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 BesselY

 http://functions.wolfram.com/03.03.20.0021.01

 Input Form

 D[BesselY[\[Nu], z], {z, \[Alpha]}] == (1/Pi) (-1)^((Abs[\[Nu]] - \[Nu])/2) (2^(1 - Abs[\[Nu]]) z^(Abs[\[Nu]] - \[Alpha]) Sum[(((-1)^k FDLogConstant[z, 2 k + Abs[\[Nu]], \[Alpha]])/ (k! (k + Abs[\[Nu]])!)) (z/2)^(2 k), {k, 0, Infinity}] - 2^(1 - Abs[\[Nu]]) Log[2] z^(Abs[\[Nu]] - \[Alpha]) Sum[(((-1)^k Gamma[2 k + Abs[\[Nu]] + 1])/(k! (k + Abs[\[Nu]])! Gamma[2 k + Abs[\[Nu]] - \[Alpha] + 1])) (z/2)^(2 k), {k, 0, Infinity}] + (-1)^Abs[\[Nu]] 2^Abs[\[Nu]] z^(-Abs[\[Nu]] - \[Alpha]) Sum[((Abs[\[Nu]] - k - 1)!/(k! (-2 k + Abs[\[Nu]] - 1)! Gamma[2 k - Abs[\[Nu]] - \[Alpha] + 1])) (Log[z] - PolyGamma[2 k - Abs[\[Nu]] - \[Alpha] + 1] + PolyGamma[-2 k + Abs[\[Nu]]]) (z/2)^(2 k), {k, 0, Floor[(Abs[\[Nu]] - 1)/2]}] - 2^Abs[\[Nu]] z^(-Abs[\[Nu]] - \[Alpha]) Sum[(((Abs[\[Nu]] - k - 1)! Gamma[2 k - Abs[\[Nu]] + 1])/ (k! Gamma[2 k - Abs[\[Nu]] - \[Alpha] + 1])) (z/2)^(2 k), {k, Floor[(Abs[\[Nu]] - 1)/2] + 1, Abs[\[Nu]] - 1}] - (z^(Abs[\[Nu]] - \[Alpha]) Sum[(((-1)^k Gamma[2 k + Abs[\[Nu]] + 1])/ (k! (k + Abs[\[Nu]])! Gamma[2 k + Abs[\[Nu]] - \[Alpha] + 1])) (PolyGamma[k + 1] + PolyGamma[k + Abs[\[Nu]] + 1]) (z/2)^(2 k), {k, 0, Infinity}])/2^Abs[\[Nu]]) /; Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", "\[Pi]"], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Nu]"]], "2"], " "]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "\[Alpha]"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], RowBox[List["Log", "[", "2", "]"]], SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Abs", "[", "\[Nu]", "]"]]], SuperscriptBox["2", RowBox[List["Abs", "[", "\[Nu]", "]"]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", " ", FractionBox[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]], " "]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["Abs", "[", "\[Nu]", "]"]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List[RowBox[List["Floor", "[", " ", FractionBox[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], "2"], "]"]], "+", "1"]]]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]], " "]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]], "-", " ", RowBox[List[SuperscriptBox["2", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]]]]]]

 MathML Form

 α Y ν ( z ) z α 1 π ( - 1 ) "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - ν 2 ( - 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" z - α - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" k = "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 2 + 1 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - k - 1 ) ! Γ ( 2 k - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) k ! Γ ( 2 k - α - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ( z 2 ) 2 k + ( - 1 ) "\[LeftBracketingBar]" ν "\[RightBracketingBar]" 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" z - α - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 2 ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - k - 1 ) ! ( log ( z ) - ψ TagBox["\[Psi]", PolyGamma] ( 2 k - α - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k ) ) k ! ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k - 1 ) ! Γ ( 2 k - α - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ( z 2 ) 2 k + 2 1 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" z "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - α k = 0 ( - 1 ) k ℱ𝒞 log ( α ) ( z , 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) k ! ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ! ( z 2 ) 2 k - 2 1 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" log ( 2 ) z "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - α k = 0 ( - 1 ) k Γ ( 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) k ! ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ! Γ ( 2 k - α + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ( z 2 ) 2 k - 2 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" z "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - α k = 0 ( - 1 ) k Γ ( 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ) k ! ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ! Γ ( 2 k - α + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ( z 2 ) 2 k ) /; ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition z α BesselY ν z 1 -1 -1 ν -1 ν 2 -1 -1 2 ν z -1 α -1 ν k ν -1 2 -1 1 ν -1 ν -1 k -1 Gamma 2 k -1 ν 1 k Gamma 2 k -1 α -1 ν 1 -1 z 2 -1 2 k -1 ν 2 ν z -1 α -1 ν k 0 ν -1 2 -1 ν -1 k -1 z -1 PolyGamma 2 k -1 α -1 ν 1 PolyGamma ν -1 2 k k ν -1 2 k -1 Gamma 2 k -1 α -1 ν 1 -1 z 2 -1 2 k 2 1 -1 ν z ν -1 α k 0 -1 k Subscript ℱ𝒞 log α z 2 k ν k k ν -1 z 2 -1 2 k -1 2 1 -1 ν 2 z ν -1 α k 0 -1 k Gamma 2 k ν 1 k k ν Gamma 2 k -1 α ν 1 -1 z 2 -1 2 k -1 2 -1 ν z ν -1 α k 0 -1 k Gamma 2 k ν 1 PolyGamma k 1 PolyGamma k ν 1 k k ν Gamma 2 k -1 α ν 1 -1 z 2 -1 2 k ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Nu]"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["FDLogConstant", "[", 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"\[Nu]", "]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], ")"]]]], "]"]], "+", "1"]]]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", 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RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]]]], ")"]]]], "\[Pi]"], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]

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

 2007-05-02