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 ArcTan

 http://functions.wolfram.com/01.14.16.0242.01

 Input Form

 ArcTan[x] - I ArcSinh[y] == ArcTan[(Sqrt[1 + x^2] Sqrt[1 - (I x y + Sqrt[1 + y^2])^2/(1 + x^2)])/ ((-1)^Floor[1/2 - Arg[((-I) y + x Sqrt[1 + y^2])/Sqrt[1 + x^2]]/Pi] (I x y + Sqrt[1 + y^2]))] + (1/2) Pi ((-1)^(Floor[1/2 - Arg[1 + x^2]/(2 Pi) + Arg[(-I) x y - Sqrt[1 + y^2]]/Pi] + Floor[1/2 + Arg[1 + x^2]/(2 Pi) - Arg[(-I) y + x Sqrt[1 + y^2]]/Pi] - 2 Floor[1/2 + Arg[1 + x^2]/(4 Pi) - Arg[(-I) y + x Sqrt[1 + y^2]]/ (2 Pi)]) + (-1)^Floor[1/2 - Arg[((-I) y + x Sqrt[1 + y^2])/Sqrt[1 + x^2]]/Pi] + 2 (1 + (-1)^Floor[1/2 - Arg[((-I) y + x Sqrt[1 + y^2])/Sqrt[1 + x^2]]/ Pi]) Floor[(Arg[(I - x)/Sqrt[1 + x^2]] + Arg[y + Sqrt[1 + y^2]])/ (2 Pi)] - 2 (-1 + (-1)^Floor[1/2 - Arg[((-I) y + x Sqrt[1 + y^2])/Sqrt[1 + x^2]]/ Pi]) Floor[1/2 - (Arg[(I - x)/Sqrt[1 + x^2]] + Arg[y + Sqrt[1 + y^2]])/(2 Pi)])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", "x", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", "y", "]"]]]]]], "\[Equal]", 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[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], 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[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], ")"]], "2"], RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "+", SuperscriptBox["x", "2"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "x", " ", "y"]], "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "+", SuperscriptBox["x", "2"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], "]"]], "\[Pi]"]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "+", SuperscriptBox["x", "2"]]], "]"]], RowBox[List["4", " ", "\[Pi]"]]], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "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[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], 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[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], 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["y", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "]"]]]], 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[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], 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["y", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], ")"]]]]]]]]]]

 MathML Form

 tan - 1 ( x ) - sinh - 1 ( y ) tan - 1 ( ( - 1 ) - 1 2 - arg ( x y 2 + 1 - y x 2 + 1 ) π x 2 + 1 1 - ( x y + y 2 + 1 ) 2 x 2 + 1 x y + y 2 + 1 ) + 1 2 π ( 2 ( 1 + ( - 1 ) 1 2 - arg ( x y 2 + 1 - y x 2 + 1 ) π ) arg ( - x x 2 + 1 ) + arg ( y + y 2 + 1 ) 2 π + ( - 1 ) 1 2 - arg ( x y 2 + 1 - y x 2 + 1 ) π + ( - 1 ) - arg ( x 2 + 1 ) 2 π + arg ( - x y - y 2 + 1 ) π + 1 2 + arg ( x 2 + 1 ) 2 π + 1 2 - arg ( x y 2 + 1 - y ) π - 2 arg ( x 2 + 1 ) 4 π + 1 2 - arg ( x y 2 + 1 - y ) 2 π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( x y 2 + 1 - y x 2 + 1 ) π ) 1 2 - arg ( - x x 2 + 1 ) + arg ( y + y 2 + 1 ) 2 π ) x -1 y -1 -1 1 2 -1 x y 2 1 1 2 -1 y x 2 1 1 2 -1 -1 x 2 1 1 2 1 -1 x y y 2 1 1 2 2 x 2 1 -1 1 2 x y y 2 1 1 2 -1 1 2 2 1 -1 1 2 -1 x y 2 1 1 2 -1 y x 2 1 1 2 -1 -1 -1 x x 2 1 1 2 -1 y y 2 1 1 2 2 -1 -1 1 2 -1 x y 2 1 1 2 -1 y x 2 1 1 2 -1 -1 -1 -1 x 2 1 2 -1 -1 x y -1 y 2 1 1 2 -1 1 2 x 2 1 2 -1 1 2 -1 x y 2 1 1 2 -1 y -1 -1 2 x 2 1 4 -1 1 2 -1 x y 2 1 1 2 -1 y 2 -1 -1 2 -1 -1 1 2 -1 x y 2 1 1 2 -1 y x 2 1 1 2 -1 -1 1 2 -1 -1 x x 2 1 1 2 -1 y y 2 1 1 2 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTan", "[", "x_", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", "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[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], 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[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], ")"]], "2"], RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "+", SuperscriptBox["x", "2"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "x", " ", "y"]], "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "+", SuperscriptBox["x", "2"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], "]"]], "\[Pi]"]]], "]"]], "-", RowBox[List["2", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "+", SuperscriptBox["x", "2"]]], "]"]], RowBox[List["4", " ", "\[Pi]"]]], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "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[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], 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[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], 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["y", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "]"]]]], 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[RowBox[List["-", "\[ImaginaryI]"]], " ", "y"]], "+", RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]], 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["y", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], ")"]]]]]]]]]]

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

 2007-05-02