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 ArcSin

 http://functions.wolfram.com/01.12.16.0134.01

 Input Form

 ArcSin[x] + ArcSinh[y] == I Pi (1 - (-1)^(Floor[-(Arg[(y + Sqrt[y^2 + 1])/(I x + Sqrt[1 - x^2])^I + 1]/ (2 Pi))] - Floor[-(Arg[(y + Sqrt[y^2 + 1])/(I x + Sqrt[1 - x^2])^I]/ (2 Pi))])) + I (-1)^(Floor[-(Arg[(y + Sqrt[y^2 + 1])/(I x + Sqrt[1 - x^2])^I]/Pi)] + Floor[Arg[(y + Sqrt[y^2 + 1])/(I x + Sqrt[1 - x^2])^I - 1]/(2 Pi) - Arg[(y + Sqrt[y^2 + 1])/(I x + Sqrt[1 - x^2])^I + 1]/(2 Pi) + 1/2]) (ArcSin[((I x + Sqrt[1 - x^2])^I ((y + Sqrt[y^2 + 1])^2/ (I x + Sqrt[1 - x^2])^(2 I) + 1))/(2 (y + Sqrt[y^2 + 1]))] - Pi/2) - 2 I Pi (Floor[(-Arg[(I x + Sqrt[1 - x^2])^(-I)] - Arg[y + Sqrt[y^2 + 1]] + Pi)/(2 Pi)] + Floor[(Pi - Im[Log[y + Sqrt[y^2 + 1]]])/(2 Pi)] + Floor[(Re[Log[I x + Sqrt[1 - x^2]]] + Pi)/(2 Pi)])

 Standard Form

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

 MathML Form

 sin - 1 ( x ) + sinh - 1 ( y ) π ( 1 - ( - 1 ) - arg ( ( y + y 2 + 1 ) ( x + 1 - x 2 ) - + 1 ) 2 π - - arg ( ( x + 1 - x 2 ) - ( y + y 2 + 1 ) ) 2 π ) + ( - 1 ) arg ( ( x + 1 - x 2 ) - ( y + y 2 + 1 ) - 1 ) 2 π + 1 2 - arg ( ( y + y 2 + 1 ) ( x + 1 - x 2 ) - + 1 ) 2 π + - arg ( ( x + 1 - x 2 ) - ( y + y 2 + 1 ) ) π ( sin - 1 ( ( x + 1 - x 2 ) ( ( y + y 2 + 1 ) 2 ( x + 1 - x 2 ) - 2 + 1 ) 2 ( y + y 2 + 1 ) ) - π 2 ) - 2 π ( - arg ( y + y 2 + 1 ) - arg ( ( x + 1 - x 2 ) - ) + π 2 π + Re ( log ( x + 1 - x 2 ) ) + π 2 π + π - Im ( log ( y + y 2 + 1 ) ) 2 π ) x y 1 -1 -1 -1 y y 2 1 1 2 x 1 -1 x 2 1 2 -1 1 2 -1 -1 -1 x 1 -1 x 2 1 2 -1 y y 2 1 1 2 2 -1 -1 x 1 -1 x 2 1 2 -1 y y 2 1 1 2 -1 2 -1 1 2 -1 y y 2 1 1 2 x 1 -1 x 2 1 2 -1 1 2 -1 -1 x 1 -1 x 2 1 2 -1 y y 2 1 1 2 -1 x 1 -1 x 2 1 2 y y 2 1 1 2 2 x 1 -1 x 2 1 2 -2 1 2 y y 2 1 1 2 -1 -1 2 -1 -1 2 -1 y y 2 1 1 2 -1 x 1 -1 x 2 1 2 -1 2 -1 x 1 -1 x 2 1 2 2 -1 -1 y y 2 1 1 2 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcSin", "[", "x_", "]"]], "+", RowBox[List["ArcSinh", "[", "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[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], ")"]], RowBox[List["-", "\[ImaginaryI]"]]], " ", RowBox[List["(", RowBox[List["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "-", 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]"]]], " ", RowBox[List["(", RowBox[List["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", 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]"]]], " ", RowBox[List["(", RowBox[List["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]]]], "]"]], "\[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]"]]], " ", RowBox[List["(", RowBox[List["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]]]], "-", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "-", 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]"]]], " ", RowBox[List["(", RowBox[List["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ArcSin", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], ")"]], "\[ImaginaryI]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]], "2"]]], "+", "1"]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], ")"]]]]], "]"]], "-", FractionBox["\[Pi]", "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]"]]], "]"]]]], "-", RowBox[List["Arg", "[", RowBox[List["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List["y", "+", SqrtBox[RowBox[List[SuperscriptBox["y", "2"], "+", "1"]]]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Re", "[", RowBox[List["Log", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]], "]"]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]]]]]]]

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

 2007-05-02