(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 303660, 6418]*) (*NotebookOutlinePosition[ 351948, 7973]*) (* CellTagsIndexPosition[ 351904, 7969]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Truncation errors in Legendre analysis of ", Cell[BoxData[ \(TraditionalForm\`N \[Rule] \[CapitalDelta]\)]], " quadrupole ratios" }], "Title", TextAlignment->Center, TextJustification->0], Cell["James J. Kelly", "Author", TextAlignment->Center, TextJustification->0], Cell["\<\ Department of Physics University of Maryland College Park, MD 20742 jjkelly@physics.umd.edu\ \>", "Address", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Most previous analyses of ", Cell[BoxData[ \(TraditionalForm\`EMR = Re[E\_\(\(1\)\(+\)\)/M\_\(\(1\)\(+\)\)]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`SMR\ = \ Re[S\_\(\(1\)\(+\)\)/M\_\(\(1\)\(+\)\)]\)]], " fit Legendre expansions to unpolarized cross sections, usually without \ Rosenbluth separation, and extract EMR and SMR from those coefficients \ assuming ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance and truncation to ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL]\_\[Pi] \[LessEqual] 1\)]], ". We derive more complete multipole expansions and test the accuracy of \ those approximations using multipole amplitudes from a realistic \ phenomenological model." }], "Abstract"], Cell[TextData[StyleBox["Created: July 31, 1998\nLast revised: July 5, 2005", FontSlant->"Italic"]], "Text", TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell["Introduction", "SectionFirst"], Cell[TextData[{ "\tThere has been long-standing interest in deformed components of the ", Cell[BoxData[ \(TraditionalForm\`N\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\)]], " wave functions. The dominant amplitude for pion electroproduction of the \ ", Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]\)]], " resonance is the ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " multipole, but smaller ", Cell[BoxData[ \(TraditionalForm\`S\_\(\(1\)\(+\)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`E\_\(\(1\)\(+\)\)\)]], " multipole amplitudes arise from configuration mixing within the quark \ core, meson or gluon exchange currents between quarks, or coupling to the \ pion cloud outside the quark core. Those mechanisms are not entirely \ independent, but recent models attempt to distinguish between the intrinsic \ deformation of the wave functions and dynamical effects. It is important to \ deduce accurate quadrupole ratios, ", Cell[BoxData[ \(TraditionalForm\`EMR = Re[E\_\(\(1\)\(+\)\)/M\_\(\(1\)\(+\)\)]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`SMR\ = \ Re[S\_\(\(1\)\(+\)\)/M\_\(\(1\)\(+\)\)]\)]], ", from pion electroproduction data with as little model dependence as \ possible. Most previous analyses relied upon the assumptions of ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance and ", StyleBox["sp", FontSlant->"Italic"], " truncation to relate the quadrupole amplitudes to Legendre coefficients \ fitted to cross section angular distributions. More recently, a recoil \ polarization experiment at Jefferson Laboratory measured angular \ distributions for 16 independent response functions and extracted complex \ multipole amplitudes in a very nearly model-independent analysis. Therefore, \ the quadrupole amplitudes for the ", Cell[BoxData[ \(TraditionalForm\`p\[VeryThinSpace]\[Pi]\^0\)]], " channel at ", Cell[BoxData[ \(TraditionalForm\`Q\^2 = 1\)]], " ", Cell[BoxData[ \(TraditionalForm\`\((GeV/c)\)\^2\)]], " were obtained without need of either assumption of the traditional \ Legendre analysis, both of which proved unreliable. In this report we \ examine in some detail the accuracy of the traditional Legendre analysis." }], "Text"], Cell[TextData[{ "\tThe unpolarized differential cross section in the ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\[VeryThinSpace]N\)]], " center of mass can be expressed in the form " }], "Text"], Cell[BoxData[ \(\[Sigma]\ = \ \(\[Nu]\_0\) \((\ \[Epsilon]\_S\ R\_L\ + \ R\_T\ + \ \(\@\(2 \( \[Epsilon]\_S\) \((1 + \[Epsilon])\)\)\) \ \(R\_LT\) Sin[\[Theta]] Cos[\[Phi]]\ + \ \[Epsilon]\ \(R\_TT\) \(Sin[\[Theta]]\^2\) Cos[2 \[Phi]]\ )\)\)], "DisplayFormula", FontFamily->"Times New Roman"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[Nu]\_0\)]], " is a phase-space factor, \[Epsilon] is the transverse polarization of the \ virtual photon, ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\_S = \[Epsilon]\ Q\^2/q\^2\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\((\[Theta], \[Phi])\)\)]], " are polar and azimuthal pion angles relative to the ", Cell[BoxData[ \(TraditionalForm\`\(q\& \[RightVector] \)\)]], " vector and scattering plane. The response functions can be expanded in \ Legendre series as" }], "Text"], Cell[BoxData[{ \(R\_L = \ \[Sum]\+\(n = 0\)\%\[Infinity]\( A\_n\%L\) P\_n[x]\), "\[IndentingNewLine]", \(R\_T = \ \[Sum]\+\(n = 0\)\%\[Infinity]\( A\_n\%T\) P\_n[x]\), "\[IndentingNewLine]", \(R\_LT = \ \[Sum]\+\(n = 0\)\%\[Infinity]\( A\_n\%LT\) P\_n[x]\), "\[IndentingNewLine]", \(R\_TT = \ \[Sum]\+\(n = 0\)\%\[Infinity]\( A\_n\%TT\) P\_n[x]\)}], "DisplayFormula", FontFamily->"Times New Roman"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`x = Cos[\[Theta]]\)]], " and where the expansion coefficients ", Cell[BoxData[ \(TraditionalForm\`A\_n\%\[Lambda]\)]], " are functions of ", Cell[BoxData[ \(TraditionalForm\`\((W, Q\^2)\)\)]], " that can be fit to cross section data. Each of the Legendre coefficients \ can in turn be expressed in terms of multipole amplitudes of the form ", Cell[BoxData[ \(TraditionalForm\`Re[\(B\_\(\(\[ScriptL]\)\(\[PlusMinus]\)\)\) \(C\_\(\ \(\[ScriptL]\)\(\[PlusMinus]\)\)\^*\)]\)]], " where ", Cell[BoxData[ \(TraditionalForm\`B, C \[Element] {M, E, S}\)]], " are magnetic, electric, or scalar multipole amplitudes for specified ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`j = \[ScriptL] \[PlusMinus] 1\/2\)]], ". The traditional Legendre analysis truncates those expansions by \ requiring ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL] \[LessEqual] 1\)]], ", described as ", StyleBox["sp", FontSlant->"Italic"], " truncation, and drops any contribution that does not contain ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], ", described as ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance. Thus, one obtains the traditional formulas" }], "Text"], Cell[BoxData[{ \(\(R\&~\)\_EM\ = \(\(\ \)\(3 A\_2\%\(L + T\) - 2 A\_0\%TT\)\)\/\(12 \ A\_0\%\(L + T\)\)\), "\[IndentingNewLine]", \(\(R\&~\)\_SM\ = \(\(\ \)\(A\_1\%LT\)\)\/\(3 A\_0\%\(L + T\)\)\)}], \ "DisplayFormula", FontFamily->"Times New Roman"], Cell[TextData[{ "where the tildes identify these as approximations and where ", Cell[BoxData[ \(TraditionalForm\`A\_n\%\(L + T\) = A\_n\%T + \(\[Epsilon]\_S\) A\_n\%L\)]], " is the linear combination used when Rosenbluth separation is not \ available. Note that ", Cell[BoxData[ \(TraditionalForm\`R\_L\)]], " would vanish if ", StyleBox["sp", FontSlant->"Italic"], " truncation and ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance were exact." }], "Text"], Cell[TextData[{ "\tBelow we derive symbolic expressions for those expansions and study the \ effects of various truncation schemes. Much of the code for performing the \ expansions and manipulating the expressions is borrowed from a more \ comprehensive analysis of recoil polarization response functions developed in \ the closely related notebook RecoilPolarization.nb. Therefore, we will not \ explain the details of ", StyleBox["Mathematica", FontSlant->"Italic"], " code here. Most readers will probably want to skip to the numerical \ analysis section without attempting to parse the somewhat intimidating \ intervening code." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Initialization", "Section"], Cell[CellGroupData[{ Cell["Defaults and packages", "Subsection"], Cell[BoxData[{ \(\(ClearAll["\"];\)\), "\n", \(\(Off[General::spell, \ General::spell1];\)\), "\n", \(\($TextStyle = {FontFamily \[Rule] "\", FontSize \[Rule] 12};\)\)}], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \(Needs["\"]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[{ RowBox[{ RowBox[{"Symbolize", "[", TagBox[\(M\_+\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(M\_-\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Symbolize", "[", TagBox[\(E\_+\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(E\_-\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Symbolize", "[", TagBox[\(S\_+\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";", RowBox[{"Symbolize", "[", TagBox[\(S\_-\), NotationBoxTag, TagStyle->"NotationTemplateStyle"], "]"}], ";"}]}], "Input", CellLabel->"In[5]:="] }, Open ]], Cell[CellGroupData[{ Cell["Manipulation of complex quantities", "Subsection"], Cell[BoxData[{ \(\(\(conjugate::usage\ = \n\t"\";\)\(\n\) \)\), "\n", \(\(conjugateRule\ = \ Complex[re_, im_] :> Complex[re, \(-im\)];\)\), "\n", \(\(conjugate[exp__]\ := \ exp /. conjugateRule;\)\)}], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(\(\(ContractAmplitudeProducts = \n\t{\n\t c_. \(A_\_a_\) \(\((A_\_a_)\)\^*\) \[RuleDelayed] c\ Abs[A\_a]\^2, \n\t d_. + c_. \(A_\_a_\) \(\((B_\_b_)\)\^*\) \[RuleDelayed] d + c\ \((Re[\(A\_a\) \(\((B\_b)\)\^*\)] + I\ Im[\(A\_a\) \(\((B\_b)\)\^*\)])\) /; b > a, \n\t d_. + c_. \ \(A_\_a_\) \(\((B_\_b_)\)\^*\) \[RuleDelayed] d + c\ \((Re[\(B\_b\) \(\((A\_a)\)\^*\)] - I\ Im[\(B\_b\) \(\((A\_a)\)\^*\)])\) /; b < a, \n\t d_. \ Re[\(A_\_a_\) \(\((B_\_b_)\)\^*\)] + \ e_. Re[\(C_\_c_\) \(\((B_\_b_)\)\^*\)] \[RuleDelayed] d\ Re[\((A\_a + e\/d\ C\_c)\) \(\((B\_b)\)\^*\)], \n\t d_. \ Im[\(A_\_a_\) \(\((B_\_b_)\)\^*\)] + \ e_. Im[\(C_\_c_\) \(\((B_\_b_)\)\^*\)] \[RuleDelayed] d\ Im[\((A\_a + e\/d\ C\_c)\) \(\((B\_b)\)\^*\)]\n\t};\)\(\t\)\)\)], \ "Input", CellLabel->"In[11]:="], Cell[BoxData[{ \(MyRules = {\(0\^*\) \[Rule] 0}; MyAssumptions = {\[Alpha] \[Element] Reals, \[Beta] \[Element] Reals, \[Gamma] \[Element] Reals, \[Delta] \[Element] Reals, \[Theta] \[Element] Reals, \[Phi] \[Element] Reals, \[Omega] > 0, q > \[Omega], Q > 0, 0 < x < 1};\), "\[IndentingNewLine]", \(\(MySimplify = Simplify[# /. MyRules, MyAssumptions] &;\)\), "\[IndentingNewLine]", \(\(MyFullSimplify = FullSimplify[# /. MyRules, MyAssumptions] &;\)\)}], "Input", CellLabel->"In[12]:="] }, Open ]], Cell[CellGroupData[{ Cell["Multipole expansion of CGLN amplitudes", "Subsection"], Cell[BoxData[ RowBox[{ RowBox[{\(CGLN1[\[ScriptL]_]\), " ", "=", " ", RowBox[{"{", "\n", "\t", RowBox[{ RowBox[{\(\[ScriptCapitalF]\_1\), "\[Rule]", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{\((\[ScriptL]\ \(M\_+\)[\[ScriptL]] + \(E\_+\)[\ \[ScriptL]])\), " ", RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] + 1\), "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], " ", "+", " ", RowBox[{\((\((\[ScriptL] + 1)\) \(M\_-\)[\[ScriptL]] + \(E\_-\)[\ \[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] - 1\), "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_2\), "\[Rule]", RowBox[{"Sum", "[", RowBox[{ RowBox[{\((\((\[ScriptL] + 1)\) \(M\_+\)[\[ScriptL]]\ + \ \[ScriptL]\ \(M\ \_-\)[\[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", "\[ScriptL]", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_3\), "\[Rule]", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{\((\(E\_+\)[\[ScriptL]]\ - \ \ \(M\_+\)[\[ScriptL]])\), " ", RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] + 1\), "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}]}], " ", "+", " ", RowBox[{\((\(E\_-\)[\[ScriptL]] + \(M\_-\)[\[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] - 1\), "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}]}]}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_4\), "\[Rule]", RowBox[{"Sum", "[", RowBox[{ RowBox[{\((\(M\_+\)[\[ScriptL]] - \(E\_+\)[\[ScriptL]] - \ \(M\_-\)[\[ScriptL]] - \(E\_-\)[\[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", "\[ScriptL]", "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}]}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_5\), "\[Rule]", RowBox[{\(\[Omega]\/q\), RowBox[{"Sum", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{\((\[ScriptL] + 1)\), \(\(S\_+\)[\[ScriptL]]\), RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] + 1\), "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], " ", "-", " ", RowBox[{"\[ScriptL]", " ", \(\(S\_-\)[\[ScriptL]]\), RowBox[{ SubsuperscriptBox["P", \(\[ScriptL] - 1\), "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}]}], ")"}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}]}], ",", "\n", "\t", RowBox[{\(\[ScriptCapitalF]\_6\), "\[Rule]", RowBox[{\(\[Omega]\/q\), RowBox[{"Sum", "[", RowBox[{ RowBox[{"(", RowBox[{\((\[ScriptL]\ \(S\_-\)[\[ScriptL]] - \ \((\ \[ScriptL] + 1)\) \(S\_+\)[\[ScriptL]])\), RowBox[{ SubsuperscriptBox["P", "\[ScriptL]", "\[Prime]", MultilineFunction->None], "[", "x", "]"}]}], ")"}], ",", \({\[ScriptL], 0, \[ScriptL]max}\)}], "]"}]}]}]}], "\n", "\t\t", "}"}]}], ";"}]], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \(\(rule[ mp] = {\(M\_-\)[0] \[Rule] 0, \(E\_-\)[0] \[Rule] 0, \(S\_-\)[0] \[Rule] 0, \(M\_+\)[0] \[Rule] 0, \(E\_-\)[1] \[Rule] 0, \(0\^*\) \[Rule] 0};\)\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(\(ExpandAmplitudeProducts = \ {\[IndentingNewLine]Abs[\[ScriptCapitalF]\_a_]\^2 \[Rule] \(\[ScriptCapitalF]\ \_a\) \(\((\[ScriptCapitalF]\_a)\)\^*\), \[IndentingNewLine]Re[\((A_. \ \[ScriptCapitalF]\_a_ + B_. \ \[ScriptCapitalF]\_b_)\)\ \(\((C_. \ \ \[ScriptCapitalF]\_c_\ + \ D_. \[ScriptCapitalF]\_d_)\)\^*\)] \[Rule] A\ C\ Re[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_c)\)\^*\ \)] + A\ D\ Re[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_d)\)\^*\)] + B\ C\ Re[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_c)\)\^*\)] + B\ D\ Re[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_d)\)\^*\)], Im[\((A_. \[ScriptCapitalF]\_a_ + B_. \ \[ScriptCapitalF]\_b_)\)\ \(\((C_. \ \ \[ScriptCapitalF]\_c_\ + \ D_. \[ScriptCapitalF]\_d_)\)\^*\)] \[Rule] A\ C\ Im[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_c)\)\^*\ \)] + A\ D\ Im[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_d)\)\^*\)] + B\ C\ Im[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_c)\)\^*\)] + B\ D\ Im[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_d)\)\^*\)], Re[\((A_. \[ScriptCapitalF]\_a_ + B_. \ \[ScriptCapitalF]\_b_)\)\ \(\((C_. \ \ \[ScriptCapitalF]\_c_)\)\^*\)\ ] \[Rule] A\ C\ Re[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_c)\)\^*\ \)] + B\ C\ Re[\(\[ScriptCapitalF]\_b\) \(\((\[ScriptCapitalF]\_c)\)\^*\)], Im[\((A_. \[ScriptCapitalF]\_a_ + B_. \ \[ScriptCapitalF]\_b_)\)\ \(\((C_. \ \ \[ScriptCapitalF]\_c_)\)\^*\)\ ] \[Rule] A\ C\ Im[\(\[ScriptCapitalF]\_a\) \(\((\[ScriptCapitalF]\_c)\)\^*\ \)] + B\ C\ Im[\(\[ScriptCapitalF]\_b\) \ \(\((\[ScriptCapitalF]\_c)\)\^*\)]};\)\)], "Input", CellLabel->"In[17]:="], Cell[BoxData[ \(\(ExpandCGLN\ = \ Join[{Sum[A_, {\[ScriptL]1_, \[ScriptL]1min_, \[ScriptL]1max_}] Sum[B_, {\[ScriptL]2_, \[ScriptL]2min_, \[ScriptL]2max_}] \ \[RuleDelayed] MySum[Expand[ A\ B], {\[ScriptL]1, \[ScriptL]1min, \[ScriptL]1max}, {\ \[ScriptL]2, \[ScriptL]2min, \[ScriptL]2max}]\ }, CGLN1[\[ScriptL]1], CGLN1[\[ScriptL]2] /. {\[ScriptCapitalF]\_a_ \[Rule] \(\((\ \[ScriptCapitalF]\_a)\)\^*\), \(M\_+\)[ a_] \[Rule] \(\((\(M\_+\)[a])\)\^*\), \(E\_+\)[ a_] \[Rule] \(\((\(E\_+\)[a])\)\^*\), \(S\_+\)[ a_] \[Rule] \(\((\(S\_+\)[a])\)\^*\), \(M\_-\)[ a_] \[Rule] \(\((\(M\_-\)[a])\)\^*\), \(E\_-\)[ a_] \[Rule] \(\((\(E\_-\)[a])\)\^*\), \(S\_-\)[ a_] \[Rule] \(\((\(S\_-\)[a])\)\^*\)}];\)\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \(AnyMP = \(M\_-\) | \(M\_+\) | \(E\_-\) | \(E\_+\) | \(S\_-\) | \ \(S\_+\); \(AnyMP\_-\) = \(M\_-\) | \(E\_-\) | \(S\_-\); \(AnyMP\_+\) = \ \(M\_+\) | \(E\_+\) | \(S\_+\);\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \(\(ExpandMultipoleProducts = {\[IndentingNewLine]\(\((a_ \((x : AnyMP)\)[ m_])\)\^*\) \[Rule] a\ \(\((x[m])\)\^*\), \[IndentingNewLine]Re[ a_. \ \ x_\ \(\((x_)\)\^*\)] \[Rule] a\ Abs[x]\^2, \ Im[a_. \ x_\ \(\((x_)\)\^*\)] \[Rule] 0, \ Abs[f_[\[Theta]]]\^2 \[Rule] f[\[Theta]]\^2, Abs[a_. \ \((b_\ + \ c_)\)]\^2 \[Rule] \(Abs[ a]\^2\) \((Abs[b]\^2 + Abs[c]\^2 + 2 Re[b\ \(c\^*\)])\), Abs[a_. \ \((x : \((AnyMP)\))\)[m_]]\^2 \[Rule] \(a\^2\) Abs[x[m]]\^2, \[IndentingNewLine]\((f : \((Re | Im)\))\)[ a_. \ \((b_\ + \ c_)\)] \[Rule] f[a\ b] + f[a\ c], \((f : \((Re | Im)\))\)[ a_. \ \(\((\((b_\ + \ c_)\) d_. )\)\^*\)] \[Rule] f[a\ \(\((b\ d)\)\^*\)] + f[a\ \(\((c\ d)\)\^*\)], \((f : \((Re | Im)\))\)[ a_\ \((x : AnyMP)\)[ m_] \(\((\((y : AnyMP)\)[n_])\)\^*\)] \[Rule] a\ f[\(\((y[n])\)\^*\) x[m]], Re[\(\((\((x : \((\(M\_-\) | \(M\_+\))\))\)[ m_])\)\^*\) \((y : \((\(E\_-\) | \(E\_+\) | \(S\_-\) \ | \(S\_+\))\))\)[n_]] \[Rule] Re[\(\((y[n])\)\^*\) x[m]], Im[\(\((\((x : \((\(M\_-\) | \(M\_+\))\))\)[ m_])\)\^*\) \((y : \((\(E\_-\) | \(E\_+\) | \(S\_-\) \ | \(S\_+\))\))\)[n_]] \[Rule] \(-Im[\(\((y[n])\)\^*\) x[m]]\), Re[\(\((\((x : \((\(E\_-\) | \(E\_+\))\))\)[ m_])\)\^*\) \((y : \((\(S\_-\) | \(S\_+\))\))\)[ n_]] \[Rule] Re[\(\((y[n])\)\^*\) x[m]], Im[\(\((\((x : \((\(E\_-\) | \(E\_+\))\))\)[ m_])\)\^*\) \((y : \((\(S\_-\) | \(S\_+\))\))\)[ n_]] \[Rule] \(-Im[\(\((y[n])\)\^*\) x[m]]\), \ Re[\(\((x : \(M\_+\)[m_])\)\^*\) \(y : \(M\_-\)[n_]\)] \[Rule] Re[\(y\^*\) x], \[IndentingNewLine]Im[\(\((x : \(M\_+\)[ m_])\)\^*\) \(y : \(M\_-\)[ n_]\)] \[Rule] \(-Im[\(y\^*\) x]\), \[IndentingNewLine]Re[\(\((x : \(E\_+\)[ m_])\)\^*\) \(y : \(E\_-\)[n_]\)] \[Rule] Re[\(y\^*\) x], \[IndentingNewLine]Im[\(\((x : \(E\_+\)[ m_])\)\^*\) \(y : \(E\_-\)[ n_]\)] \[Rule] \(-Im[\(y\^*\) x]\), \[IndentingNewLine]Re[\(\((x : \(S\_+\)[ m_])\)\^*\) \(y : \(S\_-\)[n_]\)] \[Rule] Re[\(y\^*\) x], \[IndentingNewLine]Im[\(\((x : \(S\_+\)[ m_])\)\^*\) \(y : \(S\_-\)[ n_]\)] \[Rule] \(-Im[\(y\^*\) x]\), Re[\((x : AnyMP)\)[m_]\ \(\((\((x : AnyMP)\)[n_])\)\^*\)] /; \((n < m)\) \[Rule] Re[\(x[m]\^*\) x[n]], Im[\((x : AnyMP)\)[m_]\ \(\((\((x : AnyMP)\)[n_])\)\^*\)] /; \((n < m)\) \[Rule] \(-Im[\(x[m]\^*\) x[n]]\)\n};\)\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ \(\(recombine\ = \ a_. \ Re[b_] + \ c_. \ Re[d_]\ \[Rule] Re[a\ b\ + c\ d];\)\)], "Input", CellLabel->"In[21]:="] }, Open ]], Cell[CellGroupData[{ Cell["Expansion of response functions", "Subsection"], Cell[BoxData[ RowBox[{ RowBox[{"ExpandLegendre", " ", "=", " ", RowBox[{"{", RowBox[{\(P\_\[Lambda]_[x] \[Rule] LegendreP[\[Lambda], x]\), ",", RowBox[{ RowBox[{ SubsuperscriptBox["P", "\[Lambda]_", "\[Prime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(D[LegendreP[\[Lambda], x], x]\)}], ",", RowBox[{ RowBox[{ SubsuperscriptBox["P", "\[Lambda]_", "\[DoublePrime]", MultilineFunction->None], "[", "x", "]"}], "\[Rule]", \(D[LegendreP[\[Lambda], x], x, x]\)}]}], "}"}]}], ";"}]], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \(ExpandR[R_, lmax_Integer /; \((lmax \[GreaterEqual] 0)\)] := \ Module[{\[ScriptCapitalR]}, \n\t\t\[ScriptCapitalR] = \ \(TrigExpand[\(\((\(R //. ExpandCGLN\) /. {\[ScriptL]max \[Rule] lmax, MySum \[Rule] Sum})\) //. ExpandMultipoleProducts\) //. rule[mp]] /. {Cos[\[Theta]] \[Rule] x, Sin[\[Theta]] \[Rule] \@\(1 - x\^2\)}\) /. ExpandLegendre; \[IndentingNewLine]Collect[ MySimplify[\[ScriptCapitalR]], x, MySimplify]]\)], "Input", CellLabel->"In[23]:="], Cell[BoxData[{ \(\(ToLegendreP[x_, m_?EvenQ] := \ Sum[\(\(\(2\^\(2 n\)\) \((4 n + 1)\) \(m!\) \(\((m\/2 + n)\)!\)\)\/\(\(\((m + 2 n + 1)\)!\) \(\((m\/2 - n)\)!\)\)\) P\_\(2 n\)[x], {n, 0, m\/2}];\)\), "\[IndentingNewLine]", \(\(ToLegendreP[x_, m_?OddQ] := \ Sum[\(\(\(2\^\(2 n + 1\)\) \((4 n + 3)\) \(m!\) \(\((\(m + 1\)\/2 + n)\)!\)\)\/\(\(\((m + 2 n + 2)\)!\) \(\((\(m - 1\)\/2 - n)\)!