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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





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MathML Form







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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





© 1998- Wolfram Research, Inc.