Spherical Bessel function of the first kind: Representations through more general functions (formula 03.21.26.0041)

SphericalBesselJ

$Mathematica Notation$

Bessel-Type Functions

SphericalBesselJ[nu,z]

Representations through more general functions

Through Meijer G

Classical cases involving Bessel I

http://functions.wolfram.com/03.21.26.0041.01

Input Form

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

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselI", "[", 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["\[Pi]", RowBox[List["2", " ", 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[FractionBox["1", "8"], "+", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "8"]]], ",", FractionBox["3", "8"], ",", RowBox[List[RowBox[List["-", FractionBox["3", "8"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", 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> <mi> ν </mi> <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> ⁢ </mo> <mrow> <msub> <mi> j </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mroot> <mi> z </mi> <mn> 4 </mn> </mroot> <mo> ) </mo> </mrow> </mrow> <mo>  </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mroot> <mn> 2 </mn> <mn> 4 </mn> </mroot> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 4 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mn> 64 </mn> </mfrac> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> </mrow> <mo> , </mo> <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> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 8 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["0", ",", "4"]], RowBox[List["1", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox["z", "64"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "8"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "8"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["3", "8"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "-", FractionBox["3", "8"]]], 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> <times /> <apply> <ci> BesselI </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <ci> SphericalBesselJ </ci> <ci> ν </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </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> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 8 </cn> </apply> </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> <plus /> <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 /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 8 </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["BesselI", "[", 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["\[Pi]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "8"], "+", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "8"]]], ",", FractionBox["3", "8"], ",", RowBox[List[RowBox[List["-", FractionBox["3", "8"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", FractionBox["z", "64"]]], "]"]]]], RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]]]]]]]]]

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

2007-05-02