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 Log

 http://functions.wolfram.com/01.04.16.0163.01

 Input Form

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

 Standard Form

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

 MathML Form

 a log ( x ) + b cosh - 1 ( y ) π ( 1 - ( - 1 ) - arg ( ( y + y - 1 y + 1 ) b x a + 1 ) 2 π - - arg ( x a ( y + y - 1 y + 1 ) b ) 2 π ) + ( - 1 ) arg ( x a ( y + y - 1 y + 1 ) b ) π - 2 arg ( x a ( y + y - 1 y + 1 ) b - 1 ) π + - arg ( x a ( y + y - 1 y + 1 ) b ) π - - arg ( x a ( y + y - 1 y + 1 ) b - 1 ) 2 π + - arg ( ( y + y - 1 y + 1 ) b x a + 1 ) 2 π cosh - 1 ( 1 2 x - a ( y + y - 1 y + 1 ) - b ( ( y + y - 1 y + 1 ) 2 b x 2 a + 1 ) ) - 2 π ( - arg ( ( y + y - 1 y + 1 ) b ) - arg ( x a ) + π 2 π + π - Im ( b log ( y + y - 1 y + 1 ) ) 2 π + π - Im ( a log ( x ) ) 2 π ) a x b y 1 -1 -1 -1 y y -1 1 2 y 1 1 2 b x a 1 2 -1 -1 -1 x a y y -1 1 2 y 1 1 2 b 2 -1 -1 x a y y -1 1 2 y 1 1 2 b -1 -1 2 x a y y -1 1 2 y 1 1 2 b -1 -1 -1 x a y y -1 1 2 y 1 1 2 b -1 -1 -1 x a y y -1 1 2 y 1 1 2 b -1 2 -1 -1 y y -1 1 2 y 1 1 2 b x a 1 2 -1 1 2 x -1 a y y -1 1 2 y 1 1 2 -1 b y y -1 1 2 y 1 1 2 2 b x 2 a 1 -1 2 -1 y y -1 1 2 y 1 1 2 b -1 x a 2 -1 -1 b y y -1 1 2 y 1 1 2 2 -1 -1 a x 2 -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["ArcCosh", "[", "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[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], "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["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], "b"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]], ")"]]]], "+", RowBox[List[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["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], "b"]]], "]"]], "\[Pi]"]]], "]"]], "-", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[SuperscriptBox["x", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], "b"]]], "-", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox[RowBox[List["Arg", "[", RowBox[List[SuperscriptBox["x", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], "b"]]], "]"]], "\[Pi]"], "-", FractionBox[RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List[RowBox[List[SuperscriptBox["x", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], "b"]]], "-", "1"]], "]"]]]], "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], "b"], " ", SuperscriptBox["x", "a"]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]], " ", RowBox[List["ArcCosh", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["x", RowBox[List["-", "a"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], RowBox[List["-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], RowBox[List["2", " ", "b"]]], " ", SuperscriptBox["x", RowBox[List["2", " ", "a"]]]]], "+", "1"]], ")"]]]], "]"]]]], "-", 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["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], ")"]], "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["Im", "[", RowBox[List["b", " ", RowBox[List["Log", "[", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], "]"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]]]]]]]

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

 2007-05-02