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http://functions.wolfram.com/03.21.21.0060.01
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Integrate[SphericalBesselJ[\[Nu], t], {t, 0, Infinity}] ==
(Sqrt[Pi] Gamma[(1 + \[Nu])/2])/(2 Gamma[1 + \[Nu]/2]) /; Re[\[Nu]] > -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", "1"]]]]]]]]
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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> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msub> <mi> j </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo>  </mo> <mfrac> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <ci> SphericalBesselJ </ci> <ci> ν </ci> <ci> t </ci> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], "]"]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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