\)\)\) P\_\(2 n + 1\)[x], {n, 0, m\/2 + 1}];\)\)}], "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(\(xExpand\ := \ x\^n_. \[RuleDelayed] ToLegendreP[x, n];\)\)], "Input",\ CellLabel->"In[26]:="], Cell[BoxData[ \(RtoLegendre[R_, lmax_Integer] := Collect[MySimplify[\(ExpandR[R, lmax] /. xExpand\) /. P\_0[x] \[Rule] 1], P\_\[Lambda]_[x], MySimplify]\)], "Input", CellLabel->"In[27]:="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Unpolarized response functions", "Section"], Cell[CellGroupData[{ Cell["multipole expansions", "Subsection"], Cell["\<\ \tFirst we quote the formulas relating unpolarized response functions to CGLN \ amplitudes\ \>", "Text"], Cell[BoxData[{ \(\(R\_L = \(q\^2\/\[Omega]\^2\) \((Abs[\[ScriptCapitalF]\_5]\^2 + Abs[\[ScriptCapitalF]\_6]\^2 + 2\ Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_5\ \(\((\ \[ScriptCapitalF]\_6)\)\^*\)])\);\)\), "\[IndentingNewLine]", \(\(R\_T = Abs[\[ScriptCapitalF]\_1]\^2 + Abs[\[ScriptCapitalF]\_2]\^2 - 2\ Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_2)\)\^*\)] + 1\/2\ \((Abs[\[ScriptCapitalF]\_3]\^2 + Abs[\[ScriptCapitalF]\_4]\^2 + 2\ \((Re[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + Re[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)])\))\)\ Sin[\[Theta]]\^2;\)\), "\ \[IndentingNewLine]", \(\(R\_LT = \(-\(q\/\[Omega]\)\)\ \((Re[\[ScriptCapitalF]\_2\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Re[\[ScriptCapitalF]\_3\ \(\((\[ScriptCapitalF]\_5)\)\^*\)] + Re[\[ScriptCapitalF]\_1\ \(\((\[ScriptCapitalF]\_6)\)\^*\)] + Cos[\[Theta]]\ \((Re[\[ScriptCapitalF]\_4\ \(\((\ \[ScriptCapitalF]\_5)\)\^*\)] + Re[\[ScriptCapitalF]\_3\ \ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\) + Re[\[ScriptCapitalF]\_4\ \(\((\[ScriptCapitalF]\_6)\)\^*\)])\)\ \ ;\)\), "\[IndentingNewLine]", \(\(R\_TT = 1\/2\ \((Abs[\[ScriptCapitalF]\_3]\^2 + Abs[\[ScriptCapitalF]\_4]\^2 + 2\ \((Re[\[ScriptCapitalF]\_2\ \ \(\((\[ScriptCapitalF]\_3)\)\^*\)] + Re[\[ScriptCapitalF]\_1\ \ \(\((\[ScriptCapitalF]\_4)\)\^*\)] + Cos[\[Theta]]\ Re[\[ScriptCapitalF]\_3\ \(\((\ \[ScriptCapitalF]\_4)\)\^*\)])\))\)\ ;\)\)}], "Input", CellLabel->"In[28]:="], Cell[TextData[{ "and then perform multipole expansions for ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL] \[LessEqual] 6\)]], ", which is more than sufficient to study their convergence." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Lexp = RtoLegendre[R\_L, 6]\)], "Input", CellLabel->"In[32]:="], Cell[BoxData[ \(Abs[\(S\_-\)[1]]\^2 + 8\ Abs[\(S\_-\)[2]]\^2 + 27\ Abs[\(S\_-\)[3]]\^2 + 64\ Abs[\(S\_-\)[4]]\^2 + 125\ Abs[\(S\_-\)[5]]\^2 + 216\ Abs[\(S\_-\)[6]]\^2 + Abs[\(S\_+\)[0]]\^2 + 8\ Abs[\(S\_+\)[1]]\^2 + 27\ Abs[\(S\_+\)[2]]\^2 + 64\ Abs[\(S\_+\)[3]]\^2 + 125\ Abs[\(S\_+\)[4]]\^2 + 216\ Abs[\(S\_+\)[5]]\^2 + 343\ Abs[\(S\_+\)[6]]\^2 + \((8\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_-\)[1]] + 216\/5\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_-\)[2]] + 864\/7\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_-\)[3]] + 800\/3\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[4]] + 5400\/11\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[5]] + 2\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[0]] + 8\ Re[\(\(S\_+\)[1]\^*\)\ \(S\_+\)[0]] + 16\/5\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[1]] + 216\/5\ Re[\(\(S\_+\)[2]\^*\)\ \(S\_+\)[1]] + 162\/35\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[2]] + 864\/7\ Re[\(\(S\_+\)[3]\^*\)\ \(S\_+\)[2]] + 128\/21\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[3]] + 800\/3\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[3]] + 250\/33\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[4]] + 5400\/11\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[4]] + 1296\/143\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[5]] + 10584\/13\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[5]])\)\ P\_1[ x] + \((8\ Abs[\(S\_-\)[2]]\^2 + 216\/7\ Abs[\(S\_-\)[3]]\^2 + 1600\/21\ Abs[\(S\_-\)[4]]\^2 + 5000\/33\ Abs[\(S\_-\)[5]]\^2 + 37800\/143\ Abs[\(S\_-\)[6]]\^2 + 8\ Abs[\(S\_+\)[1]]\^2 + 216\/7\ Abs[\(S\_+\)[2]]\^2 + 1600\/21\ Abs[\(S\_+\)[3]]\^2 + 5000\/33\ Abs[\(S\_+\)[4]]\^2 + 37800\/143\ Abs[\(S\_+\)[5]]\^2 + 5488\/13\ Abs[\(S\_+\)[6]]\^2 + 18\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_-\)[1]] + 576\/7\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_-\)[2]] + 1500\/7\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[3]] + 4800\/11\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[4]] + 8\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[0]] + 18\ Re[\(\(S\_+\)[2]\^*\)\ \(S\_+\)[0]] + 8\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]] + 72\/7\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[1]] + 576\/7\ Re[\(\(S\_+\)[3]\^*\)\ \(S\_+\)[1]] + 72\/7\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[2]] + 96\/7\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[2]] + 1500\/7\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[2]] + 96\/7\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[3]] + 4000\/231\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[3]] + 4800\/11\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[3]] + 4000\/231\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[4]] + 3000\/143\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[4]] + 110250\/143\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[4]] + 3000\/143\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[5]] + 3528\/143\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[6]])\)\ P\_2[ x] + \(\(1\/2145\)\((2\ \((34320\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_-\)[ 1]] + 30888\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_-\)[2]] + 143000\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[2]] + 102960\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_-\)[3]] + 351000\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[3]] + 234000\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[4]] + 441000\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[5]] + 19305\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[0]] + 34320\ Re[\(\(S\_+\)[3]\^*\)\ \(S\_+\)[0]] + 30888\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[1]] + 22880\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[1]] + 30888\ Re[\(\(S\_+\)[2]\^*\)\ \(S\_+\)[1]] + 143000\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[1]] + 19305\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[2]] + 30888\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[2]] + 29250\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[2]] + 102960\ Re[\(\(S\_+\)[3]\^*\)\ \(S\_+\)[2]] + 351000\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[2]] + 22880\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[3]] + 37440\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[3]] + 36000\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[3]] + 234000\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[3]] + 686000\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[3]] + 29250\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[4]] + 45000\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[4]] + 441000\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[4]] + 36000\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[5]] + 52920\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[5]] + 740880\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[5]] + 42875\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[6]])\)\ P\_3[ x])\)\) + \(\(1\/17017\)\((2\ \((196911\ Abs[\(S\_-\)[3]]\^2 + 572832\ Abs[\(S\_-\)[4]]\^2 + 1204875\ Abs[\(S\_-\)[5]]\^2 + 2159136\ Abs[\(S\_-\)[6]]\^2 + 196911\ Abs[\(S\_+\)[2]]\^2 + 572832\ Abs[\(S\_+\)[3]]\^2 + 1204875\ Abs[\(S\_+\)[4]]\^2 + 2159136\ Abs[\(S\_+\)[5]]\^2 + 3500658\ Abs[\(S\_+\)[6]]\^2 + 425425\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[1]] + 388960\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_-\)[2]] + 1670760\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[2]] + 1193400\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[3]] + 2570400\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[4]] + 272272\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[0]] + 425425\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[0]] + 525096\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[1]] + 309400\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[1]] + 388960\ Re[\(\(S\_+\)[3]\^*\)\ \(S\_+\)[1]] + 1670760\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[1]] + 525096\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[2]] + 477360\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[2]] + 385560\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[2]] + 1193400\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[2]] + 3935925\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[2]] + 272272\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[3]] + 477360\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[3]] + 550800\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[3]] + 2570400\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[3]] + 309400\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[4]] + 550800\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[4]] + 642600\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[4]] + 4664800\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[4]] + 385560\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[5]] + 642600\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[5]] + 466480\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[6]] + 740880\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[6]])\)\ P\_4[ x])\)\) + \((72\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[1]] + 200\/3\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[2]] + 480\/7\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_-\)[3]] + 2520\/13\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[3]] + 2400\/13\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[4]] + 