Sign[Coth[z]] == (Sinh[2 Re[z]] - I Sin[2 Im[z]])/ Sqrt[Cosh[2 Re[z]]^2 - Cos[2 Im[z]]^2]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <coth /> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", RowBox[List["Coth", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]], "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "2"], "-", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]], "]"]], "2"]]]]]]]]]

2007-05-02