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 ArcTan

 http://functions.wolfram.com/01.14.16.0246.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", "x", "]"]], "-", RowBox[List["ArcTanh", "[", "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", "[", 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", "-", "y"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Arg", "[", RowBox[List["y", "+", "1"]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", 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", "-", "y"]]]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List["y", "+", "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", "[", 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", "-", "y"]], "]"]]]], "+", "\[Pi]"]], 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", "-", "y"]], "]"]], "]"]]]]]], 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["Log", "[", 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", "-", "y"]]]]], SqrtBox[RowBox[List["y", "+", "1"]]]], "]"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTan", "[", "x_", "]"]], "-", RowBox[List["ArcTanh", "[", "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", "[", 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", "-", "y"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Arg", "[", RowBox[List["y", "+", "1"]], "]"]]]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", 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", "-", "y"]]]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List["y", "+", "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", "[", 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", "-", "y"]], "]"]]]], "+", "\[Pi]"]], 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", "-", "y"]], "]"]], "]"]]]]]], 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["Log", "[", 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", "-", "y"]]]]], SqrtBox[RowBox[List["y", "+", "1"]]]], "]"]]]]]]]]

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

 2007-05-02