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 ZernikeR

 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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", SubscriptBox["z", "0"]]], "]"]], "+", RowBox[List[FractionBox["1", RowBox[List[SubscriptBox["z", "0"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["m", "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", SubsuperscriptBox["z", "0", "2"]]]]], ")"]]]], " ", RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", SubscriptBox["z", "0"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]], " ", SubscriptBox["z", "0"], " ", RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "+", "m"]], ",", SubscriptBox["z", "0"]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", SubsuperscriptBox["z", "0", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["m", "2"], "+", RowBox[List[SubsuperscriptBox["z", "0", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "n", "+", RowBox[List["6", " ", SubsuperscriptBox["z", "0", "2"]]], "+", RowBox[List["5", " ", "n", " ", SubsuperscriptBox["z", "0", "2"]]], "+", RowBox[List[SuperscriptBox["n", "2"], " ", SubsuperscriptBox["z", "0", "2"]]]]], ")"]]]], "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["7", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", SubsuperscriptBox["z", "0", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ZernikeR", "[", RowBox[List["n", ",", "m", ",", SubscriptBox["z", "0"]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]], " ", SubscriptBox["z", "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["z", "0", "2"]]]]], ")"]], " ", RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["1", "+", "m"]], ",", SubscriptBox["z", "0"]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["4", "+", "m", "+", "n"]], ")"]], " ", SubscriptBox["z", "0"], " ", RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "m"]], ",", SubscriptBox["z", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], " ", "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]], "\[And]", 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"]]]]]]]]

 MathML Form

 R n m ( z ) R n m ( z 0 ) + 1 z 0 ( z 0 2 - 1 ) ( - ( ( n + 2 ) z 0 2 + m ) R n m ( z 0 ) + ( m + n + 2 ) z 0 R n + 1 m + 1 ( z 0 ) ) ( z - z 0 ) + 1 2 z 0 2 ( z 0 2 - 1 ) 2 ( ( m 2 + ( ( 2 n + 7 ) z 0 2 - 1 ) m + z 0 2 ( n 2 z 0 2 + 5 n z 0 2 + 6 z 0 2 + n + 2 ) ) R n m ( z 0 ) - ( m + n + 2 ) z 0 ( ( ( 2 n + 7 ) z 0 2 + 2 m + 1 ) R n + 1 m + 1 ( z 0 ) - ( m + n + 4 ) z 0 R n + 2 m + 2 ( z 0 ) ) ) ( z - z 0 ) 2 + /; ( z "\[Rule]" z 0 ) n m n m Condition Proportional ZernikeR n m z ZernikeR n m Subscript z 0 1 Subscript z 0 Subscript z 0 2 -1 -1 -1 n 2 Subscript z 0 2 m ZernikeR n m Subscript z 0 m n 2 Subscript z 0 ZernikeR n 1 m 1 Subscript z 0 z -1 Subscript z 0 1 2 Subscript z 0 2 Subscript z 0 2 -1 2 -1 m 2 2 n 7 Subscript z 0 2 -1 m Subscript z 0 2 n 2 Subscript z 0 2 5 n Subscript z 0 2 6 Subscript z 0 2 n 2 ZernikeR n m Subscript z 0 -1 m n 2 Subscript z 0 2 n 7 Subscript z 0 2 2 m 1 ZernikeR n 1 m 1 Subscript z 0 -1 m n 4 Subscript z 0 ZernikeR n 2 m 2 Subscript z 0 z -1 Subscript z 0 2 Rule z Subscript z 0 n m n m [/itex]

 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