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variants of this functions
KelvinBei






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinBei[nu,z] > Representations through more general functions > Through Meijer G > Generalized cases involving Bessel I





http://functions.wolfram.com/03.17.26.0080.01









  


  










Input Form





BesselI[\[Nu], (-1)^(1/4) z] KelvinBei[\[Nu], z] == ((-(1/2)) (-1)^(3/4) Sqrt[Pi] z^\[Nu] (E^((3 I Pi \[Nu])/2) Csc[Pi (3/4 + \[Nu])] MeijerG[{{1/2}, {1/4}}, {{\[Nu]}, {0, 1/4, -\[Nu]}}, (-1)^(1/4) z, 1/2] - MeijerG[{{}, {}}, {{\[Nu]/2}, {0, 1/2, -(\[Nu]/2)}}, ((-1)^(1/4) z)/(2 Sqrt[2]), 1/4]))/(E^((1/4) I Pi (1 + 3 \[Nu])) ((-1)^(1/4) z)^\[Nu])










Standard Form





Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]], RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", FractionBox["1", "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "\[Nu]", "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "4"], ",", RowBox[List["-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ",", FractionBox["1", "2"]]], "]"]]]], "-", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], RowBox[List["2", " ", SqrtBox["2"]]]], ",", FractionBox["1", "4"]]], "]"]]]], ")"]]]]]]]]]]










MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mroot> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;0&quot;, &quot;,&quot;, &quot;4&quot;]], RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RadicalBox[RowBox[List[&quot;-&quot;, &quot;1&quot;]], &quot;4&quot;], &quot; 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</ci> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> KelvinBei </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <cn type='rational'> 1 <sep /> 2 </cn> </list> <list> <cn type='rational'> 1 <sep /> 4 </cn> </list> </list> <list> <list> <ci> &#957; </ci> </list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </list> </list> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> MeijerG </ci> <list> <list /> <list /> </list> <list> <list> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> </list> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <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> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z_"]]]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", FractionBox["1", "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "\[Nu]", "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "4"], ",", RowBox[List["-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ",", FractionBox["1", "2"]]], "]"]]]], "-", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], RowBox[List["2", " ", SqrtBox["2"]]]], ",", FractionBox["1", "4"]]], "]"]]]], ")"]]]]]]]]










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





2007-05-02





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