html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ZernikeR

 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

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

 MathML Form

 R n m ( z ) cos ( n - m 2 π ) ( - 1 ) n - m 2 Γ ( m + n + 2 2 ) Γ ( 2 - m + n 2 ) m ! k = 0 1 k ! i = 0 k ( k i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( i - k + m + 1 ) k - i TagBox[SubscriptBox[RowBox[List["(", RowBox[List["i", "-", "k", "+", "m", "+", "1"]], ")"]], RowBox[List["k", "-", "i"]]], Pochhammer] 2 - i z 0 m - k j = 0 i ( 1 - i + 2 j ) 2 ( i - j ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "i", "+", RowBox[List["2", " ", "j"]]]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["i", "-", "j"]], ")"]]]]], Pochhammer] ( m + n + 2 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["m", "+", "n", "+", "2"]], "2"], ")"]], "j"], Pochhammer] ( m - n 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["m", "-", "n"]], "2"], ")"]], "j"], Pochhammer] ( i - j ) ! ( m + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "j"], Pochhammer] ( 2 z 0 ) 2 j 2 F 1 ( j + m + n + 2 2 , j + m - n 2 ; j + m + 1 ; z 0 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", FractionBox[RowBox[List["m", "+", "n", "+", "2"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["j", "+", FractionBox[RowBox[List["m", "-", "n"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["j", "+", "m", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SubsuperscriptBox["z", "0", "2"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ( z - z 0 ) k /; n m n m Condition ZernikeR n m z n -1 m 2 -1 -1 n -1 m 2 -1 Gamma m n 2 2 -1 Gamma 2 -1 m n 2 -1 m -1 k 0 1 k -1 i 0 k Binomial k i Pochhammer i -1 k m 1 k -1 i 2 -1 i Subscript z 0 m -1 k j 0 i Pochhammer 1 -1 i 2 j 2 i -1 j Pochhammer m n 2 2 -1 j Pochhammer m -1 n 2 -1 j i -1 j Pochhammer m 1 j -1 2 Subscript z 0 2 j Hypergeometric2F1 j m n 2 2 -1 j m -1 n 2 -1 j m 1 Subscript z 0 2 z -1 Subscript z 0 k 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[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