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 BesselY

 http://functions.wolfram.com/03.03.21.0071.01

 Input Form

 Integrate[z^(\[Alpha] - 1) BesselY[-\[Nu], a z^r] BesselY[\[Nu], a z^r], z] == (1/2) z^\[Alpha] Csc[Pi \[Nu]] (-((1/(Pi \[Alpha] \[Nu])) ((3 + Cos[2 Pi \[Nu]]) HypergeometricPFQ[{1/2, \[Alpha]/(2 r)}, {1 + \[Alpha]/(2 r), 1 - \[Nu], 1 + \[Nu]}, (-a^2) z^(2 r)])) + (2^(1 - 2 \[Nu]) Cot[Pi \[Nu]] ((16^\[Nu] HypergeometricPFQ[{1/2 - \[Nu], \[Alpha]/(2 r) - \[Nu]}, {1 - 2 \[Nu], 1 - \[Nu], 1 + \[Alpha]/(2 r) - \[Nu]}, (-a^2) z^(2 r)])/((\[Alpha] - 2 r \[Nu]) Gamma[1 - \[Nu]]^2) + ((a z^r)^(4 \[Nu]) HypergeometricPFQ[{1/2 + \[Nu], \[Alpha]/(2 r) + \[Nu]}, {1 + \[Nu], 1 + \[Alpha]/(2 r) + \[Nu], 1 + 2 \[Nu]}, (-a^2) z^(2 r)])/((\[Alpha] + 2 r \[Nu]) Gamma[1 + \[Nu]]^2)))/(a z^r)^(2 \[Nu]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["-", "\[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", "2"], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List["\[Pi]", " ", "\[Alpha]", " ", "\[Nu]"]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], ")"]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]]], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["16", "\[Nu]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "-", RowBox[List["2", " ", "r", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], "2"]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["4", " ", "\[Nu]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]], ",", RowBox[List["1", "+", 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["\[Alpha]", "+", RowBox[List["2", " ", "r", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 Y - ν ( a z r ) Y ν ( a z r ) z 1 2 z α csc ( π ν ) ( 2 1 - 2 ν ( a z r ) - 2 ν cot ( π ν ) ( ( a z r ) 4 ν ( α + 2 r ν ) Γ ( ν + 1 ) 2 2 F 3 ( ν + 1 2 , α 2 r + ν ; ν + 1 , α 2 r + ν + 1 , 2 ν + 1 ; - 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["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], 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] + 16 ν ( α - 2 r ν ) Γ ( 1 - ν ) 2 2 F 3 ( 1 2 - ν , α 2 r - ν ; 1 - 2 ν , 1 - ν , α 2 r - ν + 1 ; - a 2 z 2 r ) 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[FractionBox["1", "2"], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", "\[Nu]", "+", "1"]], 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]] ) - ( cos ( 2 π ν ) + 3 ) π α ν 2 F 3 ( 1 2 , α 2 r ; α 2 r + 1 , 1 - ν , ν + 1 ; - 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["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], 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"]]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], 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] ) z z α -1 BesselY -1 ν a z r BesselY ν a z r 1 2 z α ν 2 1 -1 2 ν a z r -2 ν ν a z r 4 ν α 2 r ν Gamma ν 1 2 -1 HypergeometricPFQ ν 1 2 α 2 r -1 ν ν 1 α 2 r -1 ν 1 2 ν 1 -1 a 2 z 2 r 16 ν α -1 2 r ν Gamma 1 -1 ν 2 -1 HypergeometricPFQ 1 2 -1 ν α 2 r -1 -1 ν 1 -1 2 ν 1 -1 ν α 2 r -1 -1 ν 1 -1 a 2 z 2 r -1 2 ν 3 α ν -1 HypergeometricPFQ 1 2 α 2 r -1 α 2 r -1 1 1 -1 ν ν 1 -1 a 2 z 2 r [/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["BesselY", "[", RowBox[List[RowBox[List["-", "\[Nu]_"]], ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], RowBox[List["\[Pi]", " ", "\[Alpha]", " ", "\[Nu]"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]]], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["16", "\[Nu]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "-", RowBox[List["2", " ", "r", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], "2"]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["4", " ", "\[Nu]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["2", " ", "r", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"]]]]]], ")"]]]]]], ")"]]]]]]]]

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

 2001-10-29

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