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 ArcTan

 http://functions.wolfram.com/01.14.16.0281.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02