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 ArcCosh

 http://functions.wolfram.com/01.26.16.0185.01

 Input Form

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

 Standard Form

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"]"]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[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[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], 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SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "y"]]], "]"]], "\[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", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "y"]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[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]"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCosh", "[", "x_", "]"]], "+", RowBox[List["ArcCsch", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "y"], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["x", "-", "1"]]], " ", SqrtBox[RowBox[List["x", "+", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]]]], "]"]], "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], " ", "x"]], "-", FractionBox[RowBox[List[SqrtBox[RowBox[List["x", "-", "1"]]], " ", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02

© 1998- Wolfram Research, Inc.