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 BesselY

 http://functions.wolfram.com/03.03.20.0012.02

 Input Form

 D[BesselY[\[Nu], z], {z, n}] == 2^(n - 2 \[Nu]) Sqrt[Pi] z^(-n - \[Nu]) Csc[Pi \[Nu]] (z^(2 \[Nu]) Cos[Pi \[Nu]] Gamma[1 + \[Nu]] HypergeometricPFQRegularized[{(\[Nu] + 1)/2, (\[Nu] + 2)/2}, {(1 + \[Nu] - n)/2, (2 + \[Nu] - n)/2, 1 + \[Nu]}, -(z^2/4)] - 16^\[Nu] Gamma[1 - \[Nu]] HypergeometricPFQRegularized[ {(1 - \[Nu])/2, (2 - \[Nu])/2}, {(1 - \[Nu] - n)/2, (2 - \[Nu] - n)/2, 1 - \[Nu]}, -(z^2/4)]) /; !Element[\[Nu], Integers] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["n", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ",", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]", "-", "n"]], "2"], " ", ",", FractionBox[RowBox[List["2", "+", "\[Nu]", "-", "n"]], "2"], " ", ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["16", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]", "-", "n"]], "2"], " ", ",", FractionBox[RowBox[List["2", "-", "\[Nu]", "-", "n"]], "2"], ",", RowBox[List["1", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n Y ν ( z ) z n 2 n - 2 ν π z - n - ν csc ( π ν ) ( z 2 ν cos ( π ν ) Γ ( ν + 1 ) 2 F ~ 3 ( ν + 1 2 , ν + 2 2 ; ν - n + 1 2 , ν - n + 2 2 , ν + 1 ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "-", "n", "+", "1"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "-", "n", "+", "2"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] - 16 ν Γ ( 1 - ν ) 2 F ~ 3 ( 1 - ν 2 , 2 - ν 2 ; 1 - n - ν 2 , 2 - n - ν 2 , 1 - ν ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "n", "-", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "n", "-", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) /; ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition z n BesselY ν z 2 n -1 2 ν 1 2 z -1 n -1 ν ν z 2 ν ν Gamma ν 1 HypergeometricPFQRegularized ν 1 2 -1 ν 2 2 -1 ν -1 n 1 2 -1 ν -1 n 2 2 -1 ν 1 -1 z 2 4 -1 -1 16 ν Gamma 1 -1 ν HypergeometricPFQRegularized 1 -1 ν 2 -1 2 -1 ν 2 -1 1 -1 n -1 ν 2 -1 2 -1 n -1 ν 2 -1 1 -1 ν -1 z 2 4 -1 ν n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["n", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ",", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]", "-", "n"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]", "-", "n"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["16", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Nu]", "-", "n"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Nu]", "-", "n"]], ")"]]]], ",", RowBox[List["1", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29