html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 10px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ArcCosh

 http://functions.wolfram.com/01.26.19.0007.01

 Input Form

 Conjugate[ArcCosh[x + I y]] == (-I) ArcTan[x + ((-1 + x)^2 + y^2)^(1/4) ((1 + x)^2 + y^2)^(1/4) Cos[(1/2) (ArcTan[-1 + x, y] + ArcTan[1 + x, y])], y + ((-1 + x)^2 + y^2)^(1/4) ((1 + x)^2 + y^2)^(1/4) Sin[(1/2) (ArcTan[-1 + x, y] + ArcTan[1 + x, y])]] + Log[Sqrt[(x + ((-1 + x)^2 + y^2)^(1/4) ((1 + x)^2 + y^2)^(1/4) Cos[(1/2) (ArcTan[-1 + x, y] + ArcTan[1 + x, y])])^2 + (y + ((-1 + x)^2 + y^2)^(1/4) ((1 + x)^2 + y^2)^(1/4) Sin[(1/2) (ArcTan[-1 + x, y] + ArcTan[1 + x, y])])^2]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Conjugate", "[", RowBox[List["ArcCosh", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["x", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ",", "y"]], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]]]], "]"]]]]]], ",", RowBox[List["y", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ",", "y"]], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]]]], "]"]]]]]]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["x", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ",", "y"]], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]]]], "]"]]]]]], ")"]], "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ",", "y"]], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]]]], "]"]]]]]], ")"]], "2"]]], ")"]]]], "]"]]]]]]]]

 MathML Form

 cosh - 1 ( x + y ) _ log ( ( ( ( x - 1 ) 2 + y 2 4 cos ( 1 2 ( tan - 1 ( x - 1 , y ) + tan - 1 ( x + 1 , y ) ) ) ( x + 1 ) 2 + y 2 4 + x ) 2 + ( ( x - 1 ) 2 + y 2 4 sin ( 1 2 ( tan - 1 ( x - 1 , y ) + tan - 1 ( x + 1 , y ) ) ) ( x + 1 ) 2 + y 2 4 + y ) 2 ) ) - tan - 1 ( ( x - 1 ) 2 + y 2 4 cos ( 1 2 ( tan - 1 ( x - 1 , y ) + tan - 1 ( x + 1 , y ) ) ) ( x + 1 ) 2 + y 2 4 + x , ( x - 1 ) 2 + y 2 4 sin ( 1 2 ( tan - 1 ( x - 1 , y ) + tan - 1 ( x + 1 , y ) ) ) ( x + 1 ) 2 + y 2 4 + y ) OverBar x y x -1 2 y 2 1 4 1 2 x -1 y x 1 y x 1 2 y 2 1 4 x 2 x -1 2 y 2 1 4 1 2 x -1 y x 1 y x 1 2 y 2 1 4 y 2 -1 x -1 2 y 2 1 4 1 2 x -1 y x 1 y x 1 2 y 2 1 4 x x -1 2 y 2 1 4 1 2 x -1 y x 1 y x 1 2 y 2 1 4 y [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Conjugate", "[", RowBox[List["ArcCosh", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["x", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ",", "y"]], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]]]], "]"]]]]]], ",", RowBox[List["y", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ",", "y"]], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]]]], "]"]]]]]]]], "]"]]]], "+", RowBox[List["Log", "[", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["x", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ",", "y"]], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]]]], "]"]]]]]], ")"]], "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ",", "y"]], "]"]], "+", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]]]], "]"]]]]]], ")"]], "2"]]]], "]"]]]]]]]]

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

 2001-10-29