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http://functions.wolfram.com/03.21.23.0007.01
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Sum[SphericalBesselJ[k - 1/2, x]^2, {k, 1, Infinity}] ==
Pi/(4 x) - (1/2) SphericalBesselJ[-(1/2), x]^2
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], " ", SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["k", "-", FractionBox["1", "2"]]], ",", "x"]], "]"]], "2"]]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", RowBox[List["4", "x"]]], "-", RowBox[List[FractionBox["1", "2"], SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "x"]], "]"]], "2"]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <msup> <mrow> <msub> <mi> j </mi> <mrow> <mi> k </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo>  </mo> <mrow> <mfrac> <mi> π </mi> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> x </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <msub> <mi> j </mi> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <power /> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> x </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <ci> SphericalBesselJ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["k_", "-", FractionBox["1", "2"]]], ",", "x_"]], "]"]], "2"]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["\[Pi]", RowBox[List["4", " ", "x"]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "x"]], "]"]], "2"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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