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variants of this functions
ArcTan






Mathematica Notation

Traditional Notation









Elementary Functions > ArcTan[z] > Transformations > Related transformations > Sums involving the direct function > Involving cos-1(z)





http://functions.wolfram.com/01.14.16.0196.01









  


  










Input Form





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










Standard Form





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MathML Form







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<apply> <times /> <ci> x </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





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