ZernikeR[n, m, z] == Piecewise[ {{(1/(z^m Gamma[1 + (n - m)/2])) Derivative[(n - m)/2][ Function[z, z^((m + n)/2) (z - 1)^((n - m)/2)]][z^2], Element[(n - m)/2, Integers]}}, 0] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0 && n >= m

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", "z"]], "]"]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List["{", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["-", "m"]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "m"]], "2"]]], "]"]]], RowBox[List[RowBox[List[RowBox[List["Derivative", "[", FractionBox[RowBox[List["n", "-", "m"]], "2"], "]"]], "[", RowBox[List["Function", "[", RowBox[List["z", ",", " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", FractionBox[RowBox[List["m", "+", "n"]], "2"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], FractionBox[RowBox[List["n", "-", "m"]], "2"]]]], ")"]]]], "]"]], "]"]], "[", SuperscriptBox["z", "2"], "]"]]]], ",", RowBox[List[FractionBox[RowBox[List["n", "-", "m"]], "2"], "\[Element]", "Integers"]]]], "}"]], "}"]], ",", "0"]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "m"]]]]]]]]

<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> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> &#62305; </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mrow> <mi> Function </mi> <mo> [ </mo> <mrow> <mi> z </mi> <mo> , </mo> <mrow> <msup> <mi> z </mi> <mfrac> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> </mrow> <mo> ] </mo> </mrow> <semantics> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, FractionBox[RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;m&quot;]], &quot;2&quot;], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <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> <eq /> <apply> <ci> ZernikeR </ci> <ci> n </ci> <ci> m </ci> <ci> z </ci> </apply> <apply> <ci> Piecewise </ci> <list> <list> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> D </ci> <apply> <apply> <ci> Function </ci> <ci> z </ci> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <list> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> </apply> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </list> </list> </apply> </apply> <apply> <and /> <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>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ZernikeR", "[", RowBox[List["n_", ",", "m_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "m"]]], " ", RowBox[List[SuperscriptBox[RowBox[List["Function", "[", RowBox[List["z", ",", RowBox[List[SuperscriptBox["z", FractionBox[RowBox[List["m", "+", "n"]], "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], FractionBox[RowBox[List["n", "-", "m"]], "2"]]]]]], "]"]], TagBox[RowBox[List["(", FractionBox[RowBox[List["n", "-", "m"]], "2"], ")"]], Derivative], Rule[MultilineFunction, None]], "[", SuperscriptBox["z", "2"], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "m"]], "2"]]], "]"]]], RowBox[List[FractionBox[RowBox[List["n", "-", "m"]], "2"], "\[Element]", "Integers"]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List[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"]]]]]]]]]]

2007-05-02