Bessel function of the first kind: Representations through more general functions (formula 03.01.26.0076)

BesselJ

$Mathematica Notation$

Bessel-Type Functions

BesselJ[nu,z]

Representations through more general functions

Through Meijer G

Generalized cases involving Bessel K

http://functions.wolfram.com/03.01.26.0076.01

Input Form

(BesselJ[-\[Nu], z] - BesselJ[\[Nu], z]) BesselK[\[Nu], z] == (1/(2 Sqrt[Pi])) Sin[(Pi \[Nu])/2] MeijerG[{{}, {}}, {{0, -(\[Nu]/2), \[Nu]/2}, {1/2}}, z/(2 Sqrt[2]), 1/4]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "z"]], "]"]], "-", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], ")"]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]]]], "}"]], ",", FractionBox["z", RowBox[List["2", SqrtBox["2"]]]], ",", FractionBox["1", "4"]]], "]"]], " "]]]]]]

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> <mo> ( </mo> <mrow> <mrow> <msub> <mi> J </mi> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> sin </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 4 </mn> </mrow> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["0", ",", "4"]], RowBox[List["3", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[FractionBox["z", RowBox[List["2", " ", SqrtBox["2"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["\[Nu]", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> J </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> J </ci> <ci> ν </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> K </ci> <ci> ν </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <ci> sin </ci> <apply> <times /> <pi /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list /> <list /> </list> <list> <list> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> </list> </list> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]_"]], ",", "z_"]], "]"]], "-", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], ")"]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]]]], "}"]], ",", FractionBox["z", RowBox[List["2", " ", SqrtBox["2"]]]], ",", FractionBox["1", "4"]]], "]"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]]]]]]

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

2001-10-29