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 ArcCosh

 http://functions.wolfram.com/01.26.16.0188.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02