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SphericalBesselJ




Mathematica Notation

Traditional Notation









Bessel-Type Functions > SphericalBesselJ[nu,z] > Representations through more general functions > Through Meijer G > Classical cases involving Bessel J,Y,y





http://functions.wolfram.com/03.21.26.0040.01









  


  








Input Form





BesselJ[\[Mu], Sqrt[z]] SphericalBesselJ[\[Nu], Sqrt[z]] + BesselY[\[Mu], Sqrt[z]] SphericalBesselY[\[Nu], Sqrt[z]] == (1/(Sqrt[2] Pi^2)) (Cos[Pi \[Mu]] MeijerG[{{-(1/4), 1/4}, {}}, {{-((\[Mu] + \[Nu] + 1)/2), (1/2) (-1 + \[Mu] - \[Nu]), (\[Mu] + \[Nu])/2}, {(1/2) (-\[Mu] + \[Nu])}}, z] - Sin[Pi \[Nu]] MeijerG[{{-(1/4), 1/4}, {}}, {{-((\[Mu] + \[Nu] + 1)/2), (1/2) (-\[Mu] + \[Nu]), (\[Mu] + \[Nu])/2}, {(1/2) (-1 + \[Mu] - \[Nu])}}, z])








Standard Form





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MathML Form







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</mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> &#956; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;4&quot;]], RowBox[List[&quot;3&quot;, &quot;,&quot;, &quot;2&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;4&quot;], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[&quot;\[Mu]&quot;, &quot;+&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; 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</ci> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </list> <list /> </list> <list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#956; </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> &#956; </ci> <ci> &#957; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; 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</ci> <ci> &#957; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </list> </list> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Mu]_", ",", SqrtBox["z_"]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", SqrtBox["z_"]]], "]"]]]], "+", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Mu]_", ",", SqrtBox["z_"]]], "]"]], " ", RowBox[List["SphericalBesselY", "[", RowBox[List["\[Nu]_", ",", SqrtBox["z_"]]], "]"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Mu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", FractionBox["1", "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]], ",", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "+", "\[Nu]"]], ")"]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], ",", FractionBox["1", "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "+", "\[Nu]"]], ")"]]]], ",", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox["\[Pi]", "2"]]]]]]]]


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





2007-05-02