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ArcTanh




Mathematica Notation

Traditional Notation









Elementary Functions > ArcTanh[z] > Transformations > Products, sums, and powers of the direct function > Sums of the direct function





http://functions.wolfram.com/01.27.16.0007.01









  


  








Input Form





ArcTanh[x] + ArcTanh[y] == ArcTanh[(x + y)/(x y + 1)] - (1/2) Pi I Sign[x + y] (Sign[x y + 1] + 1) /; Abs[x] > 1 && Abs[y] > 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArcTanh", "[", "x", "]"]], "+", RowBox[List["ArcTanh", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["x", "+", "y"]], RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Sign", "[", RowBox[List["x", "+", "y"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Sign", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], "]"]], "+", "1"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "x", "]"]], ">", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", "y", "]"]], ">", "1"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> x </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> y </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arctanh /> <ci> x </ci> </apply> <apply> <arctanh /> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <arctanh /> <apply> <times /> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Sign </ci> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <ci> Sign </ci> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <abs /> <ci> x </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <apply> <abs /> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTanh", "[", "x_", "]"]], "+", RowBox[List["ArcTanh", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["x", "+", "y"]], RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Sign", "[", RowBox[List["x", "+", "y"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Sign", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], "]"]], "+", "1"]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "x", "]"]], ">", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", "y", "]"]], ">", "1"]]]]]]]]]]


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





2001-10-29