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Root






Mathematica Notation

Traditional Notation









Elementary Functions > Root[polynomialInzk] > Representations through equivalent functions > With related functions





http://functions.wolfram.com/01.33.27.0003.01









  


  










Input Form





Sum[Subscript[a, j] z^j, {j, 0, n}] == Product[z - Subscript[\[Alpha], k], {k, 1, n}] /; Subscript[a, 0] == (-1)^n Product[Subscript[\[Alpha], k], {k, 1, n}] && Subscript[a, 1] == (-1)^(n - 1) Sum[Product[If[j != k, Subscript[\[Alpha], j], 1], {j, 1, n}], {k, 1, n}] && \[Ellipsis] && Subscript[a, n - 2] == Sum[If[j != k, Subscript[\[Alpha], j] Subscript[\[Alpha], k], 0], {k, 1, n}, {j, 1, k - 1}] && Subscript[a, n - 1] == -Sum[Subscript[\[Alpha], k], {k, 1, n}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SubscriptBox["a", "j"], " ", SuperscriptBox["z", "j"]]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["\[Alpha]", "k"]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "0"], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SubscriptBox["\[Alpha]", "k"]]]]]]], "\[And]", RowBox[List[SubscriptBox["a", "1"], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[NotEqual]", "k"]], ",", SubscriptBox["\[Alpha]", "j"], ",", "1"]], "]"]]]]]]]]]], "\[And]", "\[Ellipsis]", "\[And]", RowBox[List[SubscriptBox["a", RowBox[List["n", "-", "2"]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[NotEqual]", "k"]], ",", RowBox[List[SubscriptBox["\[Alpha]", "j"], " ", SubscriptBox["\[Alpha]", "k"]]], ",", "0"]], "]"]]]]]]]], "\[And]", RowBox[List[SubscriptBox["a", RowBox[List["n", "-", "1"]]], "\[Equal]", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], SubscriptBox["\[Alpha]", "k"]]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> j </mi> </msup> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> If </mi> <mo> [ </mo> <mrow> <mrow> <mi> j </mi> <mo> &#8800; </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> &#945; </mi> <mi> j </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ] </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mo> &#8230; </mo> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> If </mi> <mo> [ </mo> <mrow> <mrow> <mi> j </mi> <mo> &#8800; </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <msub> <mi> &#945; </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ] </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <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> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> If </ci> <apply> <neq /> <ci> j </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <ci> &#8230; </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> If </ci> <apply> <neq /> <ci> j </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j_", "=", "0"]], "n_"], RowBox[List[SubscriptBox["a_", "j_"], " ", SuperscriptBox["z_", "j_"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["\[Alpha]", "k"]]], ")"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "0"], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SubscriptBox["\[Alpha]", "k"]]]]]]], "&&", RowBox[List[SubscriptBox["a", "1"], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[NotEqual]", "k"]], ",", SubscriptBox["\[Alpha]", "j"], ",", "1"]], "]"]]]]]]]]]], "&&", "\[Ellipsis]", "&&", RowBox[List[SubscriptBox["a", RowBox[List["n", "-", "2"]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[NotEqual]", "k"]], ",", RowBox[List[SubscriptBox["\[Alpha]", "j"], " ", SubscriptBox["\[Alpha]", "k"]]], ",", "0"]], "]"]]]]]]]], "&&", RowBox[List[SubscriptBox["a", RowBox[List["n", "-", "1"]]], "\[Equal]", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], SubscriptBox["\[Alpha]", "k"]]]]]]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29





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