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 ArcSin

 http://functions.wolfram.com/01.12.16.0143.01

 Input Form

 ArcSin[x] + I ArcTanh[y] == ArcSin[((-1)^Floor[1/2 - Arg[(x + I Sqrt[1 - x^2] y)/Sqrt[1 - y^2]]/Pi] (-Sqrt[1 - x^2] + I x y))/Sqrt[1 - y^2]] + (1/2) Pi ((-1)^Floor[1/2 - Arg[(x + I Sqrt[1 - x^2] y)/Sqrt[1 - y^2]]/Pi] + 2 (1 + (-1)^Floor[1/2 - Arg[(x + I Sqrt[1 - x^2] y)/Sqrt[1 - y^2]]/Pi]) Floor[(Arg[I x + Sqrt[1 - x^2]] + Arg[(I - I y)/Sqrt[1 - y^2]])/ (2 Pi)] - 2 (-1 + (-1)^Floor[1/2 - Arg[(x + I Sqrt[1 - x^2] y)/Sqrt[1 - y^2]]/Pi]) Floor[1/2 - (Arg[I x + Sqrt[1 - x^2]] + Arg[(I - I y)/Sqrt[1 - y^2]])/ (2 Pi)])

 Standard Form

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

 MathML Form

 sin - 1 ( x ) + tanh - 1 ( y ) sin - 1 ( ( - 1 ) 1 2 - arg ( x + 1 - x 2 y 1 - y 2 ) π ( x y - 1 - x 2 ) 1 - y 2 ) + 1 2 π ( 2 ( 1 + ( - 1 ) 1 2 - arg ( x + 1 - x 2 y 1 - y 2 ) π ) arg ( x + 1 - x 2 ) + arg ( - y 1 - y 2 ) 2 π + ( - 1 ) 1 2 - arg ( x + 1 - x 2 y 1 - y 2 ) π - 2 ( - 1 + ( - 1 ) 1 2 - arg ( x + 1 - x 2 y 1 - y 2 ) π ) 1 2 - arg ( x + 1 - x 2 ) + arg ( - y 1 - y 2 ) 2 π ) x y -1 1 2 -1 x 1 -1 x 2 1 2 y 1 -1 y 2 1 2 -1 -1 x y -1 1 -1 x 2 1 2 1 -1 y 2 1 2 -1 1 2 2 1 -1 1 2 -1 x 1 -1 x 2 1 2 y 1 -1 y 2 1 2 -1 -1 x 1 -1 x 2 1 2 -1 y 1 -1 y 2 1 2 -1 2 -1 -1 1 2 -1 x 1 -1 x 2 1 2 y 1 -1 y 2 1 2 -1 -1 -1 2 -1 -1 1 2 -1 x 1 -1 x 2 1 2 y 1 -1 y 2 1 2 -1 -1 1 2 -1 x 1 -1 x 2 1 2 -1 y 1 -1 y 2 1 2 -1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcSin", "[", "x_", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTanh", "[", "y_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcSin", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", "y"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", "y"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", "y"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", "y"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], ")"]]]]]]]]]]

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

 2007-05-02