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 BesselK

 http://functions.wolfram.com/03.04.20.0002.01

 Input Form

 Derivative[1, 0][BesselK][\[Nu], z] == ((Pi Csc[Pi \[Nu]])/2) ((BesselI[-\[Nu], z] + BesselI[\[Nu], z]) (Log[2] - Log[z] + PolyGamma[\[Nu]]) + (BesselI[\[Nu], z] (1 + Pi \[Nu] Cot[Pi \[Nu]]))/\[Nu]) + ((2^(-3 + \[Nu]) z^(2 - \[Nu]) Gamma[-1 + \[Nu]])/(-1 + v)) HypergeometricPFQ[{{}, {1}, {1, 1 - \[Nu]}}, {{2, 2 - \[Nu]}, {}, {2 - \[Nu]}}, z^2/4, z^2/4] + ((2^(-3 - \[Nu]) z^(2 + \[Nu]) Gamma[-1 - \[Nu]])/(1 + \[Nu])) HypergeometricPFQ[{{}, {1}, {1, 1 + \[Nu]}}, {{2, 2 + \[Nu]}, {}, {2 + \[Nu]}}, z^2/4, z^2/4] /; !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "BesselK", "]"]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "2"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "z"]], "]"]], "+", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "\[Nu]", "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Pi]", " ", "\[Nu]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]]]], "\[Nu]"]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], "]"]], " "]], RowBox[List[RowBox[List["-", "1"]], "+", "v"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "-", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]], "]"]], " "]], RowBox[List["1", "+", "\[Nu]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "+", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]], ")"]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

 K TagBox["K", BesselK] ν ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) π csc ( π ν ) 2 ( ( I - ν ( z ) + I ν ( z ) ) ( log ( 2 ) - log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( ν ) ) + π ν cot ( π ν ) + 1 ν I ν ( z ) ) + 2 ν - 3 z 2 - ν Γ ( ν - 1 ) v - 1 F 2 0 1 0 1 2 ( ; 1 ; 1 , 1 - ν ; 2 , 2 - ν ; ; 2 - ν ; z 2 4 , z 2 4 ) + 2 - ν - 3 z ν + 2 Γ ( - ν - 1 ) ν + 1 F 2 0 1 0 1 2 ( ; 1 ; 1 , 1 + ν ; 2 , 2 + ν ; ; 2 + ν ; z 2 4 , z 2 4 ) /; ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] FormBox RowBox RowBox RowBox SubsuperscriptBox TagBox K BesselK ν TagBox RowBox ( RowBox 1 , 0 ) Derivative ( z ) RowBox RowBox FractionBox RowBox π RowBox csc ( RowBox π ν ) 2 RowBox ( RowBox RowBox RowBox ( RowBox RowBox SubscriptBox I RowBox - ν ( z ) + RowBox SubscriptBox I ν ( z ) ) RowBox ( RowBox RowBox log ( 2 ) - RowBox log ( z ) + RowBox TagBox ψ PolyGamma ( ν ) ) + RowBox FractionBox RowBox RowBox π ν RowBox cot ( RowBox π ν ) + 1 ν RowBox SubscriptBox I ν ( z ) ) + RowBox FractionBox RowBox SuperscriptBox 2 RowBox ν - 3 SuperscriptBox z RowBox 2 - ν RowBox Γ ( RowBox ν - 1 ) RowBox v - 1 RowBox SubsuperscriptBox F RowBox 2 0 1 RowBox 0 1 2 [ RowBox GridBox ErrorBox RowBox RowBox ; 1 ; 1 , RowBox RowBox 1 - ν ; RowBox 2 , RowBox RowBox RowBox 2 - ν ; ; RowBox 2 - ν ; FractionBox SuperscriptBox z 2 4 , FractionBox SuperscriptBox z 2 4 ] + RowBox FractionBox RowBox SuperscriptBox 2 RowBox RowBox - ν - 3 SuperscriptBox z RowBox ν + 2 RowBox Γ ( RowBox RowBox - ν - 1 ) RowBox ν + 1 RowBox SubsuperscriptBox F RowBox 2 0 1 RowBox 0 1 2 ( RowBox RowBox GridBox RowBox RowBox ; 1 ; 1 , RowBox RowBox 1 + ν ; RowBox 2 , RowBox RowBox RowBox 2 + ν ; ; RowBox 2 + ν ; FractionBox SuperscriptBox z 2 4 , FractionBox SuperscriptBox z 2 4 ) /; RowBox ν TagBox Function TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["BesselK", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "z"]], "]"]], "+", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "\[Nu]", "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Pi]", " ", "\[Nu]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]], "\[Nu]"]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "-", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", "v"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "+", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], RowBox[List["1", "+", "\[Nu]"]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]

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

 2001-10-29