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http://functions.wolfram.com/01.22.19.0003.01
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Abs[Coth[x + I y]] == Sqrt[-((Cos[2 y] + Cosh[2 x])/(Cos[2 y] - Cosh[2 x]))]
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Cell[BoxData[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Coth", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "y"]], "]"]], "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "x"]], "]"]]]], RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "y"]], "]"]], "-", RowBox[List["Cosh", "[", RowBox[List["2", " ", "x"]], "]"]]]]]]]]]]]]
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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> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> coth </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⩵ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msqrt> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <abs /> <apply> <coth /> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Abs", "[", RowBox[List["Coth", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "y"]], "]"]], "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "x"]], "]"]]]], RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "y"]], "]"]], "-", RowBox[List["Cosh", "[", RowBox[List["2", " ", "x"]], "]"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
