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 BesselJ

 http://functions.wolfram.com/03.01.21.0082.01

 Input Form

 Integrate[BesselJ[\[Mu], a t] BesselJ[\[Nu], b t], {t, 0, Infinity}] == Piecewise[{{(a^\[Mu] b^(-1 - \[Mu]) Gamma[(1/2) (1 + \[Mu] + \[Nu])] Hypergeometric2F1[(1/2) (1 + \[Mu] + \[Nu]), (1/2) (1 + \[Mu] - \[Nu]), 1 + \[Mu], a^2/b^2])/ (Gamma[1 + \[Mu]] Gamma[1 + (1/2) (-1 - \[Mu] + \[Nu])]), b > a}, {(a^(-1 - \[Nu]) b^\[Nu] Gamma[(1/2) (1 + \[Mu] + \[Nu])] Hypergeometric2F1[(1/2) (1 + \[Mu] + \[Nu]), (1/2) (1 - \[Mu] + \[Nu]), 1 + \[Nu], b^2/a^2])/ (Gamma[1 + (1/2) (-1 + \[Mu] - \[Nu])] Gamma[1 + \[Nu]]), a > b}}, Integrate[BesselJ[\[Mu], a t] BesselJ[\[Nu], b t], {t, 0, Infinity}]] /; a > 0 && b > 0 && Re[\[Mu] + \[Nu]] > -1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", RowBox[List["a", " ", "t"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["b", " ", "t"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "\[Mu]"], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "1"]], "-", "\[Mu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Mu]"]], ",", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "]"]]]], ")"]]]], ",", RowBox[List["b", ">", "a"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["b", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]]]], ",", RowBox[List["a", ">", "b"]]]], "}"]]]], "}"]], ",", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", RowBox[List["a", " ", "t"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["b", " ", "t"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "]"]]]], "/;", RowBox[List[RowBox[List["a", ">", "0"]], "\[And]", RowBox[List["b", ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "+", "\[Nu]"]], "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]

 MathML Form

 0 J μ ( a t ) J ν ( b t ) t a μ b - μ - 1 Γ ( 1 2 ( μ + ν + 1 ) ) Γ ( μ + 1 ) Γ ( 1 2 ( - μ + ν - 1 ) + 1 ) 2 F 1 ( 1 2 ( μ + ν + 1 ) , 1 2 ( μ - ν + 1 ) ; μ + 1 ; a 2 b 2 ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", "1"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Mu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]] b > a a - ν - 1 b ν Γ ( 1 2 ( μ + ν + 1 ) ) Γ ( 1 2 ( μ - ν - 1 ) + 1 ) Γ ( ν + 1 ) 2 F 1 ( 1 2 ( μ + ν + 1 ) , 1 2 ( - μ + ν + 1 ) ; ν + 1 ; b 2 a 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "1"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "+", "\[Nu]", "+", "1"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] a > b 0 J μ ( a t ) J ν ( b t ) t True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; a > 0 b > 0 Re ( μ + ν ) > - 1 Condition t 0 BesselJ μ a t BesselJ ν b t a μ b -1 μ -1 Gamma 1 2 μ ν 1 Gamma μ 1 Gamma 1 2 -1 μ ν -1 1 -1 HypergeometricPFQ 1 2 μ ν 1 1 2 μ -1 ν 1 μ 1 a 2 b 2 -1 b a a -1 ν -1 b ν Gamma 1 2 μ ν 1 Gamma 1 2 μ -1 ν -1 1 Gamma ν 1 -1 Hypergeometric2F1 1 2 μ ν 1 1 2 -1 μ ν 1 ν 1 b 2 a 2 -1 a b t 0 BesselJ μ a t BesselJ ν b t a 0 b 0 μ ν -1 [/itex]

 Rule Form

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

 2007-05-02