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ArcCoth




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCoth[z] > Transformations > Related transformations > Differences involving the direct function > Involving tan-1(z)





http://functions.wolfram.com/01.28.16.0200.01









  


  








Input Form





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








Standard Form





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MathML Form







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</mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> &#960; </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> &#8289; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCoth", "[", "x_", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", "y_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcCoth", "[", FractionBox[RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]]]]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List["1", "-", FractionBox["1", "x"]]], "]"]]]], "-", RowBox[List["Arg", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "y"]], "x"]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List["1", "+", FractionBox["1", "x"]]], "]"]]]], "-", RowBox[List["Arg", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "y"]], "x"]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]


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





2007-05-02