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 ArcSin

 http://functions.wolfram.com/01.12.16.0146.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2007-05-02