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 BesselY

 http://functions.wolfram.com/03.03.22.0005.01

 Input Form

 HankelTransform[BesselY[\[Nu], t], {t, \[Mu]}, z] == (-((Sqrt[2] z^(-1 - \[Nu]))/Pi)) (z^(2 \[Nu]) Gamma[(1/4) (3 + 2 \[Mu] - 2 \[Nu])] Gamma[\[Nu]] (Hypergeometric2F1[(1/4) (3 - 2 \[Mu] - 2 \[Nu]), (1/4) (3 + 2 \[Mu] - 2 \[Nu]), 1 - \[Nu], 1/z^2]/ Gamma[(1/4) (1 + 2 \[Mu] + 2 \[Nu])]) + Cos[Pi \[Nu]] Gamma[-\[Nu]] Gamma[(1/4) (3 + 2 \[Mu] + 2 \[Nu])] (Hypergeometric2F1[(1/4) (3 - 2 \[Mu] + 2 \[Nu]), (1/4) (3 + 2 \[Mu] + 2 \[Nu]), 1 + \[Nu], 1/z^2]/ Gamma[(1/4) (1 + 2 \[Mu] - 2 \[Nu])])) /; Re[\[Mu] - \[Nu]] > -(3/2) && Re[\[Mu] + \[Nu]] > -(3/2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HankelTransform", "[", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], ",", RowBox[List["{", RowBox[List["t", ",", "\[Mu]"]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " "]], "\[Pi]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]], " ", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]], "/", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]]]]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]], " ", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]], "/", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "-", "\[Nu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "+", "\[Nu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]]]]]]]]

 MathML Form

 t ; μ [ Y ν ( t ) ] ( z ) - 2 z - ν - 1 π ( z 2 ν Γ ( ν ) Γ ( 1 4 ( 2 μ + 2 ν + 1 ) ) Γ ( 1 4 ( 2 μ - 2 ν + 3 ) ) 2 F 1 ( 1 4 ( - 2 μ - 2 ν + 3 ) , 1 4 ( 2 μ - 2 ν + 3 ) ; 1 - ν ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + cos ( π ν ) Γ ( - ν ) Γ ( 1 4 ( 2 μ - 2 ν + 1 ) ) Γ ( 1 4 ( 2 μ + 2 ν + 3 ) ) 2 F 1 ( 1 4 ( - 2 μ + 2 ν + 3 ) , 1 4 ( 2 μ + 2 ν + 3 ) ; ν + 1 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) /; Re ( μ - ν ) > - 3 2 Re ( μ + ν ) > - 3 2 Condition Subscript CompoundExpression t μ BesselY ν t z -1 2 1 2 z -1 ν -1 -1 z 2 ν Gamma ν Gamma 1 4 2 μ 2 ν 1 -1 Gamma 1 4 2 μ -1 2 ν 3 Hypergeometric2F1 1 4 -2 μ -1 2 ν 3 1 4 2 μ -1 2 ν 3 1 -1 ν 1 z 2 -1 ν Gamma -1 ν Gamma 1 4 2 μ -1 2 ν 1 -1 Gamma 1 4 2 μ 2 ν 3 Hypergeometric2F1 1 4 -2 μ 2 ν 3 1 4 2 μ 2 ν 3 ν 1 1 z 2 -1 μ -1 ν -1 3 2 μ ν -1 3 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HankelTransform", "[", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], ",", RowBox[List["{", RowBox[List["t_", ",", "\[Mu]_"]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "]"]]]]], ")"]]]], "\[Pi]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "-", "\[Nu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Mu]", "+", "\[Nu]"]], "]"]], ">", RowBox[List["-", FractionBox["3", "2"]]]]]]]]]]]]]

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

 2001-10-29