4800\/13\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[5]] + 50\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[0]] + 72\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[0]] + 320\/3\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[1]] + 720\/13\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[1]] + 200\/3\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[1]] + 3528\/13\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[1]] + 900\/7\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[2]] + 1200\/13\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[2]] + 480\/7\ Re[\(\(S\_+\)[3]\^*\)\ \(S\_+\)[2]] + 2520\/13\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[2]] + 320\/3\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[3]] + 9600\/91\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[3]] + 1344\/13\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[3]] + 2400\/13\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[3]] + 15680\/39\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[3]] + 50\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[4]] + 1200\/13\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[4]] + 1500\/13\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[4]] + 4800\/13\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[4]] + 720\/13\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[5]] + 1344\/13\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[5]] + 28800\/221\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[5]] + 141120\/221\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[5]] + 882\/13\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[6]] + 78400\/663\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[6]])\)\ P\_5[ x] + \((1600\/33\ Abs[\(S\_-\)[4]]\^2 + 4000\/33\ Abs[\(S\_-\)[5]]\^2 + 43200\/187\ Abs[\(S\_-\)[6]]\^2 + 1600\/33\ Abs[\(S\_+\)[3]]\^2 + 4000\/33\ Abs[\(S\_+\)[4]]\^2 + 43200\/187\ Abs[\(S\_+\)[5]]\^2 + \(1372000\ \ Abs[\(S\_+\)[6]]\^2\)\/3553 + 1008\/11\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[2]] + 1050\/11\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[3]] + 2688\/11\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[4]] + 72\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[0]] + 98\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[0]] + 1800\/11\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[1]] + 1008\/11\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[1]] + 2400\/11\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[2]] + 1512\/11\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[2]] + 1050\/11\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[2]] + 14112\/55\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[2]] + 2400\/11\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[3]] + 5600\/33\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[3]] + 2688\/11\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[3]] + 1800\/11\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[4]] + 5600\/33\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[4]] + 33600\/187\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[4]] + 88200\/187\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[4]] + 72\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[5]] + 1512\/11\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[5]] + 33600\/187\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[5]] + 392\/5\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[6]] + 28224\/187\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[ 6]] + \(705600\ Re[\(\(S\_-\)[6]\^*\)\ \ \(S\_+\)[6]]\)\/3553)\)\ P\_6[ x] + \(\(1\/138567\)\((14\ \((1255824\ Re[\(\(S\_-\)[ 6]\^*\)\ \(S\_-\)[3]] + 1292000\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[4]] + 3078000\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[5]] + 2302344\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[1]] + 1193808\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[1]] + 3270375\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[2]] + 1255824\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[2]] + 3617600\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[3]] + 2462400\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[3]] + 1292000\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[3]] + 3102624\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[3]] + 3270375\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[4]] + 2660000\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[4]] + 3078000\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[4]] + 2302344\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[5]] + 2462400\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[5]] + 2721600\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[5]] + 5670000\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[5]] + 969969\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[6]] + 1896048\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[6]] + 2551500\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[6]])\)\ P\_7[ x])\)\) + \(\(1\/13585\)\((14\ \((83125\ Abs[\(S\_-\)[5]]\^2 + 189000\ Abs[\(S\_-\)[6]]\^2 + 83125\ Abs[\(S\_+\)[4]]\^2 + 189000\ Abs[\(S\_+\)[5]]\^2 + 336875\ Abs[\(S\_+\)[6]]\^2 + 164160\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[4]] + 451440\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[2]] + 158004\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[2]] + 532000\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[3]] + 164160\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[3]] + 532000\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[4]] + 378000\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[4]] + 378000\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[4]] + 451440\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[5]] + 378000\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[5]] + 304304\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[6]] + 332640\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[6]] + 378000\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[6]])\)\ P\_8[ x])\)\) + \(\(1\/2431\)\((84\ \((6300\ Re[\(\(S\_-\)[ 6]\^*\)\ \(S\_-\)[5]] + 22176\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[3]] + 6160\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[3]] + 23625\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[4]] + 6300\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[4]] + 22176\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[5]] + 16200\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[5]] + 13860\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[5]] + 18018\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[6]] + 15400\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[6]])\)\ P\_9[ x])\)\) + \(\(1\/96577\)\((588\ \((22356\ Abs[\(S\_-\)[6]]\^2 + 22356\ Abs[\(S\_+\)[5]]\^2 + 47628\ Abs[\(S\_+\)[6]]\^2 + 186300\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[4]] + 44275\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[4]] + 186300\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[5]] + 167440\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[6]] + 124740\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[6]])\)\ P\_10[ x])\)\) + \(2772\ \((2376\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[5]] + \ 504\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[5]] + 2275\ Re[\(\(S\_-\)[5]\^*\)\ \ \(S\_+\)[6]])\)\ P\_11[x]\)\/4199 + \(213444\ \((7\ Abs[\(S\_+\)[6]]\^2 + 72\ \ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[6]])\)\ P\_12[x]\)\/7429\)], "Output", CellLabel->"Out[32]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Texp = RtoLegendre[R\_T, 6]\)], "Input", CellLabel->"In[33]:="], Cell[BoxData[ RowBox[{\(Simplify::"time"\), \(\(:\)\(\ \)\), "\<\"Time spent on a \ transformation exceeded \\!\\(300\\) seconds, and the transformation was \ aborted. Increasing the value of TimeConstraint option may improve the result \ of simplification. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Simplify::time\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[33]:="], Cell[BoxData[ \(2\ Abs[\(E\_-\)[2]]\^2 + 9\ Abs[\(E\_-\)[3]]\^2 + 24\ Abs[\(E\_-\)[4]]\^2 + 50\ Abs[\(E\_-\)[5]]\^2 + 90\ Abs[\(E\_-\)[6]]\^2 + Abs[\(E\_+\)[0]]\^2 + 6\ Abs[\(E\_+\)[1]]\^2 + 18\ Abs[\(E\_+\)[2]]\^2 + 40\ Abs[\(E\_+\)[3]]\^2 + 75\ Abs[\(E\_+\)[4]]\^2 + 126\ Abs[\(E\_+\)[5]]\^2 + 196\ Abs[\(E\_+\)[6]]\^2 + Abs[\(M\_-\)[1]]\^2 + 6\ Abs[\(M\_-\)[2]]\^2 + 18\ Abs[\(M\_-\)[3]]\^2 + 40\ Abs[\(M\_-\)[4]]\^2 + 75\ Abs[\(M\_-\)[5]]\^2 + 126\ Abs[\(M\_-\)[6]]\^2 + 2\ Abs[\(M\_+\)[1]]\^2 + 9\ Abs[\(M\_+\)[2]]\^2 + 24\ Abs[\(M\_+\)[3]]\^2 + 50\ Abs[\(M\_+\)[4]]\^2 + 90\ Abs[\(M\_+\)[5]]\^2 + 147\ Abs[\(M\_+\)[6]]\^2 + \((\(-\(6\/5\)\)\ Re[\(E\_+\)[ 1]\ \(\(E\_-\)[2]\^*\)] - 2\ Re[\(M\_-\)[1]\ \(\(E\_-\)[2]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_-\)[2]\^*\)] + 54\/5\ Re[\(E\_-\)[2]\ \(\(E\_-\)[3]\^*\)] - 72\/35\ Re[\(E\_+\)[2]\ \(\(E\_-\)[3]\^*\)] - 18\/5\ Re[\(M\_-\)[2]\ \(\(E\_-\)[3]\^*\)] + 18\/5\ Re[\(M\_+\)[2]\ \(\(E\_-\)[3]\^*\)] + 288\/7\ Re[\(E\_-\)[3]\ \(\(E\_-\)[4]\^*\)] - 20\/7\ Re[\(E\_+\)[3]\ \(\(E\_-\)[4]\^*\)] - 36\/7\ Re[\(M\_-\)[3]\ \(\(E\_-\)[4]\^*\)] + 36\/7\ Re[\(M\_+\)[3]\ \(\(E\_-\)[4]\^*\)] + 100\ Re[\(E\_-\)[4]\ \(\(E\_-\)[5]\^*\)] - 40\/11\ Re[\(E\_+\)[4]\ \(\(E\_-\)[5]\^*\)] - 20\/3\ Re[\(M\_-\)[4]\ \(\(E\_-\)[5]\^*\)] + 20\/3\ Re[\(M\_+\)[4]\ \(\(E\_-\)[5]\^*\)] + 2160\/11\ Re[\(E\_-\)[5]\ \(\(E\_-\)[6]\^*\)] - 630\/143\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 90\/11\ Re[\(M\_-\)[5]\ \(\(E\_-\)[6]\^*\)] + 90\/11\ Re[\(M\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 2\ Re[\(M\_-\)[1]\ \(\(E\_+\)[0]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)] - 18\/5\ Re[\(M\_-\)[2]\ \(\(E\_+\)[1]\^*\)] + 18\/5\ Re[\(M\_+\)[2]\ \(\(E\_+\)[1]\^*\)] + 144\/5\ Re[\(E\_+\)[1]\ \(\(E\_+\)[2]\^*\)] - 36\/7\ Re[\(M\_-\)[3]\ \(\(E\_+\)[2]\^*\)] + 36\/7\ Re[\(M\_+\)[3]\ \(\(E\_+\)[2]\^*\)] + 540\/7\ Re[\(E\_+\)[2]\ \(\(E\_+\)[3]\^*\)] - 20\/3\ Re[\(M\_-\)[4]\ \(\(E\_+\)[3]\^*\)] + 20\/3\ Re[\(M\_+\)[4]\ \(\(E\_+\)[3]\^*\)] + 160\ Re[\(E\_+\)[3]\ \(\(E\_+\)[4]\^*\)] - 90\/11\ Re[\(M\_-\)[5]\ \(\(E\_+\)[4]\^*\)] + 90\/11\ Re[\(M\_+\)[5]\ \(\(E\_+\)[4]\^*\)] + 3150\/11\ Re[\(E\_+\)[4]\ \(\(E\_+\)[5]\^*\)] - 126\/13\ Re[\(M\_-\)[6]\ \(\(E\_+\)[5]\^*\)] + 126\/13\ Re[\(M\_+\)[6]\ \(\(E\_+\)[5]\^*\)] + 6048\/13\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] + 6\ Re[\(M\_-\)[1]\ \(\(M\_-\)[2]\^*\)] + 6\/5\ Re[\(M\_+\)[1]\ \(\(M\_-\)[2]\^*\)] + 144\/5\ Re[\(M\_-\)[2]\ \(\(M\_-\)[3]\^*\)] + 72\/35\ Re[\(M\_+\)[2]\ \(\(M\_-\)[3]\^*\)] + 540\/7\ Re[\(M\_-\)[3]\ \(\(M\_-\)[4]\^*\)] + 20\/7\ Re[\(M\_+\)[3]\ \(\(M\_-\)[4]\^*\)] + 160\ Re[\(M\_-\)[4]\ \(\(M\_-\)[5]\^*\)] + 40\/11\ Re[\(M\_+\)[4]\ \(\(M\_-\)[5]\^*\)] + 3150\/11\ Re[\(M\_-\)[5]\ \(\(M\_-\)[6]\^*\)] + 630\/143\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] + 54\/5\ Re[\(M\_+\)[1]\ \(\(M\_+\)[2]\^*\)] + 288\/7\ Re[\(M\_+\)[2]\ \(\(M\_+\)[3]\^*\)] + 100\ Re[\(M\_+\)[3]\ \(\(M\_+\)[4]\^*\)] + 2160\/11\ Re[\(M\_+\)[4]\ \(\(M\_+\)[5]\^*\)] + 4410\/13\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_1[ x] + \((\(-Abs[\(E\_-\)[2]]\^2\) + 36\/7\ Abs[\(E\_-\)[3]]\^2 + 150\/7\ Abs[\(E\_-\)[4]]\^2 + 1700\/33\ Abs[\(E\_-\)[5]]\^2 + 14175\/143\ Abs[\(E\_-\)[6]]\^2 + 3\ Abs[\(E\_+\)[1]]\^2 + 108\/7\ Abs[\(E\_+\)[2]]\^2 + 850\/21\ Abs[\(E\_+\)[3]]\^2 + 900\/11\ Abs[\(E\_+\)[4]]\^2 + 1575\/11\ Abs[\(E\_+\)[5]]\^2 + 2968\/13\ Abs[\(E\_+\)[6]]\^2 + 3\ Abs[\(M\_-\)[2]]\^2 + 108\/7\ Abs[\(M\_-\)[3]]\^2 + 850\/21\ Abs[\(M\_-\)[4]]\^2 + 900\/11\ Abs[\(M\_-\)[5]]\^2 + 1575\/11\ Abs[\(M\_-\)[6]]\^2 - Abs[\(M\_+\)[1]]\^2 + 36\/7\ Abs[\(M\_+\)[2]]\^2 + 150\/7\ Abs[\(M\_+\)[3]]\^2 + 1700\/33\ Abs[\(M\_+\)[4]]\^2 + 14175\/143\ Abs[\(M\_+\)[5]]\^2 + 168\ Abs[\(M\_+\)[6]]\^2 + 2\ Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)] - 24\/7\ Re[\(E\_+\)[2]\ \(\(E\_-\)[2]\^*\)] - 6\ Re[\(M\_-\)[2]\ \(\(E\_-\)[2]\^*\)] + 6\ Re[\(M\_+\)[2]\ \(\(E\_-\)[2]\^*\)] - 18\/7\ Re[\(E\_+\)[1]\ \(\(E\_-\)[3]\^*\)] - 40\/7\ Re[\(E\_+\)[3]\ \(\(E\_-\)[3]\^*\)] - 6\ Re[\(M\_-\)[1]\ \(\(E\_-\)[3]\^*\)] - 72\/7\ Re[\(M\_-\)[3]\ \(\(E\_-\)[3]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_-\)[3]\^*\)] + 72\/7\ Re[\(M\_+\)[3]\ \(\(E\_-\)[3]\^*\)] + 144\/7\ Re[\(E\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 36\/7\ Re[\(E\_+\)[2]\ \(\(E\_-\)[4]\^*\)] - 600\/77\ Re[\(E\_+\)[4]\ \(\(E\_-\)[4]\^*\)] - 72\/7\ Re[\(M\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 100\/7\ Re[\(M\_-\)[4]\ \(\(E\_-\)[4]\^*\)] + 72\/7\ Re[\(M\_+\)[2]\ \(\(E\_-\)[4]\^*\)] + 100\/7\ Re[\(M\_+\)[4]\ \(\(E\_-\)[4]\^*\)] + 500\/7\ Re[\(E\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 1700\/231\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 1400\/143\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 100\/7\ Re[\(M\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 200\/11\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] + 100\/7\ Re[\(M\_+\)[3]\ \(\(E\_-\)[5]\^*\)] + 200\/11\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 1800\/11\ Re[\(E\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 1350\/143\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 1680\/143\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 200\/11\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 3150\/143\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] + 200\/11\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 3150\/143\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 6\ Re[\(M\_-\)[2]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] - 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 72\/7\ Re[\(M\_-\)[3]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 72\/7\ Re[\(M\_+\)[3]\ \(\(E\_+\)[1]\^*\)] + 12\ Re[\(E\_+\)[0]\ \(\(E\_+\)[2]\^*\)] - 72\/7\ Re[\(M\_-\)[2]\ \(\(E\_+\)[2]\^*\)] - 100\/7\ Re[\(M\_-\)[4]\ \(\(E\_+\)[2]\^*\)] + 72\/7\ Re[\(M\_+\)[2]\ \(\(E\_+\)[2]\^*\)] + 100\/7\ Re[\(M\_+\)[4]\ \(\(E\_+\)[2]\^*\)] + 360\/7\ Re[\(E\_+\)[1]\ \(\(E\_+\)[3]\^*\)] - 100\/7\ Re[\(M\_-\)[3]\ \(\(E\_+\)[3]\^*\)] - 200\/11\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] + 100\/7\ Re[\(M\_+\)[3]\ \(\(E\_+\)[3]\^*\)] + 200\/11\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] + 900\/7\ Re[\(E\_+\)[2]\ \(\(E\_+\)[4]\^*\)] - 200\/11\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] - 3150\/143\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] + 200\/11\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] + 3150\/143\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] + 2800\/11\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 3150\/143\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] + 3150\/143\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] + 63000\/143\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 336\/13\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] + 336\/13\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] + 18\/7\ Re[\(M\_+\)[2]\ \(\(M\_-\)[2]\^*\)] + 12\ Re[\(M\_-\)[1]\ \(\(M\_-\)[3]\^*\)] + 24\/7\ Re[\(M\_+\)[1]\ \(\(M\_-\)[3]\^*\)] + 36\/7\ Re[\(M\_+\)[3]\ \(\(M\_-\)[3]\^*\)] + 360\/7\ Re[\(M\_-\)[2]\ \(\(M\_-\)[4]\^*\)] + 40\/7\ Re[\(M\_+\)[2]\ \(\(M\_-\)[4]\^*\)] + 1700\/231\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] + 900\/7\ Re[\(M\_-\)[3]\ \(\(M\_-\)[5]\^*\)] + 600\/77\ Re[\(M\_+\)[3]\ \(\(M\_-\)[5]\^*\)] + 1350\/143\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] + 2800\/11\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] + 1400\/143\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] + 126\/11\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] + 144\/7\ Re[\(M\_+\)[1]\ \(\(M\_+\)[3]\^*\)] + 500\/7\ Re[\(M\_+\)[2]\ \(\(M\_+\)[4]\^*\)] + 1800\/11\ Re[\(M\_+\)[3]\ \(\(M\_+\)[5]\^*\)] + 44100\/143\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)])\)\ P\_2[ x] + \(\(1\/2145\)\((2\ \((7722\ Re[\(E\_+\)[ 1]\ \(\(E\_-\)[2]\^*\)] - 7150\ Re[\(E\_+\)[3]\ \(\(E\_-\)[2]\^*\)] - 12870\ Re[\(M\_-\)[3]\ \(\(E\_-\)[2]\^*\)] + 12870\ Re[\(M\_+\)[3]\ \(\(E\_-\)[2]\^*\)] - 5148\ Re[\(E\_-\)[2]\ \(\(E\_-\)[3]\^*\)] + 6435\ Re[\(E\_+\)[0]\ \(\(E\_-\)[3]\^*\)] - 5148\ Re[\(E\_+\)[2]\ \(\(E\_-\)[3]\^*\)] - 11700\ Re[\(E\_+\)[4]\ \(\(E\_-\)[3]\^*\)] - 15444\ Re[\(M\_-\)[2]\ \(\(E\_-\)[3]\^*\)] - 21450\ Re[\(M\_-\)[4]\ \(\(E\_-\)[3]\^*\)] + 15444\ Re[\(M\_+\)[2]\ \(\(E\_-\)[3]\^*\)] + 21450\ Re[\(M\_+\)[4]\ \(\(E\_-\)[3]\^*\)] + 12870\ Re[\(E\_-\)[3]\ \(\(E\_-\)[4]\^*\)] - 4290\ Re[\(E\_+\)[1]\ \(\(E\_-\)[4]\^*\)] - 11700\ Re[\(E\_+\)[3]\ \(\(E\_-\)[4]\^*\)] - 15750\ Re[\(E\_+\)[5]\ \(\(E\_-\)[4]\^*\)] - 12870\ Re[\(M\_-\)[1]\ \(\(E\_-\)[4]\^*\)] - 25740\ Re[\(M\_-\)[3]\ \(\(E\_-\)[4]\^*\)] - 29250\ Re[\(M\_-\)[5]\ \(\(E\_-\)[4]\^*\)] + 12870\ Re[\(M\_+\)[1]\ \(\(E\_-\)[4]\^*\)] + 25740\ Re[\(M\_+\)[3]\ \(\(E\_-\)[4]\^*\)] + 29250\ Re[\(M\_+\)[5]\ \(\(E\_-\)[4]\^*\)] + 35750\ Re[\(E\_-\)[2]\ \(\(E\_-\)[5]\^*\)] + 58500\ Re[\(E\_-\)[4]\ \(\(E\_-\)[5]\^*\)] - 9750\ Re[\(E\_+\)[2]\ \(\(E\_-\)[5]\^*\)] - 17100\ Re[\(E\_+\)[4]\ \(\(E\_-\)[5]\^*\)] - 19600\ Re[\(E\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 21450\ Re[\(M\_-\)[2]\ \(\(E\_-\)[5]\^*\)] - 35100\ Re[\(M\_-\)[4]\ \(\(E\_-\)[5]\^*\)] - 36750\ Re[\(M\_-\)[6]\ \(\(E\_-\)[5]\^*\)] + 21450\ Re[\(M\_+\)[2]\ \(\(E\_-\)[5]\^*\)] + 35100\ Re[\(M\_+\)[4]\ \(\(E\_-\)[5]\^*\)] + 36750\ Re[\(M\_+\)[6]\ \(\(E\_-\)[5]\^*\)] + 117000\ Re[\(E\_-\)[3]\ \(\(E\_-\)[6]\^*\)] + 139650\ Re[\(E\_-\)[5]\ \(\(E\_-\)[6]\^*\)] - 14250\ Re[\(E\_+\)[3]\ \(\(E\_-\)[6]\^*\)] - 22050\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 29250\ Re[\(M\_-\)[3]\ \(\(E\_-\)[6]\^*\)] - 44100\ Re[\(M\_-\)[5]\ \(\(E\_-\)[6]\^*\)] + 29250\ Re[\(M\_+\)[3]\ \(\(E\_-\)[6]\^*\)] + 44100\ Re[\(M\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 12870\ Re[\(M\_-\)[3]\ \(\(E\_+\)[0]\^*\)] + 12870\ Re[\(M\_+\)[3]\ \(\(E\_+\)[0]\^*\)] - 15444\ Re[\(M\_-\)[2]\ \(\(E\_+\)[1]\^*\)] - 21450\ Re[\(M\_-\)[4]\ \(\(E\_+\)[1]\^*\)] + 15444\ Re[\(M\_+\)[2]\ \(\(E\_+\)[1]\^*\)] + 21450\ Re[\(M\_+\)[4]\ \(\(E\_+\)[1]\^*\)] + 7722\ Re[\(E\_+\)[1]\ \(\(E\_+\)[2]\^*\)] - 12870\ Re[\(M\_-\)[1]\ \(\(E\_+\)[2]\^*\)] - 25740\ Re[\(M\_-\)[3]\ \(\(E\_+\)[2]\^*\)] - 29250\ Re[\(M\_-\)[5]\ \(\(E\_+\)[2]\^*\)] + 12870\ Re[\(M\_+\)[1]\ \(\(E\_+\)[2]\^*\)] + 25740\ Re[\(M\_+\)[3]\ \(\(E\_+\)[2]\^*\)] + 29250\ Re[\(M\_+\)[5]\ \(\(E\_+\)[2]\^*\)] + 21450\ Re[\(E\_+\)[0]\ \(\(E\_+\)[3]\^*\)] + 42900\ Re[\(E\_+\)[2]\ \(\(E\_+\)[3]\^*\)] - 21450\ Re[\(M\_-\)[2]\ \(\(E\_+\)[3]\^*\)] - 35100\ Re[\(M\_-\)[4]\ \(\(E\_+\)[3]\^*\)] - 36750\ Re[\(M\_-\)[6]\ \(\(E\_+\)[3]\^*\)] + 21450\ Re[\(M\_+\)[2]\ \(\(E\_+\)[3]\^*\)] + 35100\ Re[\(M\_+\)[4]\ \(\(E\_+\)[3]\^*\)] + 36750\ Re[\(M\_+\)[6]\ \(\(E\_+\)[3]\^*\)] + 85800\ Re[\(E\_+\)[1]\ \(\(E\_+\)[4]\^*\)] + 111150\ Re[\(E\_+\)[3]\ \(\(E\_+\)[4]\^*\)] - 29250\ Re[\(M\_-\)[3]\ \(\(E\_+\)[4]\^*\)] - 44100\ Re[\(M\_-\)[5]\ \(\(E\_+\)[4]\^*\)] + 29250\ Re[\(M\_+\)[3]\ \(\(E\_+\)[4]\^*\)] + 44100\ Re[\(M\_+\)[5]\ \(\(E\_+\)[4]\^*\)] + 204750\ Re[\(E\_+\)[2]\ \(\(E\_+\)[5]\^*\)] + 220500\ Re[\(E\_+\)[4]\ \(\(E\_+\)[5]\^*\)] - 36750\ Re[\(M\_-\)[4]\ \(\(E\_+\)[5]\^*\)] - 52920\ Re[\(M\_-\)[6]\ \(\(E\_+\)[5]\^*\)] + 36750\ Re[\(M\_+\)[4]\ \(\(E\_+\)[5]\^*\)] + 52920\ Re[\(M\_+\)[6]\ \(\(E\_+\)[5]\^*\)] + 392000\ Re[\(E\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 379260\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 44100\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] + 44100\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 6435\ Re[\(M\_+\)[2]\ \(\(M\_-\)[1]\^*\)] - 7722\ Re[\(M\_+\)[1]\ \(\(M\_-\)[2]\^*\)] + 4290\ Re[\(M\_+\)[3]\ \(\(M\_-\)[2]\^*\)] + 7722\ Re[\(M\_-\)[2]\ \(\(M\_-\)[3]\^*\)] + 5148\ Re[\(M\_+\)[2]\ \(\(M\_-\)[3]\^*\)] + 9750\ Re[\(M\_+\)[4]\ \(\(M\_-\)[3]\^*\)] + 21450\ Re[\(M\_-\)[1]\ \(\(M\_-\)[4]\^*\)] + 42900\ Re[\(M\_-\)[3]\ \(\(M\_-\)[4]\^*\)] + 7150\ Re[\(M\_+\)[1]\ \(\(M\_-\)[4]\^*\)] + 11700\ Re[\(M\_+\)[3]\ \(\(M\_-\)[4]\^*\)] + 14250\ Re[\(M\_+\)[5]\ \(\(M\_-\)[4]\^*\)] + 85800\ Re[\(M\_-\)[2]\ \(\(M\_-\)[5]\^*\)] + 111150\ Re[\(M\_-\)[4]\ \(\(M\_-\)[5]\^*\)] + 11700\ Re[\(M\_+\)[2]\ \(\(M\_-\)[5]\^*\)] + 17100\ Re[\(M\_+\)[4]\ \(\(M\_-\)[5]\^*\)] + 18375\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] + 204750\ Re[\(M\_-\)[3]\ \(\(M\_-\)[6]\^*\)] + 220500\ Re[\(M\_-\)[5]\ \(\(M\_-\)[6]\^*\)] + 15750\ Re[\(M\_+\)[3]\ \(\(M\_-\)[6]\^*\)] + 22050\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] - 5148\ Re[\(M\_+\)[1]\ \(\(M\_+\)[2]\^*\)] + 12870\ Re[\(M\_+\)[2]\ \(\(M\_+\)[3]\^*\)] + 35750\ Re[\(M\_+\)[1]\ \(\(M\_+\)[4]\^*\)] + 58500\ Re[\(M\_+\)[3]\ \(\(M\_+\)[4]\^*\)] + 117000\ Re[\(M\_+\)[2]\ \(\(M\_+\)[5]\^*\)] + 139650\ Re[\(M\_+\)[4]\ \(\(M\_+\)[5]\^*\)] + 257250\ Re[\(M\_+\)[3]\ \(\(M\_+\)[6]\^*\)] + 264600\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_3[ x])\)\) - \(\(1\/17017\)\((2\ \((43758\ Abs[\(E\_-\)[3]]\^2 - 35802\ Abs[\(E\_-\)[4]]\^2 - 240975\ Abs[\(E\_-\)[5]]\^2 - 599760\ Abs[\(E\_-\)[6]]\^2 - 21879\ Abs[\(E\_+\)[2]]\^2 - 179010\ Abs[\(E\_+\)[3]]\^2 - 481950\ Abs[\(E\_+\)[4]]\^2 - 959616\ Abs[\(E\_+\)[5]]\^2 - 1643166\ Abs[\(E\_+\)[6]]\^2 - 21879\ Abs[\(M\_-\)[3]]\^2 - 179010\ Abs[\(M\_-\)[4]]\^2 - 481950\ Abs[\(M\_-\)[5]]\^2 - 959616\ Abs[\(M\_-\)[6]]\^2 + 43758\ Abs[\(M\_+\)[2]]\^2 - 35802\ Abs[\(M\_+\)[3]]\^2 - 240975\ Abs[\(M\_+\)[4]]\^2 - 599760\ Abs[\(M\_+\)[5]]\^2 - 1143072\ Abs[\(M\_+\)[6]]\^2 - 131274\ Re[\(E\_+\)[2]\ \(\(E\_-\)[2]\^*\)] + 92820\ Re[\(E\_+\)[4]\ \(\(E\_-\)[2]\^*\)] + 170170\ Re[\(M\_-\)[4]\ \(\(E\_-\)[2]\^*\)] - 170170\ Re[\(M\_+\)[4]\ \(\(E\_-\)[2]\^*\)] - 175032\ Re[\(E\_+\)[1]\ \(\(E\_-\)[3]\^*\)] + 59670\ Re[\(E\_+\)[3]\ \(\(E\_-\)[3]\^*\)] + 149940\ Re[\(E\_+\)[5]\ \(\(E\_-\)[3]\^*\)] + 218790\ Re[\(M\_-\)[3]\ \(\(E\_-\)[3]\^*\)] + 278460\ Re[\(M\_-\)[5]\ \(\(E\_-\)[3]\^*\)] - 218790\ Re[\(M\_+\)[3]\ \(\(E\_-\)[3]\^*\)] - 278460\ Re[\(M\_+\)[5]\ \(\(E\_-\)[3]\^*\)] + 72930\ Re[\(E\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 102102\ Re[\(E\_+\)[0]\ \(\(E\_-\)[4]\^*\)] + 39780\ Re[\(E\_+\)[2]\ \(\(E\_-\)[4]\^*\)] + 151470\ Re[\(E\_+\)[4]\ \(\(E\_-\)[4]\^*\)] + 199920\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] + 218790\ Re[\(M\_-\)[2]\ \(\(E\_-\)[4]\^*\)] + 358020\ Re[\(M\_-\)[4]\ \(\(E\_-\)[4]\^*\)] + 374850\ Re[\(M\_-\)[6]\ \(\(E\_-\)[4]\^*\)] - 218790\ Re[\(M\_+\)[2]\ \(\(E\_-\)[4]\^*\)] - 358020\ Re[\(M\_+\)[4]\ \(\(E\_-\)[4]\^*\)] - 374850\ Re[\(M\_+\)[6]\ \(\(E\_-\)[4]\^*\)] - 119340\ Re[\(E\_-\)[3]\ \(\(E\_-\)[5]\^*\)] + 46410\ Re[\(E\_+\)[1]\ \(\(E\_-\)[5]\^*\)] + 137700\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] + 224910\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 170170\ Re[\(M\_-\)[1]\ \(\(E\_-\)[5]\^*\)] + 358020\ Re[\(M\_-\)[3]\ \(\(E\_-\)[5]\^*\)] + 481950\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] - 170170\ Re[\(M\_+\)[1]\ \(\(E\_-\)[5]\^*\)] - 358020\ Re[\(M\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 481950\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 417690\ Re[\(E\_-\)[2]\ \(\(E\_-\)[6]\^*\)] - 589050\ Re[\(E\_-\)[4]\ \(\(E\_-\)[6]\^*\)] + 117810\ Re[\(E\_+\)[2]\ \(\(E\_-\)[6]\^*\)] + 214200\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 291060\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 278460\ Re[\(M\_-\)[2]\ \(\(E\_-\)[6]\^*\)] + 481950\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] + 599760\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] - 278460\ Re[\(M\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 481950\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 599760\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 170170\ Re[\(M\_-\)[4]\ \(\(E\_+\)[0]\^*\)] - 170170\ Re[\(M\_+\)[4]\ \(\(E\_+\)[0]\^*\)] + 218790\ Re[\(M\_-\)[3]\ \(\(E\_+\)[1]\^*\)] + 278460\ Re[\(M\_-\)[5]\ \(\(E\_+\)[1]\^*\)] - 218790\ Re[\(M\_+\)[3]\ \(\(E\_+\)[1]\^*\)] - 278460\ Re[\(M\_+\)[5]\ \(\(E\_+\)[1]\^*\)] + 218790\ Re[\(M\_-\)[2]\ \(\(E\_+\)[2]\^*\)] + 358020\ Re[\(M\_-\)[4]\ \(\(E\_+\)[2]\^*\)] + 374850\ Re[\(M\_-\)[6]\ \(\(E\_+\)[2]\^*\)] - 218790\ Re[\(M\_+\)[2]\ \(\(E\_+\)[2]\^*\)] - 358020\ Re[\(M\_+\)[4]\ \(\(E\_+\)[2]\^*\)] - 374850\ Re[\(M\_+\)[6]\ \(\(E\_+\)[2]\^*\)] - 72930\ Re[\(E\_+\)[1]\ \(\(E\_+\)[3]\^*\)] + 170170\ Re[\(M\_-\)[1]\ \(\(E\_+\)[3]\^*\)] + 358020\ Re[\(M\_-\)[3]\ \(\(E\_+\)[3]\^*\)] + 481950\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] - 170170\ Re[\(M\_+\)[1]\ \(\(E\_+\)[3]\^*\)] - 358020\ Re[\(M\_+\)[3]\ \(\(E\_+\)[3]\^*\)] - 481950\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] - 255255\ Re[\(E\_+\)[0]\ \(\(E\_+\)[4]\^*\)] - 437580\ Re[\(E\_+\)[2]\ \(\(E\_+\)[4]\^*\)] + 278460\ Re[\(M\_-\)[2]\ \(\(E\_+\)[4]\^*\)] + 481950\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] + 599760\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] - 278460\ Re[\(M\_+\)[2]\ \(\(E\_+\)[4]\^*\)] - 481950\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] - 599760\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] - 974610\ Re[\(E\_+\)[1]\ \(\(E\_+\)[5]\^*\)] - 1124550\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] + 374850\ Re[\(M\_-\)[3]\ \(\(E\_+\)[5]\^*\)] + 599760\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] - 374850\ Re[\(M\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 599760\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] - 2249100\ Re[\(E\_+\)[2]\ \(\(E\_+\)[6]\^*\)] - 2199120\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] + 466480\ Re[\(M\_-\)[4]\ \(\(E\_+\)[6]\^*\)] + 714420\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 466480\ Re[\(M\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 714420\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] + 102102\ Re[\(M\_+\)[3]\ \(\(M\_-\)[1]\^*\)] + 175032\ Re[\(M\_+\)[2]\ \(\(M\_-\)[2]\^*\)] - 46410\ Re[\(M\_+\)[4]\ \(\(M\_-\)[2]\^*\)] + 131274\ Re[\(M\_+\)[1]\ \(\(M\_-\)[3]\^*\)] - 39780\ Re[\(M\_+\)[3]\ \(\(M\_-\)[3]\^*\)] - 117810\ Re[\(M\_+\)[5]\ \(\(M\_-\)[3]\^*\)] - 72930\ Re[\(M\_-\)[2]\ \(\(M\_-\)[4]\^*\)] - 59670\ Re[\(M\_+\)[2]\ \(\(M\_-\)[4]\^*\)] - 137700\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] - 174930\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] - 255255\ Re[\(M\_-\)[1]\ \(\(M\_-\)[5]\^*\)] - 437580\ Re[\(M\_-\)[3]\ \(\(M\_-\)[5]\^*\)] - 92820\ Re[\(M\_+\)[1]\ \(\(M\_-\)[5]\^*\)] - 151470\ Re[\(M\_+\)[3]\ \(\(M\_-\)[5]\^*\)] - 214200\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] - 974610\ Re[\(M\_-\)[2]\ \(\(M\_-\)[6]\^*\)] - 1124550\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] - 149940\ Re[\(M\_+\)[2]\ \(\(M\_-\)[6]\^*\)] - 224910\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] - 282240\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] + 72930\ Re[\(M\_+\)[1]\ \(\(M\_+\)[3]\^*\)] - 119340\ Re[\(M\_+\)[2]\ \(\(M\_+\)[4]\^*\)] - 417690\ Re[\(M\_+\)[1]\ \(\(M\_+\)[5]\^*\)] - 589050\ Re[\(M\_+\)[3]\ \(\(M\_+\)[5]\^*\)] - 1311975\ Re[\(M\_+\)[2]\ \(\(M\_+\)[6]\^*\)] - 1399440\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)])\)\ P\_4[ x])\)\) + \((80\/3\ Re[\(E\_+\)[3]\ \(\(E\_-\)[2]\^*\)] - 210\/13\ Re[\(E\_+\)[5]\ \(\(E\_-\)[2]\^*\)] - 30\ Re[\(M\_-\)[5]\ \(\(E\_-\)[2]\^*\)] + 30\ Re[\(M\_+\)[5]\ \(\(E\_-\)[2]\^*\)] + 300\/7\ Re[\(E\_+\)[2]\ \(\(E\_-\)[3]\^*\)] - 120\/13\ Re[\(E\_+\)[4]\ \(\(E\_-\)[3]\^*\)] - 336\/13\ Re[\(E\_+\)[6]\ \(\(E\_-\)[3]\^*\)] - 40\ Re[\(M\_-\)[4]\ \(\(E\_-\)[3]\^*\)] - 630\/13\ Re[\(M\_-\)[6]\ \(\(E\_-\)[3]\^*\)] + 40\ Re[\(M\_+\)[4]\ \(\(E\_-\)[3]\^*\)] + 630\/13\ Re[\(M\_+\)[6]\ \(\(E\_-\)[3]\^*\)] - 120\/7\ Re[\(E\_-\)[3]\ \(\(E\_-\)[4]\^*\)] + 40\ Re[\(E\_+\)[1]\ \(\(E\_-\)[4]\^*\)] - 300\/91\ Re[\(E\_+\)[3]\ \(\(E\_-\)[4]\^*\)] - 336\/13\ Re[\(E\_+\)[5]\ \(\(E\_-\)[4]\^*\)] - 300\/7\ Re[\(M\_-\)[3]\ \(\(E\_-\)[4]\^*\)] - 840\/13\ Re[\(M\_-\)[5]\ \(\(E\_-\)[4]\^*\)] + 300\/7\ Re[\(M\_+\)[3]\ \(\(E\_-\)[4]\^*\)] + 840\/13\ Re[\(M\_+\)[5]\ \(\(E\_-\)[4]\^*\)] - 40\/3\ Re[\(E\_-\)[2]\ \(\(E\_-\)[5]\^*\)] + 60\/13\ Re[\(E\_-\)[4]\ \(\(E\_-\)[5]\^*\)] + 20\ Re[\(E\_+\)[0]\ \(\(E\_-\)[5]\^*\)] - 40\/13\ Re[\(E\_+\)[2]\ \(\(E\_-\)[5]\^*\)] - 300\/13\ Re[\(E\_+\)[4]\ \(\(E\_-\)[5]\^*\)] - 25760\/663\ Re[\(E\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 40\ Re[\(M\_-\)[2]\ \(\(E\_-\)[5]\^*\)] - 900\/13\ Re[\(M\_-\)[4]\ \(\(E\_-\)[5]\^*\)] - 1120\/13\ Re[\(M\_-\)[6]\ \(\(E\_-\)[5]\^*\)] + 40\ Re[\(M\_+\)[2]\ \(\(E\_-\)[5]\^*\)] + 900\/13\ Re[\(M\_+\)[4]\ \(\(E\_-\)[5]\^*\)] + 1120\/13\ Re[\(M\_+\)[6]\ \(\(E\_-\)[5]\^*\)] + 210\/13\ Re[\(E\_-\)[3]\ \(\(E\_-\)[6]\^*\)] + 800\/13\ Re[\(E\_-\)[5]\ \(\(E\_-\)[6]\^*\)] - 90\/13\ Re[\(E\_+\)[1]\ \(\(E\_-\)[6]\^*\)] - 280\/13\ Re[\(E\_+\)[3]\ \(\(E\_-\)[6]\^*\)] - 8400\/221\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 30\ Re[\(M\_-\)[1]\ \(\(E\_-\)[6]\^*\)] - 840\/13\ Re[\(M\_-\)[3]\ \(\(E\_-\)[6]\^*\)] - 1200\/13\ Re[\(M\_-\)[5]\ \(\(E\_-\)[6]\^*\)] + 30\ Re[\(M\_+\)[1]\ \(\(E\_-\)[6]\^*\)] + 840\/13\ Re[\(M\_+\)[3]\ \(\(E\_-\)[6]\^*\)] + 1200\/13\ Re[\(M\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 30\ Re[\(M\_-\)[5]\ \(\(E\_+\)[0]\^*\)] + 30\ Re[\(M\_+\)[5]\ \(\(E\_+\)[0]\^*\)] - 40\ Re[\(M\_-\)[4]\ \(\(E\_+\)[1]\^*\)] - 630\/13\ Re[\(M\_-\)[6]\ \(\(E\_+\)[1]\^*\)] + 40\ Re[\(M\_+\)[4]\ \(\(E\_+\)[1]\^*\)] + 630\/13\ Re[\(M\_+\)[6]\ \(\(E\_+\)[1]\^*\)] - 300\/7\ Re[\(M\_-\)[3]\ \(\(E\_+\)[2]\^*\)] - 840\/13\ Re[\(M\_-\)[5]\ \(\(E\_+\)[2]\^*\)] + 300\/7\ Re[\(M\_+\)[3]\ \(\(E\_+\)[2]\^*\)] + 840\/13\ Re[\(M\_+\)[5]\ \(\(E\_+\)[2]\^*\)] + 20\/7\ Re[\(E\_+\)[2]\ \(\(E\_+\)[3]\^*\)] - 40\ Re[\(M\_-\)[2]\ \(\(E\_+\)[3]\^*\)] - 900\/13\ Re[\(M\_-\)[4]\ \(\(E\_+\)[3]\^*\)] - 1120\/13\ Re[\(M\_-\)[6]\ \(\(E\_+\)[3]\^*\)] + 40\ Re[\(M\_+\)[2]\ \(\(E\_+\)[3]\^*\)] + 900\/13\ Re[\(M\_+\)[4]\ \(\(E\_+\)[3]\^*\)] + 