ArcCsch[(z^2 - 1)/(2 z)] == (2 ArcCot[I/z^(((I + z)/(I - z)) Sqrt[((I - z)/(I + z))^2])])/ I^(((I + z)/(I - z)) Sqrt[((I - z)/(I + z))^2])

Cell[BoxData[RowBox[List[RowBox[List["ArcCsch", "[", FractionBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], RowBox[List["2", " ", "z"]]], "]"]], "\[Equal]", RowBox[List["2", SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", "+", "z"]], RowBox[List["\[ImaginaryI]", "-", "z"]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], RowBox[List["\[ImaginaryI]", "+", "z"]]], ")"]], "2"]]]]], RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", "+", "z"]], RowBox[List["\[ImaginaryI]", "-", "z"]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], RowBox[List["\[ImaginaryI]", "+", "z"]]], ")"]], "2"]]]]]]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8520; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> &#8520; </mi> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> - </mo> <mi> z </mi> </mrow> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mrow> </msup> <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> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> &#8520; </mi> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> - </mo> <mi> z </mi> </mrow> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsch /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <imaginaryi /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <arccot /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsch", "[", FractionBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]], RowBox[List["2", " ", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["2", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", "z"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], RowBox[List["\[ImaginaryI]", "+", "z"]]], ")"]], "2"]]]], RowBox[List["\[ImaginaryI]", "-", "z"]]]]]], " ", RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", "z"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], RowBox[List["\[ImaginaryI]", "+", "z"]]], ")"]], "2"]]]], RowBox[List["\[ImaginaryI]", "-", "z"]]]]]]]], "]"]]]]]]]]

2003-08-21