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 BesselK

 http://functions.wolfram.com/03.04.06.0024.01

 Input Form

 BesselK[\[Nu], z] \[Proportional] BesselK[\[Nu], x]/E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) - 2 I Pi BesselI[\[Nu], x] Cos[Pi \[Nu]] Floor[Arg[-x + z]/(2 Pi)] - (1/2) ((BesselK[-1 + \[Nu], x] + BesselK[1 + \[Nu], x])/ E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) + 2 I Pi (BesselI[-1 + \[Nu], x] + BesselI[1 + \[Nu], x]) Cos[Pi \[Nu]] Floor[Arg[-x + z]/(2 Pi)]) (z - x) + (1/8) ((BesselK[-2 + \[Nu], x] + 2 BesselK[\[Nu], x] + BesselK[2 + \[Nu], x])/E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) - 2 I Pi (BesselI[-2 + \[Nu], x] + 2 BesselI[\[Nu], x] + BesselI[2 + \[Nu], x]) Cos[Pi \[Nu]] Floor[Arg[-x + z]/(2 Pi)]) (z - x)^2 + \[Ellipsis] /; (z -> x) && Element[x, Reals] && x < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselK", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "+", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "+", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselK", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "+", RowBox[List["2", " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "+", RowBox[List["2", " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

 K ν ( z ) - 2 π ν arg ( z - x ) 2 π K ν ( x ) - 2 π I ν ( x ) cos ( π ν ) arg ( z - x ) 2 π - 1 2 ( - 2 π ν arg ( z - x ) 2 π ( K ν - 1 ( x ) + K ν + 1 ( x ) ) + 2 π ( I ν - 1 ( x ) + I ν + 1 ( x ) ) cos ( π ν ) arg ( z - x ) 2 π ) ( z - x ) + 1 8 ( - 2 π ν arg ( z - x ) 2 π ( K ν - 2 ( x ) + 2 K ν ( x ) + K ν + 2 ( x ) ) - 2 π ( I ν - 2 ( x ) + 2 I ν ( x ) + I ν + 2 ( x ) ) cos ( π ν ) arg ( z - x ) 2 π ) ( z - x ) 2 + /; ( z "\[Rule]" x ) x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x < 0 Condition Proportional BesselK ν z -2 ν z -1 x 2 -1 BesselK ν x -1 2 BesselI ν x ν z -1 x 2 -1 -1 1 2 -2 ν z -1 x 2 -1 BesselK ν -1 x BesselK ν 1 x 2 BesselI ν -1 x BesselI ν 1 x ν z -1 x 2 -1 z -1 x 1 8 -2 ν z -1 x 2 -1 BesselK ν -2 x 2 BesselK ν x BesselK ν 2 x -1 2 BesselI ν -2 x 2 BesselI ν x BesselI ν 2 x ν z -1 x 2 -1 z -1 x 2 Rule z x x x 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselK", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "+", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "+", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselK", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "+", RowBox[List["2", " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "+", RowBox[List["2", " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["2", "+", "\[Nu]"]], ",", "x"]], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "x"]], "+", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]

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

 2007-05-02