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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/05.18.06.0002.01









  


  








Input Form





ZernikeR[n, m, z] \[Proportional] Cos[((n - m)/2) Pi] (Gamma[(2 + m + n)/2]/(Gamma[(2 - m + n)/2] m!)) Sum[(1/k!) Sum[(Binomial[k, i] Pochhammer[m - k + i + 1, k - i] Subscript[z, 0]^(m - k) Sum[(Pochhammer[2 j - i + 1, 2 (i - j)]/ (i - j)!) (2 Subscript[z, 0])^(2 j) ((Pochhammer[(2 + m + n)/2, j] Pochhammer[(m - n)/2, j])/Pochhammer[1 + m, j]) Hypergeometric2F1[(2 + m + n)/2 + j, (m - n)/2 + j, 1 + m + j, Subscript[z, 0]^2] (z - Subscript[z, 0])^k, {j, 0, i}])/2^i, {i, 0, k}], {k, 0, Infinity}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0 && n >= m








Standard Form





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MathML Form







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</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> <times /> <apply> <cos /> <apply> <times /> <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> <pi /> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <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> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <factorial /> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> k </ci> <ci> i </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> i </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> j </ci> <apply> <times /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> j </ci> <ci> m </ci> <cn type='integer'> 1 </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> <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> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </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>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ZernikeR", "[", RowBox[List["n_", ",", "m_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "k", "+", "i", "+", "1"]], ",", RowBox[List["k", "-", "i"]]]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", "i"]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["m", "-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "i"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "j"]], "-", "i", "+", "1"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["i", "-", "j"]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["2", " ", "j"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]]]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["m", "-", "n"]], "2"], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "+", "n"]], ")"]]]], "+", "j"]], ",", RowBox[List[FractionBox[RowBox[List["m", "-", "n"]], "2"], "+", "j"]], ",", RowBox[List["1", "+", "m", "+", "j"]], ",", SubsuperscriptBox["zz", "0", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["i", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m"]], ",", "j"]], "]"]]]]]]]]]]], RowBox[List["k", "!"]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "m", "+", "n"]], ")"]]]], "]"]], " ", RowBox[List["m", "!"]]]]], "/;", 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"]]]]]]]]]]


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





2007-05-02