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BesselJ






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/03.01.21.0083.01









  


  










Input Form





Integrate[t^(\[Alpha] - 1) BesselJ[\[Lambda], a t] BesselJ[\[Mu], b t] BesselJ[\[Nu], c t], {t, 0, Infinity}] == 2^(\[Alpha] - 1) a^\[Lambda] b^\[Mu] c^(-\[Alpha] - \[Lambda] - \[Mu]) (Gamma[(\[Alpha] + \[Lambda] + \[Mu] + \[Nu])/2]/ (Gamma[\[Lambda] + 1] Gamma[\[Mu] + 1] Gamma[1 - (\[Alpha] + \[Lambda] + \[Mu] - \[Nu])/2])) HypergeometricPFQ[{{(\[Alpha] + \[Lambda] + \[Mu] + \[Nu])/2, (\[Alpha] + \[Lambda] + \[Mu] - \[Nu])/2}, {}, {}}, {{}, {\[Lambda] + 1}, {\[Mu] + 1}}, a^2/c^2, b^2/c^2] /; Element[a, Reals] && Element[b, Reals] && Element[c, Reals] && a > 0 && b > 0 && c > a + b && Re[\[Alpha] + \[Lambda] + \[Mu] + \[Nu]] > 0 && Re[\[Alpha]] < 5/2










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["\[Alpha]_", "-", "1"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Lambda]_", ",", RowBox[List["a_", " ", "t_"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]_", ",", RowBox[List["b_", " ", "t_"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["c_", " ", "t_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["a", "\[Lambda]"], " ", SuperscriptBox["b", "\[Mu]"], " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Lambda]", "-", "\[Mu]"]]], " ", 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["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["\[Lambda]", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["\[Mu]", "+", "1"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["c", "2"]], ",", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[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["c", ">", RowBox[List["a", "+", "b"]]]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Lambda]", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", FractionBox["5", "2"]]]]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.