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/03.03.06.0077.01

 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)) HypergeometricPFQ[{1/4 - \[Nu]/2, 3/4 - \[Nu]/2, 1/4 + \[Nu]/2, 3/4 + \[Nu]/2}, {1/2}, -(1/z^2)] + (((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)) HypergeometricPFQ[ {3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 3/4 + \[Nu]/2, 5/4 + \[Nu]/2}, {3/2}, -(1/z^2)] /; (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["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], "+", 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["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]]]]]], "/;", 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 ) 2 π - z ( 2 z ( 2 - z 2 z - ) - ( - 1 ) ν - z 2 z ) 4 F 1 ( 1 4 - ν 2 , 3 4 - ν 2 , ν 2 + 1 4 , ν 2 + 3 4 ; 1 2 ; - 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + ( 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 ) 4 F 1 ( 3 4 - ν 2 , 5 4 - ν 2 , ν 2 + 3 4 , ν 2 + 5 4 ; 3 2 ; - 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition Proportional BesselY ν z -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 HypergeometricPFQ 1 4 -1 ν 2 -1 3 4 -1 ν 2 -1 ν 2 -1 1 4 ν 2 -1 3 4 1 2 -1 1 z 2 -1 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 HypergeometricPFQ 3 4 -1 ν 2 -1 5 4 -1 ν 2 -1 ν 2 -1 3 4 ν 2 -1 5 4 3 2 -1 1 z 2 -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["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], 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["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], 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