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 ArcTan

 http://functions.wolfram.com/01.14.16.0217.01

 Input Form

 ArcTan[x] + ArcCoth[y] == Pi Floor[(-Arg[1 - I x] + Arg[I x + 1] + Pi)/(2 Pi)] - 2 I Pi (Floor[(-Arg[((1 - I x)/(I x + 1))^(I/2)/Sqrt[1 - 1/y]] - (1/2) Arg[1 + 1/y] + Pi)/(2 Pi)] + Floor[(Pi - Im[Log[((1 - I x)/(I x + 1))^(I/2)/Sqrt[1 - 1/y]]])/(2 Pi)] + Floor[(Pi - (1/2) Im[Log[1 + 1/y]])/(2 Pi)]) - 2 I Pi (Floor[(-Arg[((1 - I x)/(I x + 1))^(I/2)] + (1/2) Arg[1 - 1/y] + Pi)/ (2 Pi)] + Floor[((1/2) Im[Log[1 - 1/y]] + Pi)/(2 Pi)] + Floor[(Pi - (1/2) Re[Log[(1 - I x)/(I x + 1)]])/(2 Pi)]) + I Pi (1 - (-1)^Floor[Arg[(((1 - I x)/(I x + 1))^(I/2) Sqrt[1 + 1/y])/ Sqrt[1 - 1/y] + 1]/(2 Pi) + 1/2]) + 2 I ArcTan[(I (1 - (((1 - I x)/(I x + 1))^(I/2) Sqrt[1 + 1/y])/ Sqrt[1 - 1/y]))/((((1 - I x)/(I x + 1))^(I/2) Sqrt[1 + 1/y])/ Sqrt[1 - 1/y] + 1)]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", "x", "]"]], "+", RowBox[List["ArcCoth", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], "]"]]]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Arg", "[", RowBox[List["1", "+", FractionBox["1", "y"]]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["1", "y"]]], "]"]], "]"]]]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Arg", "[", RowBox[List["1", "-", FractionBox["1", "y"]]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox["1", "y"]]], "]"]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Re", "[", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], "]"]], "]"]]]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox[RowBox[List["Arg", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]]]], ")"]]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]], "+", "1"]]], "]"]]]]]]]]]]

 MathML Form

 coth - 1 ( y ) + tan - 1 ( x ) π arg ( x + 1 ) - arg ( 1 - x ) + π 2 π - 2 π ( - 1 2 arg ( 1 + 1 y ) - arg ( ( 1 - x x + 1 ) / 2 1 - 1 y ) + π 2 π + π - 1 2 Im ( log ( 1 + 1 y ) ) 2 π + π - Im ( log ( ( 1 - x x + 1 ) / 2 1 - 1 y ) ) 2 π ) - 2 π ( - arg ( ( 1 - x x + 1 ) / 2 ) + 1 2 arg ( 1 - 1 y ) + π 2 π + 1 2 Im ( log ( 1 - 1 y ) ) + π 2 π + π - 1 2 Re ( log ( 1 - x x + 1 ) ) 2 π ) + π ( 1 - ( - 1 ) arg ( 1 + 1 y ( 1 - x x + 1 ) / 2 1 - 1 y + 1 ) 2 π + 1 2 ) + 2 tan - 1 ( ( 1 - ( 1 - x x + 1 ) / 2 1 + 1 y 1 - 1 y ) 1 + 1 y ( 1 - x x + 1 ) / 2 1 - 1 y + 1 ) y x x 1 -1 1 -1 x 2 -1 -1 2 -1 1 2 1 1 y -1 -1 1 -1 x x 1 -1 2 -1 1 -1 1 y -1 1 2 -1 2 -1 -1 1 2 1 1 y -1 2 -1 -1 1 -1 x x 1 -1 2 -1 1 -1 1 y -1 1 2 -1 2 -1 -1 2 -1 1 -1 x x 1 -1 2 -1 1 2 1 -1 1 y -1 2 -1 1 2 1 -1 1 y -1 2 -1 -1 1 2 1 -1 x x 1 -1 2 -1 1 -1 -1 1 1 y -1 1 2 1 -1 x x 1 -1 2 -1 1 -1 1 y -1 1 2 -1 1 2 -1 1 2 2 1 -1 1 -1 x x 1 -1 2 -1 1 1 y -1 1 2 1 -1 1 y -1 1 2 -1 1 1 y -1 1 2 1 -1 x x 1 -1 2 -1 1 -1 1 y -1 1 2 -1 1 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTan", "[", "x_", "]"]], "+", RowBox[List["ArcCoth", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], "]"]]]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Arg", "[", RowBox[List["1", "+", FractionBox["1", "y"]]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["1", "y"]]], "]"]], "]"]]]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Arg", "[", RowBox[List["1", "-", FractionBox["1", "y"]]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox["1", "y"]]], "]"]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Re", "[", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], "]"]], "]"]]]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox[RowBox[List["Arg", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]]]], ")"]]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "y"]]]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", "y"]]]]], "+", "1"]]], "]"]]]]]]]]]]

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

 2007-05-02