ArcCoth[z] == ArcCoth[Subscript[z, 0]] + (1/2) (Floor[Arg[-z + Subscript[z, 0]]/(2 Pi)] (Log[(1 + Subscript[z, 0])/Subscript[z, 0]] + Log[Subscript[z, 0]/(1 + Subscript[z, 0])]) - Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] (Log[(-1 + Subscript[z, 0])/Subscript[z, 0]] + Log[Subscript[z, 0]/(-1 + Subscript[z, 0])]) + Sum[(1/(k (1 - Subscript[z, 0]^2)^k)) Sum[Binomial[k, j] (1 + (-1)^(k + j - 1)) Subscript[z, 0]^j (z - Subscript[z, 0])^k, {j, 0, n}], {k, 1, Infinity}])

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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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> k </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;j&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; 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2007-05-02