ArcCoth[I Sqrt[1 + 2/Sqrt[5]]] == -((Pi I)/5)

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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> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 2 </mn> <msqrt> <mn> 5 </mn> </msqrt> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 5 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccoth /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCoth", "[", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["2", SqrtBox["5"]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "5"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]]]]]

2001-10-29