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/03.03.06.0076.01

 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)) (Sum[((Pochhammer[(1 - 2 \[Nu])/4, k] Pochhammer[(3 - 2 \[Nu])/4, k] Pochhammer[(1 + 2 \[Nu])/4, k] Pochhammer[(3 + 2 \[Nu])/4, k])/ (Pochhammer[1/2, k] k!)) (-(1/z^2))^k, {k, 0, n}] + O[1/z^(2 n + 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)) (Sum[((Pochhammer[(3 - 2 \[Nu])/4, k] Pochhammer[(5 - 2 \[Nu])/4, k] Pochhammer[(3 + 2 \[Nu])/4, k] Pochhammer[(5 + 2 \[Nu])/4, k])/ (Pochhammer[3/2, k] k!)) (-(1/z^2))^k, {k, 0, n}] + O[1/z^(2 n + 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["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["2", "\[Nu]"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "-", RowBox[List["2", "\[Nu]"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "+", RowBox[List["2", "\[Nu]"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "+", RowBox[List["2", "\[Nu]"]]]], "4"], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List[RowBox[List["2", "n"]], "+", "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["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "-", RowBox[List["2", "\[Nu]"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["5", "-", RowBox[List["2", "\[Nu]"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "+", RowBox[List["2", "\[Nu]"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["5", "+", RowBox[List["2", "\[Nu]"]]]], "4"], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List[RowBox[List["2", "n"]], "+", "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 ) ( k = 0 n ( 1 4 ( 1 - 2 ν ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 3 - 2 ν ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 2 ν + 1 ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 2 ν + 3 ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], ")"]], "k"], Pochhammer] ( - 1 z 2 ) k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! + O ( 1 z 2 n + 2 ) ) + ( 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 ) ( k = 0 n ( 1 4 ( 3 - 2 ν ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 5 - 2 ν ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 2 ν + 3 ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 2 ν + 5 ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "5"]], ")"]]]], ")"]], "k"], Pochhammer] ( - 1 z 2 ) k ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! + O ( 1 z 2 n + 2 ) ) /; ( "\[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 k 0 n Pochhammer 1 4 1 -1 2 ν k Pochhammer 1 4 3 -1 2 ν k Pochhammer 1 4 2 ν 1 k Pochhammer 1 4 2 ν 3 k -1 1 z 2 -1 k Pochhammer 1 2 k k -1 O 1 z 2 n 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 k 0 n Pochhammer 1 4 3 -1 2 ν k Pochhammer 1 4 5 -1 2 ν k Pochhammer 1 4 2 ν 3 k Pochhammer 1 4 2 ν 5 k -1 1 z 2 -1 k Pochhammer 3 2 k k -1 O 1 z 2 n 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["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List[RowBox[List["2", " ", "n"]], "+", "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"]]]]], 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")"]], 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

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