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 BesselY

 http://functions.wolfram.com/03.03.06.0074.01

 Input Form

 BesselY[\[Nu], z] \[Proportional] (-((1 + I)/(E^((1/2) I (Pi \[Nu] + 2 z)) (2 Pi^(1/2) (-z)^(1/2))))) (E^(2 I z) ((2 Sqrt[-z^2])/z - I) - I (((-1)^\[Nu] Sqrt[-z^2])/z)) (1 - ((1 - 2 \[Nu]) (3 - 2 \[Nu]) (1 + 2 \[Nu]) (3 + 2 \[Nu]))/ (128 z^2) + (11025 - 51664 \[Nu]^2 + 31584 \[Nu]^4 - 5376 \[Nu]^6 + 256 \[Nu]^8)/(98304 z^4) + \[Ellipsis]) + (((1 - I) (1 - 4 \[Nu]^2))/(E^((1/2) I (Pi \[Nu] + 2 z)) (16 Pi^(1/2) (-z)^(3/2)))) (E^(2 I z) ((2 Sqrt[-z^2])/z - I) + I (-1)^\[Nu] (Sqrt[-z^2]/z)) (1 - ((3 - 2 \[Nu]) (5 - 2 \[Nu]) (3 + 2 \[Nu]) (5 + 2 \[Nu]))/ (384 z^2) + (893025 - 656784 \[Nu]^2 + 137824 \[Nu]^4 - 10496 \[Nu]^6 + 256 \[Nu]^8)/(491520 z^4) + \[Ellipsis]) /; (Abs[z] -> Infinity) && Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]]], RowBox[List["2", SuperscriptBox["\[Pi]", RowBox[List["1", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["1", "/", "2"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"], "-", "\[ImaginaryI]"]], " ", ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["11025", "-", RowBox[List["51664", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["31584", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["5376", " ", SuperscriptBox["\[Nu]", "6"]]], "+", RowBox[List["256", " ", SuperscriptBox["\[Nu]", "8"]]]]], RowBox[List["98304", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]]]], RowBox[List["16", SuperscriptBox["\[Pi]", RowBox[List["1", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", " ", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"], "-", "\[ImaginaryI]"]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["384", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["893025", "-", RowBox[List["656784", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["137824", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["10496", " ", SuperscriptBox["\[Nu]", "6"]]], "+", RowBox[List["256", " ", SuperscriptBox["\[Nu]", "8"]]]]], RowBox[List["491520", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]

 MathML Form

 Y ν ( z ) ( 1 - ) - 1 2 ( π ν + 2 z ) ( 1 - 4 ν 2 ) 16 π ( - z ) 3 / 2 ( 2 z ( 2 - z 2 z - ) + ( - 1 ) ν - z 2 z ) ( 1 - ( 3 - 2 ν ) ( 2 ν + 3 ) ( 2 ν + 5 ) ( 5 - 2 ν ) 384 z 2 + 256 ν 8 - 10496 ν 6 + 137824 ν 4 - 656784 ν 2 + 893025 491520 z 4 + ) - ( 1 + ) - 1 2 ( π ν + 2 z ) 2 π - z ( 2 z ( 2 - z 2 z - ) - ( - 1 ) ν - z 2 z ) ( 1 - ( 1 - 2 ν ) ( 2 ν + 1 ) ( 2 ν + 3 ) ( 3 - 2 ν ) 128 z 2 + 256 ν 8 - 5376 ν 6 + 31584 ν 4 - 51664 ν 2 + 11025 98304 z 4 + ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition Proportional BesselY ν z 1 -1 -1 1 2 ν 2 z 1 -1 4 ν 2 16 1 2 -1 z 3 2 -1 2 z 2 -1 z 2 1 2 z -1 -1 -1 ν -1 z 2 1 2 z -1 1 -1 3 -1 2 ν 2 ν 3 2 ν 5 5 -1 2 ν 384 z 2 -1 256 ν 8 -1 10496 ν 6 137824 ν 4 -1 656784 ν 2 893025 491520 z 4 -1 -1 1 -1 1 2 ν 2 z 2 1 2 -1 z 1 2 -1 2 z 2 -1 z 2 1 2 z -1 -1 -1 -1 ν -1 z 2 1 2 z -1 1 -1 1 -1 2 ν 2 ν 1 2 ν 3 3 -1 2 ν 128 z 2 -1 256 ν 8 -1 5376 ν 6 31584 ν 4 -1 51664 ν 2 11025 98304 z 4 -1 Rule z ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"], "-", "\[ImaginaryI]"]], ")"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], ")"]]]], "z"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["11025", "-", RowBox[List["51664", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["31584", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["5376", " ", SuperscriptBox["\[Nu]", "6"]]], "+", RowBox[List["256", " ", SuperscriptBox["\[Nu]", "8"]]]]], RowBox[List["98304", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"], "-", "\[ImaginaryI]"]], ")"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["384", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["893025", "-", RowBox[List["656784", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["137824", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["10496", " ", SuperscriptBox["\[Nu]", "6"]]], "+", RowBox[List["256", " ", SuperscriptBox["\[Nu]", "8"]]]]], RowBox[List["491520", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["16", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02