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Cyclotomic






Mathematica Notation

Traditional Notation









Polynomials > Cyclotomic[n,z] > Primary definition





http://functions.wolfram.com/05.11.02.0001.01









  


  










Input Form





Cyclotomic[n, z] == Product[(z - E^((2 Pi I k)/n))^ KroneckerDelta[GCD[k, n], 1], {k, 1, n}] /; Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Cyclotomic", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k"]], "n"]]]], ")"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "n"]], "]"]], ",", "1"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]










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> <msub> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, Cyclotomic] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Cyclotomic </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> k </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <gcd /> <ci> k </ci> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cyclotomic", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k"]], "n"]]]], ")"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "n"]], "]"]], ",", "1"]], "]"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2001-10-29