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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 K





http://functions.wolfram.com/03.21.26.0043.01









  


  








Input Form





BesselK[1/2 + \[Nu], z^(1/4)] SphericalBesselJ[\[Nu], z^(1/4)] == (1/(8 2^(1/4))) MeijerG[{{}, {}}, {{-(1/8), 3/8, (1 + 4 \[Nu])/8}, {-((3 + 4 \[Nu])/8)}}, z/64]








Standard Form





Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["8", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "8"]]], ",", FractionBox["3", "8"], ",", FractionBox[RowBox[List["1", "+", RowBox[List["4", "\[Nu]"]]]], "8"]]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox[RowBox[List["3", "+", RowBox[List["4", "\[Nu]"]]]], "8"]]], "}"]]]], "}"]], ",", FractionBox["z", "64"]]], "]"]]]]]]]]]]








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> <msub> <mi> I </mi> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mroot> <mi> z </mi> <mn> 4 </mn> </mroot> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mroot> <mi> z </mi> <mn> 4 </mn> </mroot> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mroot> <mn> 2 </mn> <mn> 4 </mn> </mroot> </mrow> </mfrac> <mo> &#8290; </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> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mn> 64 </mn> </mfrac> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 8 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </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;3&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[FractionBox[&quot;z&quot;, &quot;64&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;8&quot;]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;8&quot;], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;8&quot;], &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;4&quot;, &quot; &quot;, &quot;\[Nu]&quot;]], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, FractionBox[&quot;1&quot;, &quot;8&quot;]]], &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;4&quot;, &quot; &quot;, &quot;\[Nu]&quot;]], &quot;+&quot;, &quot;3&quot;]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <ci> BesselI </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <ci> SphericalBesselJ </ci> <ci> &#957; </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list /> <list /> </list> <list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 8 </cn> </apply> <cn type='rational'> 3 <sep /> 8 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </list> <list> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 8 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </list> </list> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 64 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselK", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]_"]], ",", SuperscriptBox["z_", RowBox[List["1", "/", "4"]]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", SuperscriptBox["z_", RowBox[List["1", "/", "4"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "8"]]], ",", FractionBox["3", "8"], ",", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "8"]]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "}"]]]], "}"]], ",", FractionBox["z", "64"]]], "]"]], RowBox[List["8", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]]]]]]]]]


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





2007-05-02