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 BesselJ

 http://functions.wolfram.com/03.01.21.0081.01

 Input Form

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

 Standard Form

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

 MathML Form

 0 t α - 1 J μ ( a t ) J ν ( b t ) t 2 α - 1 b - α - μ a μ Γ ( 1 2 ( α + μ + ν ) ) Γ ( 1 2 ( - α - μ + ν ) + 1 ) Γ ( μ + 1 ) 2 F 1 ( 1 2 ( α + μ + ν ) , 1 2 ( α + μ - ν ) ; μ + 1 ; a 2 b 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["\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]", "-", "\[Nu]"]], ")"]]]], 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]], Hypergeometric2F1] b > a Re ( α ) < 2 2 α - 1 a - α - ν b ν Γ ( 1 2 ( α + μ + ν ) ) Γ ( 1 2 ( - α + μ - ν ) + 1 ) Γ ( ν + 1 ) 2 F 1 ( 1 2 ( α + μ + ν ) , 1 2 ( α - μ + ν ) ; 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["\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "-", "\[Mu]", "+", "\[Nu]"]], ")"]]]], 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 Re ( α ) < 2 2 α - 1 a - α Γ ( 1 2 ( α + μ + ν ) ) Γ ( 1 - α ) Γ ( 1 2 ( - α + μ - ν ) + 1 ) Γ ( 1 2 ( - α - μ + ν ) + 1 ) Γ ( 1 2 ( - α + μ + ν ) + 1 ) b a Re ( α ) < 1 0 t α - 1 J μ ( a t ) J ν ( b t ) t True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; a > 0 b > 0 Re ( α + μ + ν ) > 0 Condition t 0 t α -1 BesselJ μ a t BesselJ ν b t 2 α -1 b -1 α -1 μ a μ Gamma 1 2 α μ ν Gamma 1 2 -1 α -1 μ ν 1 Gamma μ 1 -1 Hypergeometric2F1 1 2 α μ ν 1 2 α μ -1 ν μ 1 a 2 b 2 -1 b a α 2 2 α -1 a -1 α -1 ν b ν Gamma 1 2 α μ ν Gamma 1 2 -1 α μ -1 ν 1 Gamma ν 1 -1 Hypergeometric2F1 1 2 α μ ν 1 2 α -1 μ ν ν 1 b 2 a 2 -1 a b α 2 2 α -1 a -1 α Gamma 1 2 α μ ν Gamma 1 -1 α Gamma 1 2 -1 α μ -1 ν 1 Gamma 1 2 -1 α -1 μ ν 1 Gamma 1 2 -1 α μ ν 1 -1 b a α 1 t 0 t α -1 BesselJ μ a t BesselJ ν b t a 0 b 0 α μ ν 0 [/itex]

 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["\[Mu]_", ",", RowBox[List["a_", " ", "t_"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["b_", " ", "t_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Alpha]"]]], " ", SuperscriptBox["a", "\[Mu]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", "\[Alpha]"]], ")"]]]], ",", RowBox[List["\[Mu]", "+", "1"]], ",", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "-", "\[Alpha]"]], ")"]]]], "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Mu]", "+", "1"]], "]"]]]]], RowBox[List[RowBox[List["b", ">", "a"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "2"]]]]], List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "\[Alpha]"]]], " ", SuperscriptBox["b", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "\[Alpha]"]], ")"]]]], ",", RowBox[List["\[Nu]", "+", "1"]], ",", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "-", "\[Alpha]"]], ")"]]]], "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]]], RowBox[List[RowBox[List["a", ">", "b"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "2"]]]]], List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["a", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "-", "\[Alpha]"]], ")"]]]], "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", "\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "]"]]]]], RowBox[List[RowBox[List["b", "\[Equal]", "a"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", "1"]]]]], List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", RowBox[List["a", " ", "t"]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["b", " ", "t"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List[RowBox[List["a", ">", "0"]], "&&", RowBox[List["b", ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Alpha]", "+", "\[Mu]", "+", "\[Nu]"]], "]"]], ">", "0"]]]]]]]]]]

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

 2007-05-02