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SphericalBesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > SphericalBesselJ[nu,z] > Integral representations > Contour integral representations





http://functions.wolfram.com/03.21.07.0009.01









  


  










Input Form





SphericalBesselJ[\[Nu], z] == (1/(2 Pi I)) Sqrt[Pi/2] z^\[Nu] (z^2)^(-(1/4) - \[Nu]/2) ContourIntegral[ Gamma[1/4 + \[Nu]/2 + s]/Gamma[5/4 + \[Nu]/2 - s]/(z^2/4)^s, \[ScriptCapitalL]["s"]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["1", " "]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], SqrtBox[FractionBox["\[Pi]", "2"]], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "-", FractionBox["\[Nu]", "2"]]]], " ", RowBox[List[SubscriptBox["\[Integral]", "\[ScriptCapitalL]"], RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"], "+", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "2"], "-", "s"]], "]"]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> j </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mo> &#8747; </mo> <mi> &#8466; </mi> </msub> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> s </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> &#957; </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#63449; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#960; </ms> <ms> &#8520; </ms> </list> </apply> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> FractionBox </ci> <ms> &#960; </ms> <ms> 2 </ms> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> &#957; </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> &#957; </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8747; </ms> </apply> <ms> &#8466; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> s </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> &#957; </ms> <ms> 2 </ms> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> &#957; </ms> <ms> 2 </ms> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 5 </ms> <ms> 4 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> 4 </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> s </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8518; </ms> <ms> s </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "-", FractionBox["\[Nu]", "2"]]]], " ", RowBox[List["ContourIntegral", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"], "+", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "2"], "-", "s"]], "]"]]], ",", RowBox[List["\[ScriptCapitalL]", "[", "\"s\"", "]"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]]]]










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





2007-05-02





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