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ZernikeR






Mathematica Notation

Traditional Notation









Polynomials > ZernikeR[n,m,z] > Differentiation > Low-order differentiation





http://functions.wolfram.com/05.18.20.0002.01









  


  










Input Form





D[ZernikeR[n, m, z], z] == ((m^2 + n^2 - 2 n^2 z^2)/(2 n z (1 - z^2))) ZernikeR[n, m, z] - (((n + m) (n - m))/(2 n z (z^2 - 1))) ZernikeR[n - 2, m, z] /; Element[n, Integers] && n >= 2 && Element[m, Integers] && m >= 0 && m <= n










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["m", "2"], "+", SuperscriptBox["n", "2"], "-", RowBox[List["2", " ", SuperscriptBox["n", "2"], " ", SuperscriptBox["z", "2"]]]]], RowBox[List["2", " ", "n", " ", "z", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], " "]]], RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]]]], RowBox[List["2", " ", "n", " ", "z", " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]]]]], RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["n", "-", "2"]], ",", "m", ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "2"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[LessEqual]", "n"]]]]]]]]










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> <mo> &#8706; </mo> <mrow> <msubsup> <mi> R </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> R </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> R </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </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> 2 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> ZernikeR </ci> <ci> n </ci> <ci> m </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ZernikeR </ci> <ci> n </ci> <ci> m </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <ci> z </ci> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ZernikeR </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <ci> m </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <geq /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> <apply> <leq /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], RowBox[List["ZernikeR", "[", RowBox[List["n_", ",", "m_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "+", SuperscriptBox["n", "2"], "-", RowBox[List["2", " ", SuperscriptBox["n", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", "z"]], "]"]]]], RowBox[List["2", " ", "n", " ", "z", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]]]], ")"]], " ", RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["n", "-", "2"]], ",", "m", ",", "z"]], "]"]]]], RowBox[List["2", " ", "n", " ", "z", " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "2"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[LessEqual]", "n"]]]]]]]]]]










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





2007-05-02





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