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/03.03.20.0016.01

 Input Form

 D[BesselY[\[Nu], z], {z, \[Alpha]}] == ((2^(\[Alpha] - 2 \[Nu] + 1) z^(-\[Alpha] + \[Nu]))/Sqrt[Pi]) Gamma[1 + \[Nu]] Log[z/2] HypergeometricPFQRegularized[ {1/2 + \[Nu]/2, 1 + \[Nu]/2}, {1/2 - \[Alpha]/2 + \[Nu]/2, 1 - \[Alpha]/2 + \[Nu]/2, 1 + \[Nu]}, -(z^2/4)] - (1/(z^\[Alpha] Pi)) Sum[(((-1)^k (2 k + \[Nu])!)/(k! (k + \[Nu])! Gamma[1 + 2 k - \[Alpha] + \[Nu]])) (z/2)^(2 k + \[Nu]) (PolyGamma[1 + k] + PolyGamma[1 + k + \[Nu]] - 2 PolyGamma[1 + 2 k + \[Nu]] + 2 PolyGamma[1 + 2 k - \[Alpha] + \[Nu]]), {k, 0, Infinity}] - Sum[((\[Nu] - k - 1)! FDPowerConstant[z, 2 k - \[Nu], \[Alpha]] z^(2 k - \[Nu] - \[Alpha]))/(Pi 2^(2 k - \[Nu]) k!), {k, 0, Floor[(\[Nu] - 1)/2]}] - (1/(z^\[Alpha] Pi)) Sum[(((\[Nu] - k - 1)! (2 k - \[Nu])!)/ (k! Gamma[2 k - \[Nu] + 1 - \[Alpha]])) (z/2)^(2 k - \[Nu]), {k, Floor[(\[Nu] + 1)/2], \[Nu] - 1}] /; Element[\[Nu], Integers] && \[Nu] >= 0

 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[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", "\[Nu]"]]], " "]], SqrtBox["\[Pi]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], "-", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], "\[Pi]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Nu]"]], ")"]], "!"]]]], RowBox[List[" ", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "\[Nu]"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]], "]"]]]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", "k"]], "+", "\[Nu]"]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", "\[Nu]"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", "\[Nu]"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]], "]"]]]]]], ")"]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Nu]", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "k", "-", "1"]], ")"]], "!"]], RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]"]], ",", "\[Alpha]"]], "]"]], SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]", "-", "\[Alpha]"]]]]], RowBox[List["\[Pi]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]"]]], " ", RowBox[List["k", "!"]]]]]]], "-", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], "\[Pi]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], "]"]]]], RowBox[List["\[Nu]", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]"]], ")"]], "!"]], " "]], RowBox[List[" ", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]", "+", "1", "-", "\[Alpha]"]], "]"]]]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", "k"]], "-", "\[Nu]"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 α Y ν ( z ) z α 2 α - 2 ν + 1 Γ ( ν + 1 ) z ν - α π log ( z 2 ) 2 F ~ 3 ( ν 2 + 1 2 , ν 2 + 1 ; - α 2 + ν 2 + 1 2 , - α 2 + ν 2 + 1 , ν + 1 ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Alpha]", "2"]]], "+", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Alpha]", "2"]]], "+", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] - z - α π k = ν + 1 2 ν - 1 ( ν - k - 1 ) ! ( 2 k - ν ) ! k ! Γ ( 2 k - α - ν + 1 ) ( z 2 ) 2 k - ν - k = 0 ν - 1 2 ( ν - k - 1 ) ! ℱ𝒞 exp ( α ) ( z , 2 k - ν ) z 2 k - α - ν π 2 2 k - ν k ! - z - α π k = 0 ( - 1 ) k ( 2 k + ν ) ! ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + ν + 1 ) - 2 ψ TagBox["\[Psi]", PolyGamma] ( 2 k + ν + 1 ) + 2 ψ TagBox["\[Psi]", PolyGamma] ( 2 k - α + ν + 1 ) ) k ! ( k + ν ) ! Γ ( 2 k - α + ν + 1 ) ( z 2 ) 2 k + ν /; ν TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition z α BesselY ν z 2 α -1 2 ν 1 Gamma ν 1 z ν -1 α 1 2 -1 z 2 -1 HypergeometricPFQRegularized ν 2 -1 1 2 ν 2 -1 1 -1 α 2 -1 ν 2 -1 1 2 -1 α 2 -1 ν 2 -1 1 ν 1 -1 z 2 4 -1 -1 z -1 α -1 k ν 1 2 -1 ν -1 ν -1 k -1 2 k -1 ν k Gamma 2 k -1 α -1 ν 1 -1 z 2 -1 2 k -1 ν -1 k 0 ν -1 2 -1 ν -1 k -1 Subscript ℱ𝒞 exp α z 2 k -1 ν z 2 k -1 α -1 ν 2 2 k -1 ν k -1 -1 z -1 α -1 k 0 -1 k 2 k ν PolyGamma k 1 PolyGamma k ν 1 -1 2 PolyGamma 2 k ν 1 2 PolyGamma 2 k -1 α ν 1 k k ν Gamma 2 k -1 α ν 1 -1 z 2 -1 2 k ν 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[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", "\[Nu]"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], SqrtBox["\[Pi]"]], "-", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[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[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Nu]"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", "\[Nu]"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", "\[Nu]"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "\[Nu]"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "-", "\[Alpha]", "+", "\[Nu]"]], "]"]]]]]]]]], "\[Pi]"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Nu]", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]"]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]", "-", "\[Alpha]"]]]]], RowBox[List["\[Pi]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]"]]], " ", RowBox[List["k", "!"]]]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], "]"]]]], RowBox[List["\[Nu]", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Nu]", "+", "1", "-", "\[Alpha]"]], "]"]]]]]]]]], "\[Pi]"]]], "/;", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29