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 ArcCosh

 http://functions.wolfram.com/01.26.16.0220.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02