK[nu,z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/03.04.27.0008.01

Input Form

BesselK[\[Nu], z] == (-(1/2)) Pi I^\[Nu] BesselY[\[Nu], I z] + (-1)^\[Nu] (Log[I z] - Log[z]) BesselI[\[Nu], z] /; Element[\[Nu], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[Pi]", " ", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]], "-", RowBox[List["Log", "[", "z", "]"]]]], ")"]], RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]

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> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> ⅈ </mi> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> ν </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BesselK </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <power /> <imaginaryi /> <ci> ν </ci> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> BesselI </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <in /> <ci> ν </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[Pi]", " ", SuperscriptBox["\[ImaginaryI]", "\[Nu]"], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]], "-", RowBox[List["Log", "[", "z", "]"]]]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]

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

2007-05-02