ArcTanh[1/z] == I ArcCot[I z]

Cell[BoxData[RowBox[List[RowBox[List["ArcTanh", "[", FractionBox["1", "z"], "]"]], "\[Equal]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arctanh /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <arccot /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", FractionBox["1", "z_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]]]]

2003-08-21