Root[Function[z, Sum[Subscript[a, j] z^j, {j, 0, n}]], k] == Limit[Subscript[w, k]^m, m -> Infinity] /; Subscript[w, k]^(m + 1) == Subscript[w, k]^m - Sum[Subscript[a, j] ((Subscript[w, k]^m)^j/ Product[If[i != j, Subscript[w, i]^m - Subscript[w, j]^m, 1], {i, 1, m}, {j, 1, m}]), {j, 0, n}] && Element[Subscript[w, k]^0, Complexes] && Subscript[w, k]^0 != Subscript[w, l]^0

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</mo> <mrow> <msubsup> <mi> w </mi> <mi> k </mi> <mi> m </mi> </msubsup> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msubsup> <mi> w </mi> <mi> k </mi> <mi> m </mi> </msubsup> <mo> ) </mo> </mrow> <mi> j </mi> </msup> </mrow> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mi> If </mi> <mo> [ </mo> <mrow> <mrow> <mi> i </mi> <mo> &#8800; </mo> <mi> j </mi> </mrow> <mo> , </mo> <mrow> <msubsup> <mi> w </mi> <mi> i </mi> <mi> m </mi> </msubsup> <mo> - </mo> <msubsup> <mi> w </mi> <mi> j </mi> <mi> m </mi> </msubsup> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ] </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msubsup> <mi> w </mi> <mi> k </mi> <mn> 0 </mn> </msubsup> <mo> &#8712; </mo> <semantics> <mi> &#8450; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalC]&quot;, Function[Complexes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msubsup> <mi> w </mi> <mi> k </mi> <mn> 0 </mn> </msubsup> <mo> &#8800; </mo> <msubsup> <mi> w </mi> <mi> l </mi> <mn> 0 </mn> </msubsup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> CompoundExpression </ci> <ci> z </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <power /> <ci> z </ci> <ci> j </ci> </apply> </apply> </apply> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <limit /> <bvar> <ci> m </ci> </bvar> <condition> <apply> <tendsto /> <ci> m </ci> <infinity /> </apply> </condition> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> k </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> k </ci> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> k </ci> </apply> <ci> m </ci> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <product /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> If </ci> <apply> <neq /> <ci> i </ci> <ci> j </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> i </ci> </apply> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> j </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> k </ci> </apply> <cn type='integer'> 0 </cn> </apply> <complexes /> </apply> <apply> <neq /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> k </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> l </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29