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ZernikeR






Mathematica Notation

Traditional Notation









Polynomials > ZernikeR[n,m,z] > Specific values > Specialized values > For fixed n, z





http://functions.wolfram.com/05.18.03.0012.01









  


  










Input Form





ZernikeR[n, n - 6, z] == (1/6) (n - 2) (n - 1) n z^n - (1/2) (n - 3) (n - 2) (n - 1) z^(n - 2) + (1/2) (n - 4) (n - 3) (n - 2) z^(n - 4) - (1/6) (n - 5) (n - 4) (n - 3) z^(n - 6) /; Element[n, Integers] && n >= 6










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ZernikeR", "[", RowBox[List["n", ",", RowBox[List["n", "-", "6"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", SuperscriptBox["z", "n"]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "3"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "2"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "4"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "3"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "4"]]]]], "-", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["n", "-", "5"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "4"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "3"]], ")"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "6"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "6"]]]]]]]]










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> <msubsup> <mi> R </mi> <mi> n </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 6 </mn> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 6 </mn> </mrow> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8805; </mo> <mn> 6 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> ZernikeR </ci> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -6 </cn> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -4 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -5 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -4 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -3 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -6 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <geq /> <ci> n </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ZernikeR", "[", RowBox[List["n_", ",", RowBox[List["n_", "-", "6"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", SuperscriptBox["z", "n"]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "3"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "2"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "4"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "3"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "4"]]]]], "-", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["n", "-", "5"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "4"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "3"]], ")"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "6"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "6"]]]]]]]]]]










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





2007-05-02





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