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 BesselY

 http://functions.wolfram.com/03.03.21.0024.01

 Input Form

 Integrate[z^(\[Alpha] - 1) E^(I a z) BesselY[\[Nu], a z], z] == (1/Pi) ((z^\[Alpha] ((1/(-\[Alpha] + \[Nu])) (4^\[Nu] Gamma[\[Nu]] HypergeometricPFQ[{1/2 - \[Nu], \[Alpha] - \[Nu]}, {1 - 2 \[Nu], 1 + \[Alpha] - \[Nu]}, 2 I a z]) - (1/(\[Alpha] + \[Nu])) ((a z)^(2 \[Nu]) Cos[Pi \[Nu]] Gamma[-\[Nu]] HypergeometricPFQ[{1/2 + \[Nu], \[Alpha] + \[Nu]}, {1 + \[Alpha] + \[Nu], 1 + 2 \[Nu]}, 2 I a z])))/ (2^\[Nu] (a z)^\[Nu]))

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29