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 ArcTan

 http://functions.wolfram.com/01.14.16.0222.01

 Input Form

 ArcTan[x] + I ArcCsch[y] == ArcTan[((Sqrt[1 + x^2]/(Sqrt[1 + 1/y^2] - (I x)/y)) Sqrt[1 - (Sqrt[1 + 1/y^2] - (I x)/y)^2/(1 + x^2)])/ (-1)^Floor[1/2 - Arg[(x Sqrt[1 + 1/y^2] + I/y)/Sqrt[1 + x^2]]/Pi]] + (1/2) Pi ((-1)^Floor[1/2 - Arg[(x Sqrt[1 + 1/y^2] + I/y)/Sqrt[1 + x^2]]/Pi] E^(I Pi Floor[1/2 + Arg[-Sqrt[1 + 1/y^2] + (I x)/y]/Pi - Arg[1 + x^2]/(2 Pi)]) + (-1)^Floor[1/2 - Arg[(x Sqrt[1 + 1/y^2] + I/y)/Sqrt[1 + x^2]]/Pi] + 2 (1 + (-1)^Floor[1/2 - Arg[(x Sqrt[1 + 1/y^2] + I/y)/Sqrt[1 + x^2]]/Pi]) Floor[(Arg[(I - x)/Sqrt[1 + x^2]] + Arg[Sqrt[1 + 1/y^2] - 1/y])/ (2 Pi)] - 2 (-1 + (-1)^Floor[1/2 - Arg[(x Sqrt[1 + 1/y^2] + I/y)/Sqrt[1 + x^2]]/ Pi]) Floor[1/2 - (Arg[(I - x)/Sqrt[1 + x^2]] + Arg[Sqrt[1 + 1/y^2] - 1/y])/(2 Pi)])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", "x", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCsch", "[", "y", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], FractionBox[RowBox[List[" ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "y"]]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "y"]]], ")"]], "2"], RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]]]]], ")"]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "y"]]], "]"]], "\[Pi]"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "+", SuperscriptBox["x", "2"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", "x"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "-", FractionBox["1", "y"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", "x"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "-", FractionBox["1", "y"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], ")"]]]]]]]]]]

 MathML Form

 tan - 1 ( x ) + csch - 1 ( y ) tan - 1 ( ( - 1 ) - 1 2 - arg ( x 1 + 1 y 2 + y x 2 + 1 ) π x 2 + 1 1 - ( 1 + 1 y 2 - x y ) 2 x 2 + 1 1 + 1 y 2 - x y ) + 1 2 π ( 2 ( 1 + ( - 1 ) 1 2 - arg ( x 1 + 1 y 2 + y x 2 + 1 ) π ) arg ( - x x 2 + 1 ) + arg ( 1 + 1 y 2 - 1 y ) 2 π + ( - 1 ) 1 2 - arg ( x 1 + 1 y 2 + y x 2 + 1 ) π + ( - 1 ) 1 2 - arg ( x 1 + 1 y 2 + y x 2 + 1 ) π π - arg ( x 2 + 1 ) 2 π + arg ( x y - 1 + 1 y 2 ) π + 1 2 - 2 ( - 1 + ( - 1 ) 1 2 - arg ( x 1 + 1 y 2 + y x 2 + 1 ) π ) 1 2 - arg ( - x x 2 + 1 ) + arg ( 1 + 1 y 2 - 1 y ) 2 π ) x y -1 -1 1 2 -1 x 1 1 y 2 -1 1 2 y -1 x 2 1 1 2 -1 -1 x 2 1 1 2 1 -1 1 1 y 2 -1 1 2 -1 x y -1 2 x 2 1 -1 1 2 1 1 y 2 -1 1 2 -1 x y -1 -1 1 2 2 1 -1 1 2 -1 x 1 1 y 2 -1 1 2 y -1 x 2 1 1 2 -1 -1 -1 x x 2 1 1 2 -1 1 1 y 2 -1 1 2 -1 1 y -1 2 -1 -1 1 2 -1 x 1 1 y 2 -1 1 2 y -1 x 2 1 1 2 -1 -1 -1 1 2 -1 x 1 1 y 2 -1 1 2 y -1 x 2 1 1 2 -1 -1 -1 x 2 1 2 -1 x y -1 -1 1 1 y 2 -1 1 2 -1 1 2 -1 2 -1 -1 1 2 -1 x 1 1 y 2 -1 1 2 y -1 x 2 1 1 2 -1 -1 1 2 -1 -1 x x 2 1 1 2 -1 1 1 y 2 -1 1 2 -1 1 y -1 2 -1 [/itex]

 Rule Form

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

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

 2007-05-02