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 BesselY

 http://functions.wolfram.com/03.03.21.0106.01

 Input Form

 Integrate[z^(\[Alpha] - 1) BesselJ[1 + \[Nu], a z^r] BesselY[\[Nu], a z^r], z] == (1/(4 r Gamma[2 + \[Nu]])) (z^\[Alpha] Csc[Pi \[Nu]] ((2^(1 - 2 \[Nu]) r (a z^r)^(1 + 2 \[Nu]) Cos[Pi \[Nu]] HypergeometricPFQ[{3/2 + \[Nu], 1/2 + \[Alpha]/(2 r) + \[Nu]}, {2 + \[Nu], 3/2 + \[Alpha]/(2 r) + \[Nu], 2 + 2 \[Nu]}, (-a^2) z^(2 r)])/((r + \[Alpha] + 2 r \[Nu]) Gamma[1 + \[Nu]]) + (4 r Csc[Pi \[Nu]] (Pi (r + \[Alpha]) \[Nu] (1 + \[Nu]) BesselJ[-\[Nu], a z^r] BesselJ[1 + \[Nu], a z^r] - a r z^r HypergeometricPFQ[{3/2, 1/2 + \[Alpha]/(2 r)}, {3/2 + \[Alpha]/(2 r), 1 - \[Nu], 2 + \[Nu]}, (-a^2) z^(2 r)] Sin[Pi \[Nu]]))/((r - \[Alpha]) (r + \[Alpha]) Gamma[1 - \[Nu]])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", "r", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", "r", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]], ",", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["r", "+", "\[Alpha]", "+", RowBox[List["2", " ", "r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["4", " ", "r", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["r", "+", "\[Alpha]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "-", RowBox[List["a", " ", "r", " ", SuperscriptBox["z", "r"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["r", "-", "\[Alpha]"]], ")"]], " ", RowBox[List["(", RowBox[List["r", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 J ν + 1 ( a z r ) Y ν ( a z r ) z 1 4 r Γ ( ν + 2 ) ( z α csc ( π ν ) ( 2 1 - 2 ν r ( a z r ) 2 ν + 1 cos ( π ν ) ( 2 ν r + r + α ) Γ ( ν + 1 ) 2 F 3 ( ν + 3 2 , α 2 r + ν + 1 2 ; ν + 2 , α 2 r + ν + 3 2 , 2 ν + 2 ; - a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 ( r - α ) ( r + α ) Γ ( 1 - ν ) ( 4 r csc ( π ν ) ( π ( r + α ) ν ( ν + 1 ) J - ν ( a z r ) J ν + 1 ( a z r ) - a r z r 2 F 3 ( 3 2 , α 2 r + 1 2 ; α 2 r + 3 2 , 1 - ν , ν + 2 ; - a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] sin ( π ν ) ) ) ) ) z z α -1 BesselJ ν 1 a z r BesselY ν a z r 1 4 r Gamma ν 2 -1 z α ν 2 1 -1 2 ν r a z r 2 ν 1 ν 2 ν r r α Gamma ν 1 -1 HypergeometricPFQ ν 3 2 α 2 r -1 ν 1 2 ν 2 α 2 r -1 ν 3 2 2 ν 2 -1 a 2 z 2 r 1 r -1 α r α Gamma 1 -1 ν -1 4 r ν r α ν ν 1 BesselJ -1 ν a z r BesselJ ν 1 a z r -1 a r z r HypergeometricPFQ 3 2 α 2 r -1 1 2 α 2 r -1 3 2 1 -1 ν ν 2 -1 a 2 z 2 r ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]_"]], ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", "r", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]], ",", RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["r", "+", "\[Alpha]", "+", RowBox[List["2", " ", "r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["r", "+", "\[Alpha]"]], ")"]], " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "-", RowBox[List["a", " ", "r", " ", SuperscriptBox["z", "r"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["r", "-", "\[Alpha]"]], ")"]], " ", RowBox[List["(", RowBox[List["r", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]]]]], ")"]]]], RowBox[List["4", " ", "r", " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]]]]]]]]]

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

 2001-10-29