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

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<mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> R </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#8290; </mo> <mrow> <msubsup> <mi> R </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> m </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> R </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> R </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#8290; </mo> <mrow> <msubsup> <mi> R </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <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 /> <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> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 4 </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'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> 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>

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2007-05-02