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: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ArcCos

 http://functions.wolfram.com/01.13.16.0120.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

 2007-05-02