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

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

<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> coth </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> <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> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <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> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> y </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> y </mi> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arccoth /> <ci> x </ci> </apply> <apply> <arctanh /> <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 /> <pi /> <imaginaryi /> <apply> <ci> Sign </ci> <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> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <gt /> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <and /> <apply> <lt /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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

2001-10-29