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 EllipticK

 http://functions.wolfram.com/08.02.06.0023.01

 Input Form

 EllipticK[z] \[Proportional] EllipticK[x] - 2 I EllipticK[1 - x] Floor[Arg[-z + x]/(2 Pi)] + (1/(2 Pi)) (((2 I Pi)/((-1 + x) x)) Floor[Arg[-z + x]/(2 Pi)] (-EllipticE[1 - x] + EllipticK[1 - x] x) + MeijerG[{{-(1/2), -(1/2)}, {}}, {{0, -1}, {}}, 1 - x]) (z - x) + (1/(4 Pi)) (((I Pi)/((-1 + x)^2 x^2)) Floor[Arg[-z + x]/(2 Pi)] (-2 EllipticE[1 - x] + (4 EllipticE[1 - x] + EllipticK[1 - x]) x - 3 EllipticK[1 - x] x^2) + MeijerG[{{-(3/2), -(3/2)}, {}}, {{0, -2}, {}}, 1 - x]) (z - x)^2 + O[(z - x)^3] /; Element[x, Reals] && x > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticK", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["EllipticK", "[", "x", "]"]], "-", RowBox[List["2", "\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", "\[Pi]"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], " ", "x"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]]]], "+", RowBox[List[RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]], " ", "x"]]]], ")"]]]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "1"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["1", "-", "x"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["4", " ", "\[Pi]"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " "]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], " ", SuperscriptBox["x", "2"]]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]]]], "+", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]]]], ")"]], " ", "x"]], "-", RowBox[List["3", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]], " ", SuperscriptBox["x", "2"]]]]], ")"]]]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], ",", RowBox[List["-", FractionBox["3", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "2"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["1", "-", "x"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "3"], "]"]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", ">", "1"]]]]]]]]

 MathML Form

 K ( z ) K ( x ) - 2 K ( 1 - x ) arg ( x - z ) 2 π + 1 2 π ( 2 π ( x - 1 ) x ( K ( 1 - x ) x - E ( 1 - x ) ) arg ( x - z ) 2 π + G 2 , 2 2 , 2 ( 1 - x - 1 2 , - 1 2 0 , - 1 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["1", "-", "x"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["-", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "1"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ( z - x ) + 1 4 π ( π ( x - 1 ) 2 x 2 arg ( x - z ) 2 π ( - 3 K ( 1 - x ) x 2 + ( 4 E ( 1 - x ) + K ( 1 - x ) ) x - 2 E ( 1 - x ) ) + G 2 , 2 2 , 2 ( 1 - x - 3 2 , - 3 2 0 , - 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["1", "-", "x"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["-", FractionBox["3", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["3", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "2"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ( z - x ) 2 + O ( ( z - x ) 3 ) /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x > 1 Condition Proportional EllipticK z K x -1 2 EllipticK 1 -1 x x -1 z 2 -1 1 2 -1 2 x -1 x -1 EllipticK 1 -1 x x -1 EllipticE 1 -1 x x -1 z 2 -1 MeijerG -1 1 2 -1 1 2 0 -1 1 -1 x z -1 x 1 4 -1 x -1 2 x 2 -1 x -1 z 2 -1 -3 EllipticK 1 -1 x x 2 4 EllipticE 1 -1 x EllipticK 1 -1 x x -1 2 EllipticE 1 -1 x MeijerG -1 3 2 -1 3 2 0 -2 1 -1 x z -1 x 2 O z -1 x 3 x x 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticK", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["EllipticK", "[", "x", "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]]]], "+", RowBox[List[RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]], " ", "x"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], " ", "x"]]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "1"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["1", "-", "x"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "z"]], "+", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "x"]], "]"]]]], "+", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]]]], ")"]], " ", "x"]], "-", RowBox[List["3", " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "x"]], "]"]], " ", SuperscriptBox["x", "2"]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], " ", SuperscriptBox["x", "2"]]]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], ",", RowBox[List["-", FractionBox["3", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["-", "2"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["1", "-", "x"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], RowBox[List["4", " ", "\[Pi]"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", "x"]], "]"]], "3"]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "1"]]]]]]]]]]

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

 2007-05-02

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