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ArcSin






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSin[z] > Transformations > Products, sums, and powers of the direct function > Linear combinations of the direct function





http://functions.wolfram.com/01.12.16.0113.01









  


  










Input Form





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










Standard Form





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










MathML Form







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Rule Form





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










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





2007-05-02