Coth[z] == 1/z + (1/z) ContinueFraction[{z^2, 1 + 2 k}, {k, 1, Infinity}]

Cell[BoxData[RowBox[List[RowBox[List["Coth", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["1", "z"], "+", RowBox[List[FractionBox["1", "z"], RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[SuperscriptBox["z", "2"], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <msubsup> <mrow> <msub> <mi> &#922; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> &#8734; </mi> </msubsup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <coth /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <apply> <ci> Subscript </ci> <ci> &#922; </ci> <ci> k </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <infinity /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Coth", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "z"], "+", FractionBox[RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[SuperscriptBox["z", "2"], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]], "z"]]]]]]]

Andreas Lauschke (2006)

2007-05-02