ArcTanh[Sqrt[1 - z]] == ArcCosh[1/Sqrt[z]]

Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List["ArcTanh", "[", SqrtBox[RowBox[List["1", "-", "z"]]], "]"]], "\[Equal]", RowBox[List["ArcCosh", "[", FractionBox["1", SqrtBox["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> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> z </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arctanh /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <arccosh /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", SqrtBox[RowBox[List["1", "-", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["ArcCosh", "[", FractionBox["1", SqrtBox["z"]], "]"]]]]]]

2003-08-21