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 BesselY

 http://functions.wolfram.com/03.03.20.0002.01

 Input Form

 Derivative[1, 0][BesselY][\[Nu], z] == -((BesselJ[\[Nu], z] (Cot[Pi \[Nu]] + Pi \[Nu] Csc[Pi \[Nu]]^2))/\[Nu]) - (BesselJ[\[Nu], z] Cot[Pi \[Nu]] + BesselJ[-\[Nu], z] Csc[Pi \[Nu]]) (Log[2] - Log[z] + PolyGamma[\[Nu]]) - ((2^(-2 + \[Nu]) z^(2 - \[Nu]) Csc[Pi \[Nu]])/ ((-1 + \[Nu]) Gamma[2 - \[Nu]])) HypergeometricPFQ[ {{}, {1}, {1, 1 - \[Nu]}}, {{2, 2 - \[Nu]}, {}, {2 - \[Nu]}}, -(z^2/4), -(z^2/4)] + ((2^(-2 - \[Nu]) z^(2 + \[Nu]) Cot[Pi \[Nu]])/ ((1 + \[Nu]) Gamma[2 + \[Nu]])) HypergeometricPFQ[ {{}, {1}, {1, 1 + \[Nu]}}, {{2, 2 + \[Nu]}, {}, {2 + \[Nu]}}, -(z^2/4), -(z^2/4)] /; !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "BesselY", "]"]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[Nu]", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"]]]]], ")"]]]], "\[Nu]"]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "z"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "\[Nu]", "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Nu]"]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "-", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Nu]"]]], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "+", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

 Y TagBox["Y", BesselY] ν ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) - 2 ν - 2 z 2 - ν csc ( π ν ) ( ν - 1 ) Γ ( 2 - ν ) F 2 0 1 0 1 2 ( ; 1 ; 1 , 1 - ν ; 2 , 2 - ν ; ; 2 - ν ; - z 2 4 , - z 2 4 ) + 2 - ν - 2 z ν + 2 cot ( π ν ) ( ν + 1 ) Γ ( ν + 2 ) F 2 0 1 0 1 2 ( ; 1 ; 1 , 1 + ν ; 2 , 2 + ν ; ; 2 + ν ; - z 2 4 , - z 2 4 ) - ( J ν ( z ) cot ( π ν ) + J - ν ( z ) csc ( π ν ) ) ( log ( 2 ) - log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( ν ) ) - π ν csc 2 ( π ν ) + cot ( π ν ) ν J ν ( z ) /; ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] FormBox RowBox RowBox RowBox SubsuperscriptBox TagBox Y BesselY ν TagBox RowBox ( RowBox 1 , 0 ) Derivative ( z ) RowBox RowBox RowBox - FractionBox RowBox SuperscriptBox 2 RowBox ν - 2 SuperscriptBox z RowBox 2 - ν RowBox csc ( RowBox π ν ) RowBox RowBox ( RowBox ν - 1 ) RowBox Γ ( RowBox 2 - ν ) RowBox SubsuperscriptBox F RowBox 2 0 1 RowBox 0 1 2 [ RowBox GridBox ErrorBox RowBox RowBox ; 1 ; 1 , RowBox RowBox 1 - ν ; RowBox 2 , RowBox RowBox RowBox 2 - ν ; ; RowBox 2 - ν ; - FractionBox SuperscriptBox z 2 4 , RowBox - FractionBox SuperscriptBox z 2 4 ] + RowBox FractionBox RowBox SuperscriptBox 2 RowBox RowBox - ν - 2 SuperscriptBox z RowBox ν + 2 RowBox cot ( RowBox π ν ) RowBox RowBox ( RowBox ν + 1 ) RowBox Γ ( RowBox ν + 2 ) RowBox SubsuperscriptBox F RowBox 2 0 1 RowBox 0 1 2 ( RowBox RowBox GridBox RowBox RowBox ; 1 ; 1 , RowBox RowBox 1 + ν ; RowBox 2 , RowBox RowBox RowBox 2 + ν ; ; RowBox 2 + ν ; - FractionBox SuperscriptBox z 2 4 , RowBox - FractionBox SuperscriptBox z 2 4 ) - RowBox RowBox ( RowBox RowBox RowBox SubscriptBox J ν ( z ) RowBox cot ( RowBox π ν ) + RowBox RowBox SubscriptBox J RowBox - ν ( z ) RowBox csc ( RowBox π ν ) ) RowBox ( RowBox RowBox log ( 2 ) - RowBox log ( z ) + RowBox TagBox ψ PolyGamma ( ν ) ) - RowBox FractionBox RowBox RowBox π ν RowBox SuperscriptBox csc 2 ( RowBox π ν ) + RowBox cot ( RowBox π ν ) ν RowBox SubscriptBox J ν ( z ) /; RowBox ν TagBox Function TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["BesselY", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[Nu]", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"]]]]], ")"]]]], "\[Nu]"]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "z"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "\[Nu]", "]"]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "-", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Nu]"]]], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "+", "\[Nu]"]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]

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

 2001-10-29