Integrate[SphericalBesselJ[\[Nu], a t] SphericalBesselJ[\[Nu], b t], {t, 0, Infinity}] == (Pi a^(-1 - \[Nu]) b^\[Nu])/(2 + 4 \[Nu]) /; Re[\[Nu]] > -(1/2) && 0 < b < a

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "t"]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["b", " ", "t"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["b", "\[Nu]"]]], RowBox[List["2", "+", RowBox[List["4", " ", "\[Nu]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]], "&&", RowBox[List["0", "<", "b", "<", "a"]]]]]]]]

<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> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <mrow> <msub> <mi> j </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> b </mi> <mi> &#957; </mi> </msup> </mrow> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> b </mi> <mo> &lt; </mo> <mi> a </mi> </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> <times /> <apply> <ci> SphericalBesselJ </ci> <ci> &#957; </ci> <apply> <times /> <ci> a </ci> <ci> t </ci> </apply> </apply> <apply> <ci> SphericalBesselJ </ci> <ci> &#957; </ci> <apply> <times /> <ci> b </ci> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <ci> &#957; </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> b </ci> <ci> a </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", "t_"]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["b_", " ", "t_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["b", "\[Nu]"]]], RowBox[List["2", "+", RowBox[List["4", " ", "\[Nu]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", FractionBox["1", "2"]]]]], "&&", RowBox[List["0", "<", "b", "<", "a"]]]]]]]]]]

2007-05-02