html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 BesselY

 http://functions.wolfram.com/03.03.06.0028.01

 Input Form

 BesselY[\[Nu], z] \[Proportional] BesselY[\[Nu], x]/E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) + 4 I BesselJ[\[Nu], x] Cos[Pi \[Nu]]^2 Floor[Arg[-x + z]/(2 Pi)] - (1/2) ((-BesselY[-1 + \[Nu], x] + BesselY[1 + \[Nu], x])/ E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) - 4 I (BesselJ[-1 + \[Nu], x] - BesselJ[1 + \[Nu], x]) Cos[Pi \[Nu]]^2 Floor[Arg[-x + z]/(2 Pi)]) (z - x) + (1/8) ((BesselY[-2 + \[Nu], x] - 2 BesselY[\[Nu], x] + BesselY[2 + \[Nu], x])/E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) + 4 I (BesselJ[-2 + \[Nu], x] - 2 BesselJ[\[Nu], x] + BesselJ[2 + \[Nu], x]) Cos[Pi \[Nu]]^2 Floor[Arg[-x + z]/(2 Pi)]) (z - x)^2 + \[Ellipsis] /; (z -> x) && Element[x, Reals] && x < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]]]], "+", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "-", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

 Y ν ( z ) - 2 π ν arg ( z - x ) 2 π Y ν ( x ) + 4 J ν ( x ) cos 2 ( π ν ) arg ( z - x ) 2 π - 1 2 ( - 2 π ν arg ( z - x ) 2 π ( Y ν + 1 ( x ) - Y ν - 1 ( x ) ) - 4 ( J ν - 1 ( x ) - J ν + 1 ( x ) ) cos 2 ( π ν ) arg ( z - x ) 2 π ) ( z - x ) + 1 8 ( 4 ( J ν - 2 ( x ) - 2 J ν ( x ) + J ν + 2 ( x ) ) arg ( z - x ) 2 π cos 2 ( π ν ) + - 2 π ν arg ( z - x ) 2 π ( Y ν - 2 ( x ) - 2 Y ν ( x ) + Y ν + 2 ( x ) ) ) ( z - x ) 2 + /; ( z "\[Rule]" x ) x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x < 0 Condition Proportional BesselY ν z -2 ν z -1 x 2 -1 BesselY ν x 4 BesselJ ν x ν 2 z -1 x 2 -1 -1 1 2 -2 ν z -1 x 2 -1 BesselY ν 1 x -1 BesselY ν -1 x -1 4 BesselJ ν -1 x -1 BesselJ ν 1 x ν 2 z -1 x 2 -1 z -1 x 1 8 4 BesselJ ν -2 x -1 2 BesselJ ν x BesselJ ν 2 x z -1 x 2 -1 ν 2 -2 ν z -1 x 2 -1 BesselY ν -2 x -1 2 BesselY ν x BesselY ν 2 x z -1 x 2 Rule z x x x 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]]]], "+", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "-", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]

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

 2007-05-02