Kelvin function of the first kind: Representations through more general functions (formula 03.17.26.0068)

KelvinBei

$Mathematica Notation$

Bessel-Type Functions

KelvinBei[nu,z]

Representations through more general functions

Through Meijer G

Generalized cases involving ber

http://functions.wolfram.com/03.17.26.0068.01

Input Form

KelvinBer[\[Mu], z] KelvinBer[\[Nu], z] - KelvinBei[\[Nu], z] KelvinBei[\[Mu], z] == 2^(3/2) Pi^(5/2) MeijerG[{{0, 1/4, 3/4}, {1/6, 1/2, 5/6}}, {{(\[Mu] + \[Nu])/4, (\[Mu] + \[Nu] + 2)/4}, {1/6, 5/6, -((\[Mu] + \[Nu])/4), (\[Mu] - \[Nu])/4, (\[Nu] - \[Mu])/4, (2 - \[Mu] - \[Nu])/4, (\[Mu] - \[Nu] + 2)/4, (\[Nu] - \[Mu] + 2)/4}}, z/2, 1/4]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["1", "2"], ",", FractionBox["5", "6"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]", "+", "2"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["5", "6"], ",", RowBox[List["-", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "4"]]], ",", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "4"], ",", FractionBox[RowBox[List["2", "-", "\[Mu]", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]", "+", "2"]], "4"], ",", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]", "+", "2"]], "4"]]], "}"]]]], "}"]], ",", FractionBox["z", "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> <mrow> <msub> <mi> ber </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> ber </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <msub> <mi> bei </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> bei </mi> <mi> μ </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 6 </mn> <mo> , </mo> <mn> 10 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> ν </mi> <mo> - </mo> <mi> μ </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["6", ",", "10"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[FractionBox["z", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["3", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]", "+", "2"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "\[Mu]", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]", "+", "2"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "\[Mu]", "+", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <ci> KelvinBer </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <ci> KelvinBer </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> KelvinBei </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <ci> KelvinBei </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 5 <sep /> 2 </cn> </apply> <apply> <ci> MeijerG </ci> <list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 4 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </list> <list> <cn type='rational'> 1 <sep /> 6 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 5 <sep /> 6 </cn> </list> </list> <list> <list> <apply> <times /> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <cn type='rational'> 1 <sep /> 6 </cn> <cn type='rational'> 5 <sep /> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <ci> ν </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> </list> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <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["KelvinBer", "[", RowBox[List["\[Mu]_", ",", "z_"]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "-", RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List["\[Mu]_", ",", "z_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["1", "2"], ",", FractionBox["5", "6"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "4"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "2"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["5", "6"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "4"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]", "-", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", "2"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "2"]], ")"]]]]]], "}"]]]], "}"]], ",", FractionBox["z", "2"], ",", FractionBox["1", "4"]]], "]"]]]]]]]]

Contributed by

Brychkov Yu.A. (2006)

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

2007-05-02

KelvinBei[z]