SphericalBesselJ[\[Nu], z] == 2^(-1 - \[Nu]) Pi^(3/2) z^\[Nu] Sum[Residue[(1/((-(z^2/4))^s (Gamma[1/2 + s] Gamma[1/2 - s] Gamma[3/2 + \[Nu] - s]))) Gamma[s], {s, -j}], {j, 0, Infinity}]

Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], RowBox[List["-", "s"]]], " "]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]", "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", "s", "]"]]]], " ", ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]]]]]]]]

<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> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SphericalBesselJ </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <ci> &#957; </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], RowBox[List["-", "s"]]], " ", RowBox[List["Gamma", "[", "s", "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]", "-", "s"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", "j"]]]], "}"]]]], "]"]]]]]]]]]]

2007-05-02