Bessel function of the first kind: Representations through more general functions (formula 03.01.26.0113)

BesselJ

$Mathematica Notation$

Bessel-Type Functions

BesselJ[nu,z]

Representations through more general functions

Through Meijer G

Classical cases involving Bessel Y

http://functions.wolfram.com/03.01.26.0113.01

Input Form

Cos[a Pi] BesselJ[\[Nu], Sqrt[z]] - Sin[a Pi] BesselY[\[Nu], Sqrt[z]] == MeijerG[{{}, {-(\[Nu]/2) + a}}, {{-(\[Nu]/2), \[Nu]/2}, {-(\[Nu]/2) + a}}, z/4]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["a", " ", "Pi"]], "]"]], RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", SqrtBox["z"]]], "]"]]]], "-", RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", "Pi"]], "]"]], RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", SqrtBox["z"]]], "]"]]]]]], "\[Equal]", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "+", "a"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "+", "a"]], "}"]]]], "}"]], ",", FractionBox["z", "4"]]], "]"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mtext> </mtext> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mtext> </mtext> <mrow> <mi> sin </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mn> 4 </mn> </mfrac> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mi> a </mi> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mi> a </mi> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["2", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox["z", "4"], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[RowBox[List["a", "-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["\[Nu]", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <ci> cos </ci> <apply> <times /> <ci> a </ci> <pi /> </apply> <apply> <ci> Subscript </ci> <ci> J </ci> <ci> ν </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> sin </ci> <apply> <times /> <ci> a </ci> <pi /> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> Y </ci> <ci> ν </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list /> <list> <apply> <plus /> <ci> a </ci> <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> </list> </list> <list> <list> <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> <apply> <plus /> <ci> a </ci> <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> </list> </list> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["a_", " ", "\[Pi]"]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", SqrtBox["z_"]]], "]"]]]], "-", RowBox[List[RowBox[List["Sin", "[", RowBox[List["a_", " ", "\[Pi]"]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", SqrtBox["z_"]]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "+", "a"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "+", "a"]], "}"]]]], "}"]], ",", FractionBox["z", "4"]]], "]"]]]]]]

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

2007-05-02