Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











SphericalBesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > SphericalBesselJ[nu,z] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/03.21.21.0066.01









  


  










Input Form





Integrate[t^(-1 + \[Alpha]) SphericalBesselJ[\[Lambda], a t] SphericalBesselJ[\[Mu], b t] SphericalBesselJ[\[Nu], c t], {t, 0, Infinity}] == (2^(-4 + \[Alpha]) a^\[Lambda] b^\[Mu] c^(-\[Alpha] - \[Lambda] - \[Mu]) Pi^(3/2) Gamma[(1/2) (\[Alpha] + \[Lambda] + \[Mu] + \[Nu])] HypergeometricPFQ[{{(1/2) (\[Alpha] + \[Lambda] + \[Mu] + \[Nu]), (1/2) (-1 + \[Alpha] + \[Lambda] + \[Mu] - \[Nu])}, {}, {}}, {{}, {3/2 + \[Lambda]}, {3/2 + \[Mu]}}, a^2/c^2, b^2/c^2])/ (Gamma[3/2 + \[Lambda]] Gamma[3/2 + \[Mu]] Gamma[(1/2) (3 - \[Alpha] - \[Lambda] - \[Mu] + \[Nu])]) /; Element[a, Reals] && Element[b, Reals] && Element[c, Reals] && a > 0 && b > 0 && Re[\[Alpha] + \[Lambda] + \[Mu] + \[Nu]] > 0 && a + b < c && Re[\[Alpha]] < 4










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Lambda]", ",", RowBox[List["a", " ", "t"]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Mu]", ",", RowBox[List["b", " ", "t"]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["c", " ", "t"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], "+", "\[Alpha]"]]], " ", SuperscriptBox["a", "\[Lambda]"], " ", SuperscriptBox["b", "\[Mu]"], " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Lambda]", "-", "\[Mu]"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "\[Lambda]"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["c", "2"]], ",", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "-", "\[Lambda]", "-", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["a", "\[Element]", "Reals"]], "&&", RowBox[List["b", "\[Element]", "Reals"]], "&&", RowBox[List["c", "\[Element]", "Reals"]], "&&", RowBox[List["a", ">", "0"]], "&&", RowBox[List["b", ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["a", "+", "b"]], "<", "c"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "4"]]]]]]]]










MathML Form







<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> <msup> <mi> t </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mi> &#955; </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> &#956; </mi> </msub> <mo> ( </mo> <mrow> <mi> b </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> c </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> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 4 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mi> &#955; </mi> </msup> <mo> &#8290; </mo> <msup> <mi> b </mi> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <msup> <mi> c </mi> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> - </mo> <mi> &#955; </mi> <mo> - </mo> <mi> &#956; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mi> &#955; </mi> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mi> &#955; </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> F </mi> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mi> &#955; </mi> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mi> &#955; </mi> <mo> + </mo> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ; </mo> <mo> ; </mo> </mrow> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> ; </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> </mrow> </mtd> </mtr> </mtable> <mo> &#8290; </mo> <mfrac> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> , </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> c </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mi> &#955; </mi> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> &lt; </mo> <mi> c </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 4 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> &#8747; </ms> <ms> 0 </ms> <ms> &#8734; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> t </ms> <apply> <ci> RowBox </ci> <list> <ms> &#945; </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> &#955; </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> t </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> &#956; </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> b </ms> <ms> t </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> &#957; </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> c </ms> <ms> t </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8518; </ms> <ms> t </ms> </list> </apply> </list> </apply> </list> </apply> <ms> &#63449; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> &#945; </ms> <ms> - </ms> <ms> 4 </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> a </ms> <ms> &#955; </ms> </apply> <apply> <ci> SuperscriptBox </ci> <ms> b </ms> <ms> &#956; </ms> </apply> <apply> <ci> SuperscriptBox </ci> <ms> c </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> &#945; </ms> </list> </apply> <ms> - </ms> <ms> &#955; </ms> <ms> - </ms> <ms> &#956; </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <ms> / </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> &#915; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#945; </ms> <ms> + </ms> <ms> &#955; </ms> <ms> + </ms> <ms> &#956; </ms> <ms> + </ms> <ms> &#957; </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#955; </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <ms> - </ms> <ms> &#945; </ms> <ms> - </ms> <ms> &#955; </ms> <ms> - </ms> <ms> &#956; </ms> <ms> + </ms> <ms> &#957; </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> , </ms> <ms> 1 </ms> <ms> , </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> , </ms> <ms> 0 </ms> <ms> , </ms> <ms> 0 </ms> </list> </apply> </apply> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#945; </ms> <ms> + </ms> <ms> &#955; </ms> <ms> + </ms> <ms> &#956; </ms> <ms> + </ms> <ms> &#957; </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#945; </ms> <ms> + </ms> <ms> &#955; </ms> <ms> + </ms> <ms> &#956; </ms> <ms> - </ms> <ms> &#957; </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ; </ms> <ms> ; </ms> </list> </apply> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#955; </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#956; </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ; </ms> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> a </ms> <ms> 2 </ms> </apply> <apply> <ci> SuperscriptBox </ci> <ms> c </ms> <ms> 2 </ms> </apply> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> <apply> <ci> SuperscriptBox </ci> <ms> c </ms> <ms> 2 </ms> </apply> </apply> </list> </apply> <ms> ] </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8477; </ms> <apply> <ci> Function </ci> <list /> <reals /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> b </ms> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8477; </ms> <apply> <ci> Function </ci> <list /> <reals /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> c </ms> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8477; </ms> <apply> <ci> Function </ci> <list /> <reals /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> &gt; </ms> <ms> 0 </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> b </ms> <ms> &gt; </ms> <ms> 0 </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#945; </ms> <ms> + </ms> <ms> &#955; </ms> <ms> + </ms> <ms> &#956; </ms> <ms> + </ms> <ms> &#957; </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &gt; </ms> <ms> 0 </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> b </ms> </list> </apply> <ms> &lt; </ms> <ms> c </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> &#945; </ms> <ms> ) </ms> </list> </apply> <ms> &lt; </ms> <ms> 4 </ms> </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[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]_"]]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Lambda]_", ",", RowBox[List["a_", " ", "t_"]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Mu]_", ",", RowBox[List["b_", " ", "t_"]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["c_", " ", "t_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], "+", "\[Alpha]"]]], " ", SuperscriptBox["a", "\[Lambda]"], " ", SuperscriptBox["b", "\[Mu]"], " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Lambda]", "-", "\[Mu]"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "\[Lambda]"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["c", "2"]], ",", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "-", "\[Lambda]", "-", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List["a", "\[Element]", "Reals"]], "&&", RowBox[List["b", "\[Element]", "Reals"]], "&&", RowBox[List["c", "\[Element]", "Reals"]], "&&", RowBox[List["a", ">", "0"]], "&&", RowBox[List["b", ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["a", "+", "b"]], "<", "c"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "4"]]]]]]]]]]










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





2007-05-02