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/03.03.06.0061.01

 Input Form

 BesselY[\[Nu], z] \[Proportional] (Sqrt[2]/Sqrt[(-Pi) z]) Csc[Pi \[Nu]] (((-1)^\[Nu] Cos[\[Nu] Pi] Cos[z + (Pi (2 \[Nu] + 1))/4] - Cos[z + (Pi (-2 \[Nu] + 1))/4]/(-1)^\[Nu]) HypergeometricPFQ[ {(1 - 2 \[Nu])/4, (3 - 2 \[Nu])/4, (1 + 2 \[Nu])/4, (3 + 2 \[Nu])/4}, {1/2}, -(1/z^2)] + ((1 - 4 \[Nu]^2)/(8 z)) ((-1)^\[Nu] Cos[\[Nu] Pi] Sin[z + (Pi (2 \[Nu] + 1))/4] - Sin[z + (Pi (-2 \[Nu] + 1))/4]/(-1)^\[Nu]) HypergeometricPFQ[ {(3 - 2 \[Nu])/4, (5 - 2 \[Nu])/4, (3 + 2 \[Nu])/4, (5 + 2 \[Nu])/4}, {3/2}, -(1/z^2)]) /; Inequality[0, Less, Arg[z], LessEqual, Pi] && (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[SqrtBox["2"], SqrtBox[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "z"]]]], RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List["Cos", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], "4"]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "\[Nu]"]]], RowBox[List["Cos", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], "+", "1"]], ")"]]]], "4"]]], "]"]]]]]], ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["2", "\[Nu]"]]]], "4"], ",", FractionBox[RowBox[List["3", "-", RowBox[List["2", "\[Nu]"]]]], "4"], ",", FractionBox[RowBox[List["1", "+", RowBox[List["2", "\[Nu]"]]]], "4"], ",", FractionBox[RowBox[List["3", "+", RowBox[List["2", "\[Nu]"]]]], "4"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"], " "]]]], RowBox[List["8", " ", "z"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List["Sin", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], "4"]]], "]"]]]], " ", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "\[Nu]"]]], RowBox[List["Sin", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], "+", "1"]], ")"]]]], "4"]]], "]"]]]]]], " ", ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "-", RowBox[List["2", "\[Nu]"]]]], "4"], ",", FractionBox[RowBox[List["5", "-", RowBox[List["2", "\[Nu]"]]]], "4"], ",", FractionBox[RowBox[List["3", "+", RowBox[List["2", "\[Nu]"]]]], "4"], ",", FractionBox[RowBox[List["5", "+", RowBox[List["2", "\[Nu]"]]]], "4"]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

 MathML Form

 Y ν ( z ) 2 csc ( π ν ) - π z ( ( ( - 1 ) ν cos ( ν π ) cos ( π ( 2 ν + 1 ) 4 + z ) - ( - 1 ) - ν cos ( π ( 1 - 2 ν ) 4 + z ) ) 4 F 1 ( 1 - 2 ν 4 , 3 - 2 ν 4 , 2 ν + 1 4 , 2 ν + 3 4 ; 1 2 ; - 1 z 2 ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", RowBox[List["2", "\[Nu]"]]]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", "-", RowBox[List["2", "\[Nu]"]]]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "1"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "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]] + 1 - 4 ν 2 8 z ( ( - 1 ) ν cos ( ν π ) sin ( π ( 2 ν + 1 ) 4 + z ) - ( - 1 ) - ν sin ( π ( 1 - 2 ν ) 4 + z ) ) 4 F 1 ( 3 - 2 ν 4 , 5 - 2 ν 4 , 2 ν + 3 4 , 2 ν + 5 4 ; 3 2 ; - 1 z 2 ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["3", "-", RowBox[List["2", "\[Nu]"]]]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["5", "-", RowBox[List["2", "\[Nu]"]]]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "3"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "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]] ) /; 0 < arg ( z ) π ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional BesselY ν z 2 1 2 ν -1 z 1 2 -1 -1 ν ν 2 ν 1 4 -1 z -1 -1 -1 ν 1 -1 2 ν 4 -1 z HypergeometricPFQ 1 -1 2 ν 4 -1 3 -1 2 ν 4 -1 2 ν 1 4 -1 2 ν 3 4 -1 1 2 -1 1 z 2 -1 1 -1 4 ν 2 8 z -1 -1 ν ν 2 ν 1 4 -1 z -1 -1 -1 ν 1 -1 2 ν 4 -1 z HypergeometricPFQ 3 -1 2 ν 4 -1 5 -1 2 ν 4 -1 2 ν 3 4 -1 2 ν 5 4 -1 3 2 -1 1 z 2 -1 Inequality 0 z Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["z", "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Cos", "[", RowBox[List["z", "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], "+", "1"]], ")"]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["z", "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["z", "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]], "+", "1"]], ")"]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], RowBox[List["8", " ", "z"]]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "z"]]]], "/;", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]

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

 2007-05-02