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ZernikeR






Mathematica Notation

Traditional Notation









Polynomials > ZernikeR[n,m,z] > Series representations > Generalized power series > Expansions at generic point z==z0





http://functions.wolfram.com/05.18.06.0001.01









  


  










Input Form





ZernikeR[n, m, z] \[Proportional] ZernikeR[n, m, Subscript[z, 0]] + (1/(Subscript[z, 0] (-1 + Subscript[z, 0]^2))) ((-(m + (2 + n) Subscript[z, 0]^2)) ZernikeR[n, m, Subscript[z, 0]] + (2 + m + n) Subscript[z, 0] ZernikeR[1 + n, 1 + m, Subscript[z, 0]]) (z - Subscript[z, 0]) + (1/(2 Subscript[z, 0]^2 (-1 + Subscript[z, 0]^2)^2)) ((m^2 + Subscript[z, 0]^2 (2 + n + 6 Subscript[z, 0]^2 + 5 n Subscript[z, 0]^2 + n^2 Subscript[z, 0]^2) + m (-1 + (7 + 2 n) Subscript[z, 0]^2)) ZernikeR[n, m, Subscript[z, 0]] - (2 + m + n) Subscript[z, 0] ((1 + 2 m + (7 + 2 n) Subscript[z, 0]^2) ZernikeR[1 + n, 1 + m, Subscript[z, 0]] - (4 + m + n) Subscript[z, 0] ZernikeR[2 + n, 2 + m, Subscript[z, 0]])) (z - Subscript[z, 0])^2 + \[Ellipsis] /; (z -> Subscript[z, 0]) && Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0 && n >= m










Standard Form





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MathML Form







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<mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8805; </mo> <mi> m </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> ZernikeR </ci> <ci> n </ci> <ci> m </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> ZernikeR </ci> <ci> n </ci> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <ci> ZernikeR </ci> <ci> n </ci> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> ZernikeR </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <ci> m </ci> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <ci> n </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> ZernikeR </ci> <ci> n </ci> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> 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Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> <apply> <geq /> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ZernikeR", "[", RowBox[List["n_", ",", "m_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", SubscriptBox["zz", "0"]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["m", "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", SubsuperscriptBox["zz", "0", "2"]]]]], ")"]]]], " ", RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", SubscriptBox["zz", "0"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]], " ", SubscriptBox["zz", "0"], " ", RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "+", "m"]], ",", SubscriptBox["zz", "0"]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], RowBox[List[SubscriptBox["zz", "0"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["zz", "0", "2"]]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "+", RowBox[List[SubsuperscriptBox["zz", "0", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "n", "+", RowBox[List["6", " ", SubsuperscriptBox["zz", "0", "2"]]], "+", RowBox[List["5", " ", "n", " ", SubsuperscriptBox["zz", "0", "2"]]], "+", RowBox[List[SuperscriptBox["n", "2"], " ", SubsuperscriptBox["zz", "0", "2"]]]]], ")"]]]], "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["7", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", SubsuperscriptBox["zz", "0", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", SubscriptBox["zz", "0"]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]], " ", SubscriptBox["zz", "0"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]], "+", RowBox[List[RowBox[List["(", RowBox[List["7", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", SubsuperscriptBox["zz", "0", "2"]]]]], ")"]], " ", RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "+", "m"]], ",", SubscriptBox["zz", "0"]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["4", "+", "m", "+", "n"]], ")"]], " ", SubscriptBox["zz", "0"], " ", RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "m"]], ",", SubscriptBox["zz", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], RowBox[List["2", " ", SubsuperscriptBox["zz", "0", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["zz", "0", "2"]]], ")"]], "2"]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[GreaterEqual]", "m"]]]]]]]]]]










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





2007-05-02





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