Log[f[z]] \[Proportional] 2 I Pi Floor[(Pi - Arg[Subscript[c, 0]] - Arg[f[z]/Subscript[c, 0]])/ (2 Pi)] + Log[Subscript[c, 0]] + (Subscript[c, 1]/Subscript[c, 0]) z + (Subscript[c, 2]/Subscript[c, 0] - Subscript[c, 1]^2/ (2 Subscript[c, 0]^2)) z^2 + (Subscript[c, 1]^3/(3 Subscript[c, 0]^3) - (Subscript[c, 1] Subscript[c, 2])/Subscript[c, 0]^2 + Subscript[c, 3]/Subscript[c, 0]) z^3 + \[Ellipsis] /; (z -> 0) && f[z] == Sum[Subscript[c, k] z^k, {k, 0, Infinity}] && Subscript[c, 0] != 0

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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> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <msub> <mi> c </mi> <mn> 0 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> c </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> c </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <msub> <mi> c </mi> <mn> 0 </mn> </msub> </mfrac> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <msub> <mi> c </mi> <mn> 0 </mn> </msub> </mfrac> <mo> - </mo> <mfrac> <msubsup> <mi> c </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msubsup> <mi> c </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <msubsup> <mi> c </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msubsup> <mi> c </mi> <mn> 0 </mn> <mn> 3 </mn> </msubsup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 1 </mn> </msub> </mrow> <msubsup> <mi> c </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mfrac> <mo> + </mo> <mfrac> <msub> <mi> c </mi> <mn> 3 </mn> </msub> <msub> <mi> c </mi> <mn> 0 </mn> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> c </mi> <mn> 0 </mn> </msub> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ln /> <apply> <ci> f </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <times /> <apply> <ci> f </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <ci> &#8230; 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2007-05-02