ArcTanh[x] + ArcCoth[y] == ArcTanh[(x y + 1)/(x + y)] - (1/2) Pi I Sign[x y + 1] (1 - Sign[x + y]) /; y < -1 || (Abs[x] < 1 && y < 0) || (x > 1 && 0 < y < 1)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArcTanh", "[", "x", "]"]], "+", RowBox[List["ArcCoth", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], RowBox[List["x", "+", "y"]]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", RowBox[List["x", "+", "y"]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["y", "<", RowBox[List["-", "1"]]]], "||", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "x", "]"]], "<", "1"]], "&&", RowBox[List["y", "<", "0"]]]], "||", RowBox[List[RowBox[List["x", ">", "1"]], "&&", RowBox[List["0", "<", "y", "<", "1"]]]]]]]]]]

<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> coth </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> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </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> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> y </mi> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8744; </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> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> y </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> x </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> y </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arctanh /> <ci> x </ci> </apply> <apply> <arccoth /> <ci> y </ci> </apply> </apply> <apply> <plus /> <apply> <arctanh /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <ci> y </ci> </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 /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <lt /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <ci> x </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> y </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTanh", "[", "x_", "]"]], "+", RowBox[List["ArcCoth", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], RowBox[List["x", "+", "y"]]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Sign", "[", RowBox[List["x", "+", "y"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["y", "<", RowBox[List["-", "1"]]]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "x", "]"]], "<", "1"]], "&&", RowBox[List["y", "<", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["x", ">", "1"]], "&&", RowBox[List["0", "<", "y", "<", "1"]]]], ")"]]]]]]]]]]

2001-10-29