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ArcCoth






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCoth[z] > Complex characteristics > Argument





http://functions.wolfram.com/01.28.19.0006.01









  


  










Input Form





Arg[ArcCoth[x + I y]] == ArcTan[-Log[(1 - 2 x + x^2 + y^2)/(x^2 + y^2)] + Log[(1 + 2 x + x^2 + y^2)/(x^2 + y^2)], -2 ArcTan[1 - x/(x^2 + y^2), y/(x^2 + y^2)] + 2 ArcTan[1 + x/(x^2 + y^2), -(y/(x^2 + y^2))]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Arg", "[", RowBox[List["ArcCoth", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", RowBox[List["2", " ", "x"]], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]], "]"]]]], "+", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "x"]], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]], "]"]]]], ",", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", FractionBox["x", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]]], ",", FractionBox["y", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", FractionBox["x", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]]], ",", RowBox[List["-", FractionBox["y", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]]]]]], "]"]]]]]]]], "]"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> x </mi> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> y </mi> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> x </mi> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mi> y </mi> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arg /> <apply> <arccoth /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <arctan /> <apply> <plus /> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctan /> <apply> <plus /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> y </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> y </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2001-10-29





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