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 Log

 http://functions.wolfram.com/01.04.16.0151.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02