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 BesselY

 http://functions.wolfram.com/03.03.03.0010.01

 Input Form

 BesselY[\[Nu], z] == (-(((-1)^((1/2) (Sign[\[Nu]] + 1) (Abs[\[Nu]] - 4/3)) 2^(Abs[\[Nu]] - 2) Gamma[-(1/3)])/(z^Abs[\[Nu]] (3^(5/6) Gamma[1 - Abs[\[Nu]]])))) (2^(1/3) 3^(1/6) z^(2/3) (Sqrt[3] AiryAiPrime[(-(3/2)^(2/3)) z^(2/3)] + Sign[\[Nu]] 3 AiryBiPrime[(-(3/2)^(2/3)) z^(2/3)]) Sum[((Abs[\[Nu]] - k - 4/3)!/(k! (Abs[\[Nu]] - 2 k - 4/3)! Pochhammer[4/3, k] Pochhammer[1 - Abs[\[Nu]], k])) (z^2/4)^k, {k, 0, Abs[\[Nu]] - 4/3}] + 2 (AiryAi[(-(3/2)^(2/3)) z^(2/3)] + Sqrt[3] Sign[\[Nu]] AiryBi[(-(3/2)^(2/3)) z^(2/3)]) Sum[((Abs[\[Nu]] - k - 1/3)!/(k! (Abs[\[Nu]] - 2 k - 1/3)! Pochhammer[1/3, k] Pochhammer[1 - Abs[\[Nu]], k])) (z^2/4)^k, {k, 0, Abs[\[Nu]] - 1/3}]) /; Element[Abs[\[Nu]] - 1/3, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Sign", "[", "\[Nu]", "]"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["4", "3"]]], ")"]]]]], SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "2"]]], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["1", "3"]]], "]"]], " "]], RowBox[List[SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryAiPrime", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", "3", " ", RowBox[List["AiryBiPrime", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["4", "3"]]]], RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["4", "3"]]], ")"]], "!"]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["4", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"]]]]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["AiryAi", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], "+", RowBox[List[SqrtBox["3"], " ", RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", RowBox[List["AiryBi", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["1", "3"]]]], " ", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["1", "3"]]], ")"]], "!"]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["1", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]], " ", ")"]]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"]]]]]]]]], ")"]]]]]], "/;", RowBox[List["Element", "[", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["1", "3"]]], ",", "Integers"]], "]"]]]]]]

 MathML Form

 Y ν ( z ) - ( - 1 ) 1 2 ( sgn ( ν ) + 1 ) ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 4 3 ) 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 z - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" Γ ( - 1 3 ) 3 5 / 6 Γ ( 1 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ( 2 3 3 6 ( 3 Ai ( - ( 3 2 ) 2 / 3 z 2 / 3 ) + 3 Bi ( - ( 3 2 ) 2 / 3 z 2 / 3 ) sgn ( ν ) ) ( k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 4 3 ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - k - 4 3 ) ! k ! ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k - 4 3 ) ! ( 4 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["4", "3"], ")"]], "k"], Pochhammer] ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ( z 2 4 ) k ) z 2 / 3 + 2 ( Ai ( - ( 3 2 ) 2 / 3 z 2 / 3 ) + 3 Bi ( - ( 3 2 ) 2 / 3 z 2 / 3 ) sgn ( ν ) ) k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 3 ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - k - 1 3 ) ! k ! ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k - 1 3 ) ! ( 1 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "3"], ")"]], "k"], Pochhammer] ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ( z 2 4 ) k ) /; "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 3 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition BesselY ν z -1 -1 1 2 Sign ν 1 ν -1 4 3 2 ν -2 z -1 ν Gamma -1 1 3 3 5 6 Gamma 1 -1 ν -1 2 1 3 3 1 6 3 1 2 AiryAiPrime -1 3 2 2 3 z 2 3 3 AiryBiPrime -1 3 2 2 3 z 2 3 Sign ν k 0 ν -1 4 3 ν -1 k -1 4 3 k ν -1 2 k -1 4 3 Pochhammer 4 3 k Pochhammer 1 -1 ν k -1 z 2 4 -1 k z 2 3 2 AiryAi -1 3 2 2 3 z 2 3 3 1 2 AiryBi -1 3 2 2 3 z 2 3 Sign ν k 0 ν -1 1 3 ν -1 k -1 1 3 k ν -1 2 k -1 1 3 Pochhammer 1 3 k Pochhammer 1 -1 ν k -1 z 2 4 -1 k ν -1 1 3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Sign", "[", "\[Nu]", "]"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["4", "3"]]], ")"]]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "2"]]], " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["1", "3"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryAiPrime", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", "3", " ", RowBox[List["AiryBiPrime", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["4", "3"]]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["4", "3"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["4", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["AiryAi", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], "+", RowBox[List[SqrtBox["3"], " ", RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", RowBox[List["AiryBi", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["1", "3"]]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["1", "3"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["1", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["1", "3"]]], "\[Element]", "Integers"]]]]]]]]

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

 2001-10-29