Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Cyclotomic




Mathematica Notation

Traditional Notation









Polynomials > Cyclotomic[n,z] > Differentiation > Fractional integro-differentiation





http://functions.wolfram.com/05.11.20.0004.01









  


  








Input Form





D[Cyclotomic[n, z], {z, \[Alpha]}] == Sum[((a[n, j] (EulerPhi[n] - j)!)/Gamma[EulerPhi[n] - j - \[Alpha] + 1]) z^(EulerPhi[n] - j - \[Alpha]), {j, 0, EulerPhi[n]}] /; a[n, j] == (-(1/j)) (MoebiusMu[n] Sum[a[n, m] MoebiusMu[GCD[n, j - m]] EulerPhi[GCD[n, j - m]], {m, 0, j - 1}]) && a[n, 0] == 1 && !(Element[Sqrt[n], Integers] && n >= 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["Cyclotomic", "[", RowBox[List["n", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["EulerPhi", "[", "n", "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "j"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["EulerPhi", "[", "n", "]"]], "-", "j"]], ")"]], "!"]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["EulerPhi", "[", "n", "]"]], "-", "j", "-", "\[Alpha]", "+", "1"]], "]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["EulerPhi", "[", "n", "]"]], "-", "j", "-", "\[Alpha]"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "j"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "j"]]], RowBox[List["(", RowBox[List[RowBox[List["MoebiusMu", "[", "n", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "m"]], "]"]], " ", RowBox[List["MoebiusMu", "[", RowBox[List["GCD", "[", RowBox[List["n", ",", RowBox[List["j", "-", "m"]]]], "]"]], "]"]], " ", RowBox[List["EulerPhi", "[", RowBox[List["GCD", "[", RowBox[List["n", ",", RowBox[List["j", "-", "m"]]]], "]"]], "]"]]]]]]]], ")"]]]]]], "\[And]", RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "0"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["Sqrt", "[", "n", "]"]], ",", "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> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <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> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </munderover> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mi> j </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mi> j </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <semantics> <mi> &#956; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Mu]&quot;, MoebiusMu] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mi> j </mi> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> a </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#956; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Mu]&quot;, MoebiusMu] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msqrt> <mi> n </mi> </msqrt> <mo> &#8713; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ci> Cyclotomic </ci> <ci> n </ci> <ci> z </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> a </ci> <ci> n </ci> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> EulerPhi </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> a </ci> <ci> n </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> MoebiusMu </ci> <ci> n </ci> </apply> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> a </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <ci> MoebiusMu </ci> <apply> <gcd /> <ci> n </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <ci> EulerPhi </ci> <apply> <gcd /> <ci> n </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> a </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <notin /> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Cyclotomic", "[", RowBox[List["n_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["EulerPhi", "[", "n", "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["EulerPhi", "[", "n", "]"]], "-", "j"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["EulerPhi", "[", "n", "]"]], "-", "j", "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["EulerPhi", "[", "n", "]"]], "-", "j", "-", "\[Alpha]", "+", "1"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "j"]], "]"]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["MoebiusMu", "[", "n", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "m"]], "]"]], " ", RowBox[List["MoebiusMu", "[", RowBox[List["GCD", "[", RowBox[List["n", ",", RowBox[List["j", "-", "m"]]]], "]"]], "]"]], " ", RowBox[List["EulerPhi", "[", RowBox[List["GCD", "[", RowBox[List["n", ",", RowBox[List["j", "-", "m"]]]], "]"]], "]"]]]]]]]], "j"]]]]], "&&", RowBox[List[RowBox[List["a", "[", RowBox[List["n", ",", "0"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["n"], "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]


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





2001-10-29