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ArcTanh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcTanh[z] > Complex characteristics > Signum value





http://functions.wolfram.com/01.27.19.0008.01









  


  










Input Form





Sign[ArcTanh[x + I y]] == ((-I) ArcTan[1 - x, -y] + I ArcTan[1 + x, y] - (1/2) Log[1 - 2 x + x^2 + y^2] + (1/2) Log[(1 + x)^2 + y^2])/ Sqrt[(ArcTan[1 - x, -y] - ArcTan[1 + x, y])^2 + (1/4) (Log[1 - 2 x + x^2 + y^2] - Log[(1 + x)^2 + y^2])^2]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Sign", "[", RowBox[List["ArcTanh", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", "x"]], ",", RowBox[List["-", "y"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "x"]], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", "x"]], ",", RowBox[List["-", "y"]]]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]], "2"], "+", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "x"]], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], "]"]]]], ")"]], "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> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> tanh </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> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <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> <mi> x </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <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> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> x </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <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> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <arctanh /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <arctan /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <ci> y </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ln /> <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> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ln /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <ci> y </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <power /> <apply> <plus /> <apply> <ln /> <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> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", RowBox[List["ArcTanh", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", "x"]], ",", RowBox[List["-", "y"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "x"]], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", "x"]], ",", RowBox[List["-", "y"]]]], "]"]], "-", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", "x"]], ",", "y"]], "]"]]]], ")"]], "2"], "+", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "x"]], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "x"]], ")"]], "2"], "+", SuperscriptBox["y", "2"]]], "]"]]]], ")"]], "2"]]]]]]]]]]]










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





2007-05-02