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 BesselK

 http://functions.wolfram.com/03.04.20.0022.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], 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["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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", RowBox[List["Log", "[", "2", "]"]], 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["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[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], 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[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], " ", 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", "[", 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RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[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 K ν ( z ) z α 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 z - α - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" k = "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 2 + 1 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 ( - 1 ) k ( "\[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]" - 1 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 z - α - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 2 ( - 1 ) k ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - k - 1 ) ! k ! ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k - 1 ) ! Γ ( 2 k - α - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ( log ( z ) - ψ TagBox["\[Psi]", PolyGamma] ( 2 k - α - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 k ) ) ( z 2 ) 2 k + ( - 1 ) ν - 1 2 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" z "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - α k = 0 ℱ𝒞 log ( α ) ( z , 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) k ! ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ! ( z 2 ) 2 k - ( - 1 ) ν - 1 log ( 2 ) 2 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" z "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - α k = 0 Γ ( 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) k ! ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ! Γ ( 2 k - α + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ( z 2 ) 2 k + ( - 1 ) ν 2 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 z "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - α k = 0 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 BesselK ν z 2 ν -1 z -1 α -1 ν k ν -1 2 -1 1 ν -1 -1 k 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 -1 2 ν -1 z -1 α -1 ν k 0 ν -1 2 -1 -1 k ν -1 k -1 k ν -1 2 k -1 Gamma 2 k -1 α -1 ν 1 -1 log z -1 PolyGamma 2 k -1 α -1 ν 1 PolyGamma ν -1 2 k z 2 -1 2 k -1 ν -1 2 -1 ν z ν -1 α k 0 Subscript ℱ𝒞 log α z 2 k ν k k ν -1 z 2 -1 2 k -1 -1 ν -1 2 2 -1 ν z ν -1 α k 0 Gamma 2 k ν 1 k k ν Gamma 2 k -1 α ν 1 -1 z 2 -1 2 k -1 ν 2 -1 ν -1 z ν -1 α k 0 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["BesselK", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", 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]"], FractionBox[RowBox[List[RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["k", 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RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]]]], ")"]], " ", RowBox[List["(", 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RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]

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

 2007-05-02