1120\/13\ Re[\(M\_+\)[6]\ \(\(E\_+\)[3]\^*\)] + 10\ Re[\(E\_+\)[1]\ \(\(E\_+\)[4]\^*\)] + 600\/13\ Re[\(E\_+\)[3]\ \(\(E\_+\)[4]\^*\)] - 30\ Re[\(M\_-\)[1]\ \(\(E\_+\)[4]\^*\)] - 840\/13\ Re[\(M\_-\)[3]\ \(\(E\_+\)[4]\^*\)] - 1200\/13\ Re[\(M\_-\)[5]\ \(\(E\_+\)[4]\^*\)] + 30\ Re[\(M\_+\)[1]\ \(\(E\_+\)[4]\^*\)] + 840\/13\ Re[\(M\_+\)[3]\ \(\(E\_+\)[4]\^*\)] + 1200\/13\ Re[\(M\_+\)[5]\ \(\(E\_+\)[4]\^*\)] + 42\ Re[\(E\_+\)[0]\ \(\(E\_+\)[5]\^*\)] + 840\/13\ Re[\(E\_+\)[2]\ \(\(E\_+\)[5]\^*\)] + 1680\/13\ Re[\(E\_+\)[4]\ \(\(E\_+\)[5]\^*\)] - 630\/13\ Re[\(M\_-\)[2]\ \(\(E\_+\)[5]\^*\)] - 1120\/13\ Re[\(M\_-\)[4]\ \(\(E\_+\)[5]\^*\)] - 25200\/221\ Re[\(M\_-\)[6]\ \(\(E\_+\)[5]\^*\)] + 630\/13\ Re[\(M\_+\)[2]\ \(\(E\_+\)[5]\^*\)] + 1120\/13\ Re[\(M\_+\)[4]\ \(\(E\_+\)[5]\^*\)] + 25200\/221\ Re[\(M\_+\)[6]\ \(\(E\_+\)[5]\^*\)] + 2016\/13\ Re[\(E\_+\)[1]\ \(\(E\_+\)[6]\^*\)] + 6440\/39\ Re[\(E\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 3360\/13\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 840\/13\ Re[\(M\_-\)[3]\ \(\(E\_+\)[6]\^*\)] - 23520\/221\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] + 840\/13\ Re[\(M\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 23520\/221\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 20\ Re[\(M\_+\)[4]\ \(\(M\_-\)[1]\^*\)] - 40\ Re[\(M\_+\)[3]\ \(\(M\_-\)[2]\^*\)] + 90\/13\ Re[\(M\_+\)[5]\ \(\(M\_-\)[2]\^*\)] - 300\/7\ Re[\(M\_+\)[2]\ \(\(M\_-\)[3]\^*\)] + 40\/13\ Re[\(M\_+\)[4]\ \(\(M\_-\)[3]\^*\)] + 252\/13\ Re[\(M\_+\)[6]\ \(\(M\_-\)[3]\^*\)] + 20\/7\ Re[\(M\_-\)[3]\ \(\(M\_-\)[4]\^*\)] - 80\/3\ Re[\(M\_+\)[1]\ \(\(M\_-\)[4]\^*\)] + 300\/91\ Re[\(M\_+\)[3]\ \(\(M\_-\)[4]\^*\)] + 280\/13\ Re[\(M\_+\)[5]\ \(\(M\_-\)[4]\^*\)] + 10\ Re[\(M\_-\)[2]\ \(\(M\_-\)[5]\^*\)] + 600\/13\ Re[\(M\_-\)[4]\ \(\(M\_-\)[5]\^*\)] + 120\/13\ Re[\(M\_+\)[2]\ \(\(M\_-\)[5]\^*\)] + 300\/13\ Re[\(M\_+\)[4]\ \(\(M\_-\)[5]\^*\)] + 7840\/221\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] + 42\ Re[\(M\_-\)[1]\ \(\(M\_-\)[6]\^*\)] + 840\/13\ Re[\(M\_-\)[3]\ \(\(M\_-\)[6]\^*\)] + 1680\/13\ Re[\(M\_-\)[5]\ \(\(M\_-\)[6]\^*\)] + 210\/13\ Re[\(M\_+\)[1]\ \(\(M\_-\)[6]\^*\)] + 336\/13\ Re[\(M\_+\)[3]\ \(\(M\_-\)[6]\^*\)] + 8400\/221\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] - 120\/7\ Re[\(M\_+\)[2]\ \(\(M\_+\)[3]\^*\)] - 40\/3\ Re[\(M\_+\)[1]\ \(\(M\_+\)[4]\^*\)] + 60\/13\ Re[\(M\_+\)[3]\ \(\(M\_+\)[4]\^*\)] + 210\/13\ Re[\(M\_+\)[2]\ \(\(M\_+\)[5]\^*\)] + 800\/13\ Re[\(M\_+\)[4]\ \(\(M\_+\)[5]\^*\)] + 882\/13\ Re[\(M\_+\)[1]\ \(\(M\_+\)[6]\^*\)] + 1120\/13\ Re[\(M\_+\)[3]\ \(\(M\_+\)[6]\^*\)] + 35280\/221\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_5[ x] + \((\(-\(150\/11\)\)\ Abs[\(E\_-\)[4]]\^2 - 80\/33\ Abs[\(E\_-\)[5]]\^2 + 5400\/187\ Abs[\(E\_-\)[6]]\^2 - 50\/33\ Abs[\(E\_+\)[3]]\^2 + 240\/11\ Abs[\(E\_+\)[4]]\^2 + 12600\/187\ Abs[\(E\_+\)[5]]\^2 + \(490000\ Abs[\(E\_+\)[6]]\^2\)\ \/3553 - 50\/33\ Abs[\(M\_-\)[4]]\^2 + 240\/11\ Abs[\(M\_-\)[5]]\^2 + 12600\/187\ Abs[\(M\_-\)[6]]\^2 - 150\/11\ Abs[\(M\_+\)[3]]\^2 - 80\/33\ Abs[\(M\_+\)[4]]\^2 + 5400\/187\ Abs[\(M\_+\)[5]]\^2 + \(294000\ \ Abs[\(M\_+\)[6]]\^2\)\/3553 + 450\/11\ Re[\(E\_+\)[4]\ \(\(E\_-\)[2]\^*\)] - 112\/5\ Re[\(E\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 42\ Re[\(M\_-\)[6]\ \(\(E\_-\)[2]\^*\)] + 42\ Re[\(M\_+\)[6]\ \(\(E\_-\)[2]\^*\)] + 800\/11\ Re[\(E\_+\)[3]\ \(\(E\_-\)[3]\^*\)] - 126\/11\ Re[\(E\_+\)[5]\ \(\(E\_-\)[3]\^*\)] - 630\/11\ Re[\(M\_-\)[5]\ \(\(E\_-\)[3]\^*\)] + 630\/11\ Re[\(M\_+\)[5]\ \(\(E\_-\)[3]\^*\)] + 900\/11\ Re[\(E\_+\)[2]\ \(\(E\_-\)[4]\^*\)] - 6552\/187\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] - 700\/11\ Re[\(M\_-\)[4]\ \(\(E\_-\)[4]\^*\)] - 1008\/11\ Re[\(M\_-\)[6]\ \(\(E\_-\)[4]\^*\)] + 700\/11\ Re[\(M\_+\)[4]\ \(\(E\_-\)[4]\^*\)] + 1008\/11\ Re[\(M\_+\)[6]\ \(\(E\_-\)[4]\^*\)] - 280\/11\ Re[\(E\_-\)[3]\ \(\(E\_-\)[5]\^*\)] + 720\/11\ Re[\(E\_+\)[1]\ \(\(E\_-\)[5]\^*\)] + 140\/33\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 5600\/187\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 700\/11\ Re[\(M\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 1120\/11\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] + 700\/11\ Re[\(M\_+\)[3]\ \(\(E\_-\)[5]\^*\)] + 1120\/11\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 210\/11\ Re[\(E\_-\)[2]\ \(\(E\_-\)[6]\^*\)] + 30\ Re[\(E\_+\)[0]\ \(\(E\_-\)[6]\^*\)] - 5040\/187\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 16800\/323\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 630\/11\ Re[\(M\_-\)[2]\ \(\(E\_-\)[6]\^*\)] - 1120\/11\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 25200\/187\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] + 630\/11\ Re[\(M\_+\)[2]\ \(\(E\_-\)[6]\^*\)] + 1120\/11\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 25200\/187\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 42\ Re[\(M\_-\)[6]\ \(\(E\_+\)[0]\^*\)] + 42\ Re[\(M\_+\)[6]\ \(\(E\_+\)[0]\^*\)] - 630\/11\ Re[\(M\_-\)[5]\ \(\(E\_+\)[1]\^*\)] + 630\/11\ Re[\(M\_+\)[5]\ \(\(E\_+\)[1]\^*\)] - 700\/11\ Re[\(M\_-\)[4]\ \(\(E\_+\)[2]\^*\)] - 1008\/11\ Re[\(M\_-\)[6]\ \(\(E\_+\)[2]\^*\)] + 700\/11\ Re[\(M\_+\)[4]\ \(\(E\_+\)[2]\^*\)] + 1008\/11\ Re[\(M\_+\)[6]\ \(\(E\_+\)[2]\^*\)] - 700\/11\ Re[\(M\_-\)[3]\ \(\(E\_+\)[3]\^*\)] - 1120\/11\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] + 700\/11\ Re[\(M\_+\)[3]\ \(\(E\_+\)[3]\^*\)] + 1120\/11\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] - 630\/11\ Re[\(M\_-\)[2]\ \(\(E\_+\)[4]\^*\)] - 1120\/11\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] - 25200\/187\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] + 630\/11\ Re[\(M\_+\)[2]\ \(\(E\_+\)[4]\^*\)] + 1120\/11\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] + 25200\/187\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] + 126\/11\ Re[\(E\_+\)[1]\ \(\(E\_+\)[5]\^*\)] + 560\/11\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 42\ Re[\(M\_-\)[1]\ \(\(E\_+\)[5]\^*\)] - 1008\/11\ Re[\(M\_-\)[3]\ \(\(E\_+\)[5]\^*\)] - 25200\/187\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] + 42\ Re[\(M\_+\)[1]\ \(\(E\_+\)[5]\^*\)] + 1008\/11\ Re[\(M\_+\)[3]\ \(\(E\_+\)[5]\^*\)] + 25200\/187\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] + 56\ Re[\(E\_+\)[0]\ \(\(E\_+\)[6]\^*\)] + 4368\/55\ Re[\(E\_+\)[2]\ \(\(E\_+\)[6]\^*\)] + 2520\/17\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 336\/5\ Re[\(M\_-\)[2]\ \(\(E\_+\)[6]\^*\)] - 22680\/187\ Re[\(M\_-\)[ 4]\ \(\(E\_+\)[ 6]\^*\)] - \(588000\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\ \)]\)\/3553 + 336\/5\ Re[\(M\_+\)[2]\ \(\(E\_+\)[6]\^*\)] + 22680\/187\ Re[\(M\_+\)[ 4]\ \(\(E\_+\)[ 6]\^*\)] + \(588000\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\ \)]\)\/3553 - 30\ Re[\(M\_+\)[5]\ \(\(M\_-\)[1]\^*\)] - 720\/11\ Re[\(M\_+\)[4]\ \(\(M\_-\)[2]\^*\)] + 42\/5\ Re[\(M\_+\)[6]\ \(\(M\_-\)[2]\^*\)] - 900\/11\ Re[\(M\_+\)[3]\ \(\(M\_-\)[3]\^*\)] - 800\/11\ Re[\(M\_+\)[2]\ \(\(M\_-\)[4]\^*\)] - 140\/33\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] + 5040\/187\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] - 450\/11\ Re[\(M\_+\)[1]\ \(\(M\_-\)[5]\^*\)] + 5040\/187\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] + 126\/11\ Re[\(M\_-\)[2]\ \(\(M\_-\)[6]\^*\)] + 560\/11\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] + 126\/11\ Re[\(M\_+\)[2]\ \(\(M\_-\)[6]\^*\)] + 5600\/187\ Re[\(M\_+\)[ 4]\ \(\(M\_-\)[ 6]\^*\)] + \(176400\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\ \)]\)\/3553 - 280\/11\ Re[\(M\_+\)[2]\ \(\(M\_+\)[4]\^*\)] - 210\/11\ Re[\(M\_+\)[1]\ \(\(M\_+\)[5]\^*\)] + 1008\/55\ Re[\(M\_+\)[2]\ \(\(M\_+\)[6]\^*\)] + 12600\/187\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)])\)\ P\_6[ x] + \(\(1\/138567\)\((14\ \((575586\ Re[\(E\_+\)[ 5]\ \(\(E\_-\)[2]\^*\)] + 1090125\ Re[\(E\_+\)[4]\ \(\(E\_-\)[3]\^*\)] - 135432\ Re[\(E\_+\)[6]\ \(\(E\_-\)[3]\^*\)] - 767448\ Re[\(M\_-\)[6]\ \(\(E\_-\)[3]\^*\)] + 767448\ Re[\(M\_+\)[6]\ \(\(E\_-\)[3]\^*\)] + 1356600\ Re[\(E\_+\)[3]\ \(\(E\_-\)[4]\^*\)] + 51300\ Re[\(E\_+\)[5]\ \(\(E\_-\)[4]\^*\)] - 872100\ Re[\(M\_-\)[5]\ \(\(E\_-\)[4]\^*\)] + 872100\ Re[\(M\_+\)[5]\ \(\(E\_-\)[4]\^*\)] - 387600\ Re[\(E\_-\)[4]\ \(\(E\_-\)[5]\^*\)] + 1308150\ Re[\(E\_+\)[2]\ \(\(E\_-\)[5]\^*\)] + 159600\ Re[\(E\_+\)[4]\ \(\(E\_-\)[5]\^*\)] - 364500\ Re[\(E\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 904400\ Re[\(M\_-\)[4]\ \(\(E\_-\)[5]\^*\)] - 1385100\ Re[\(M\_-\)[6]\ \(\(E\_-\)[5]\^*\)] + 904400\ Re[\(M\_+\)[4]\ \(\(E\_-\)[5]\^*\)] + 1385100\ Re[\(M\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 348840\ Re[\(E\_-\)[3]\ \(\(E\_-\)[6]\^*\)] - 153900\ Re[\(E\_-\)[5]\ \(\(E\_-\)[6]\^*\)] + 959310\ Re[\(E\_+\)[1]\ \(\(E\_-\)[6]\^*\)] + 153900\ Re[\(E\_+\)[3]\ \(\(E\_-\)[6]\^*\)] - 302400\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 872100\ Re[\(M\_-\)[3]\ \(\(E\_-\)[6]\^*\)] - 1436400\ Re[\(M\_-\)[5]\ \(\(E\_-\)[6]\^*\)] + 872100\ Re[\(M\_+\)[3]\ \(\(E\_-\)[6]\^*\)] + 1436400\ Re[\(M\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 767448\ Re[\(M\_-\)[6]\ \(\(E\_+\)[1]\^*\)] + 767448\ Re[\(M\_+\)[6]\ \(\(E\_+\)[1]\^*\)] - 872100\ Re[\(M\_-\)[5]\ \(\(E\_+\)[2]\^*\)] + 872100\ Re[\(M\_+\)[5]\ \(\(E\_+\)[2]\^*\)] - 904400\ Re[\(M\_-\)[4]\ \(\(E\_+\)[3]\^*\)] - 1385100\ Re[\(M\_-\)[6]\ \(\(E\_+\)[3]\^*\)] + 904400\ Re[\(M\_+\)[4]\ \(\(E\_+\)[3]\^*\)] + 1385100\ Re[\(M\_+\)[6]\ \(\(E\_+\)[3]\^*\)] - 96900\ Re[\(E\_+\)[3]\ \(\(E\_+\)[4]\^*\)] - 872100\ Re[\(M\_-\)[3]\ \(\(E\_+\)[4]\^*\)] - 1436400\ Re[\(M\_-\)[5]\ \(\(E\_+\)[4]\^*\)] + 872100\ Re[\(M\_+\)[3]\ \(\(E\_+\)[4]\^*\)] + 1436400\ Re[\(M\_+\)[5]\ \(\(E\_+\)[4]\^*\)] - 34884\ Re[\(E\_+\)[2]\ \(\(E\_+\)[5]\^*\)] + 410400\ Re[\(E\_+\)[4]\ \(\(E\_+\)[5]\^*\)] - 767448\ Re[\(M\_-\)[2]\ \(\(E\_+\)[5]\^*\)] - 1385100\ Re[\(M\_-\)[4]\ \(\(E\_+\)[5]\^*\)] - 1890000\ Re[\(M\_-\)[6]\ \(\(E\_+\)[5]\^*\)] + 767448\ Re[\(M\_+\)[2]\ \(\(E\_+\)[5]\^*\)] + 1385100\ Re[\(M\_+\)[4]\ \(\(E\_+\)[5]\^*\)] + 1890000\ Re[\(M\_+\)[6]\ \(\(E\_+\)[5]\^*\)] + 127908\ Re[\(E\_+\)[1]\ \(\(E\_+\)[6]\^*\)] + 554040\ Re[\(E\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 1417500\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 554268\ Re[\(M\_-\)[1]\ \(\(E\_+\)[6]\^*\)] - 1218888\ Re[\(M\_-\)[3]\ \(\(E\_+\)[6]\^*\)] - 1822500\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] + 554268\ Re[\(M\_+\)[1]\ \(\(E\_+\)[6]\^*\)] + 1218888\ Re[\(M\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 1822500\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 415701\ Re[\(M\_+\)[6]\ \(\(M\_-\)[1]\^*\)] - 959310\ Re[\(M\_+\)[5]\ \(\(M\_-\)[2]\^*\)] - 1308150\ Re[\(M\_+\)[4]\ \(\(M\_-\)[3]\^*\)] - 45144\ Re[\(M\_+\)[6]\ \(\(M\_-\)[3]\^*\)] - 1356600\ Re[\(M\_+\)[3]\ \(\(M\_-\)[4]\^*\)] - 153900\ Re[\(M\_+\)[5]\ \(\(M\_-\)[4]\^*\)] - 96900\ Re[\(M\_-\)[4]\ \(\(M\_-\)[5]\^*\)] - 1090125\ Re[\(M\_+\)[2]\ \(\(M\_-\)[5]\^*\)] - 159600\ Re[\(M\_+\)[4]\ \(\(M\_-\)[5]\^*\)] + 291600\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] - 34884\ Re[\(M\_-\)[3]\ \(\(M\_-\)[6]\^*\)] + 410400\ Re[\(M\_-\)[5]\ \(\(M\_-\)[6]\^*\)] - 575586\ Re[\(M\_+\)[1]\ \(\(M\_-\)[6]\^*\)] - 51300\ Re[\(M\_+\)[3]\ \(\(M\_-\)[6]\^*\)] + 302400\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] - 387600\ Re[\(M\_+\)[3]\ \(\(M\_+\)[4]\^*\)] - 348840\ Re[\(M\_+\)[2]\ \(\(M\_+\)[5]\^*\)] - 153900\ Re[\(M\_+\)[4]\ \(\(M\_+\)[5]\^*\)] - 255816\ Re[\(M\_+\)[1]\ \(\(M\_+\)[6]\^*\)] - 55404\ Re[\(M\_+\)[3]\ \(\(M\_+\)[6]\^*\)] + 540000\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_7[ x])\)\) - \(\(1\/13585\)\((14\ \((26600\ Abs[\(E\_-\)[5]]\^2 + 15750\ Abs[\(E\_-\)[6]]\^2 + 9975\ Abs[\(E\_+\)[4]]\^2 - 15750\ Abs[\(E\_+\)[5]]\^2 - 68750\ Abs[\(E\_+\)[6]]\^2 + 9975\ Abs[\(M\_-\)[5]]\^2 - 15750\ Abs[\(M\_-\)[6]]\^2 + 26600\ Abs[\(M\_+\)[4]]\^2 + 15750\ Abs[\(M\_+\)[5]]\^2 - 20625\ Abs[\(M\_+\)[6]]\^2 - 76076\ Re[\(E\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 150480\ Re[\(E\_+\)[5]\ \(\(E\_-\)[3]\^*\)] - 199500\ Re[\(E\_+\)[4]\ \(\(E\_-\)[4]\^*\)] - 11880\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] + 112860\ Re[\(M\_-\)[6]\ \(\(E\_-\)[4]\^*\)] - 112860\ Re[\(M\_+\)[6]\ \(\(E\_-\)[4]\^*\)] - 212800\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 31500\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 119700\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] - 119700\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 51300\ Re[\(E\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 188100\ Re[\(E\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 37800\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 31500\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 119700\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] + 189000\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] - 119700\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 189000\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 112860\ Re[\(M\_-\)[6]\ \(\(E\_+\)[2]\^*\)] - 112860\ Re[\(M\_+\)[6]\ \(\(E\_+\)[2]\^*\)] + 119700\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] - 119700\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] + 119700\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] + 189000\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] - 119700\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] - 189000\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] + 17100\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] + 112860\ Re[\(M\_-\)[3]\ \(\(E\_+\)[5]\^*\)] + 189000\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] - 112860\ Re[\(M\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 189000\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] + 7524\ Re[\(E\_+\)[2]\ \(\(E\_+\)[6]\^*\)] - 37800\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] + 97812\ Re[\(M\_-\)[2]\ \(\(E\_+\)[6]\^*\)] + 178200\ Re[\(M\_-\)[4]\ \(\(E\_+\)[6]\^*\)] + 247500\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 97812\ Re[\(M\_+\)[2]\ \(\(E\_+\)[6]\^*\)] - 178200\ Re[\(M\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 247500\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] + 130416\ Re[\(M\_+\)[6]\ \(\(M\_-\)[2]\^*\)] + 188100\ Re[\(M\_+\)[5]\ \(\(M\_-\)[3]\^*\)] + 212800\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] + 29700\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] + 199500\ Re[\(M\_+\)[3]\ \(\(M\_-\)[5]\^*\)] + 37800\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] + 17100\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] + 150480\ Re[\(M\_+\)[2]\ \(\(M\_-\)[6]\^*\)] + 31500\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] - 27000\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] + 51300\ Re[\(M\_+\)[3]\ \(\(M\_+\)[5]\^*\)] + 45144\ Re[\(M\_+\)[2]\ \(\(M\_+\)[6]\^*\)] + 27000\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)])\)\ P\_8[ x])\)\) + \(\(1\/2431\)\((84\ \((6006\ Re[\(E\_+\)[ 6]\ \(\(E\_-\)[3]\^*\)] + 8316\ Re[\(E\_+\)[5]\ \(\(E\_-\)[4]\^*\)] + 9450\ Re[\(E\_+\)[4]\ \(\(E\_-\)[5]\^*\)] + 1540\ Re[\(E\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 4620\ Re[\(M\_-\)[6]\ \(\(E\_-\)[5]\^*\)] + 4620\ Re[\(M\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 2100\ Re[\(E\_-\)[5]\ \(\(E\_-\)[6]\^*\)] + 9240\ Re[\(E\_+\)[3]\ \(\(E\_-\)[6]\^*\)] + 2025\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 4725\ Re[\(M\_-\)[5]\ \(\(E\_-\)[6]\^*\)] + 4725\ Re[\(M\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 4620\ Re[\(M\_-\)[6]\ \(\(E\_+\)[3]\^*\)] + 4620\ Re[\(M\_+\)[6]\ \(\(E\_+\)[3]\^*\)] - 4725\ Re[\(M\_-\)[5]\ \(\(E\_+\)[4]\^*\)] + 4725\ Re[\(M\_+\)[5]\ \(\(E\_+\)[4]\^*\)] - 945\ Re[\(E\_+\)[4]\ \(\(E\_+\)[5]\^*\)] - 4620\ Re[\(M\_-\)[4]\ \(\(E\_+\)[5]\^*\)] - 7425\ Re[\(M\_-\)[6]\ \(\(E\_+\)[5]\^*\)] + 4620\ Re[\(M\_+\)[4]\ \(\(E\_+\)[5]\^*\)] + 7425\ Re[\(M\_+\)[6]\ \(\(E\_+\)[5]\^*\)] - 770\ Re[\(E\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 660\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 4290\ Re[\(M\_-\)[3]\ \(\(E\_+\)[6]\^*\)] - 7260\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] + 4290\ Re[\(M\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 7260\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 7722\ Re[\(M\_+\)[6]\ \(\(M\_-\)[3]\^*\)] - 9240\ Re[\(M\_+\)[5]\ \(\(M\_-\)[4]\^*\)] - 9450\ Re[\(M\_+\)[4]\ \(\(M\_-\)[5]\^*\)] - 1980\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] - 945\ Re[\(M\_-\)[5]\ \(\(M\_-\)[6]\^*\)] - 8316\ Re[\(M\_+\)[3]\ \(\(M\_-\)[6]\^*\)] - 2025\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] - 2100\ Re[\(M\_+\)[4]\ \(\(M\_+\)[5]\^*\)] - 1980\ Re[\(M\_+\)[3]\ \(\(M\_+\)[6]\^*\)] - 1485\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_9[ x])\)\) - \(\(1\/96577\)\((294\ \((15525\ Abs[\(E\_-\)[6]]\^2 + 8073\ Abs[\(E\_+\)[5]]\^2 - 972\ Abs[\(E\_+\)[6]]\^2 + 8073\ Abs[\(M\_-\)[6]]\^2 + 15525\ Abs[\(M\_+\)[5]]\^2 + 12636\ Abs[\(M\_+\)[6]]\^2 - 125580\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] - 149040\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 155250\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 35640\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 68310\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] - 68310\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 68310\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] - 68310\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] + 68310\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] - 68310\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] + 15180\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] + 65780\ Re[\(M\_-\)[4]\ \(\(E\_+\)[6]\^*\)] + 106920\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 65780\ Re[\(M\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 106920\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] + 143520\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] + 155250\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] + 149040\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] + 38610\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] + 30360\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)])\)\ P\_10[ x])\)\) + \(\(1\/4199\)\((2772\ \((910\ Re[\(E\_+\)[ 6]\ \(\(E\_-\)[5]\^*\)] + 990\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 396\ Re[\(M\_-\)[6]\ \(\(E\_+\)[5]\^*\)] + 396\ Re[\(M\_+\)[6]\ \(\(E\_+\)[5]\^*\)] - 102\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 390\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] + 390\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 975\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] - 990\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] - 180\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_11[ x])\)\) - \(\(1\/7429\)\((30492\ \((11\ Abs[\(E\_+\)[6]]\^2 + 6\ \((3\ Abs[\(M\_+\)[6]]\^2 - 35\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 13\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 13\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] + 36\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)])\))\)\ P\_12[ x])\)\)\)], "Output", CellLabel->"Out[33]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LTexp = RtoLegendre[R\_LT, 6]\)], "Input", CellLabel->"In[34]:="], Cell[BoxData[ RowBox[{\(Simplify::"time"\), \(\(:\)\(\ \)\), "\<\"Time spent on a \ transformation exceeded \\!\\(300\\) seconds, and the transformation was \ aborted. Increasing the value of TimeConstraint option may improve the result \ of simplification. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Simplify::time\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[34]:="], Cell[BoxData[ \(2\ Re[\(E\_-\)[2]\ \(\(S\_-\)[1]\^*\)] + 4\ Re[\(E\_-\)[4]\ \(\(S\_-\)[1]\^*\)] + 6\ Re[\(E\_-\)[6]\ \(\(S\_-\)[1]\^*\)] - Re[\(E\_+\)[0]\ \(\(S\_-\)[1]\^*\)] - 3\ Re[\(E\_+\)[2]\ \(\(S\_-\)[1]\^*\)] - 5\ Re[\(E\_+\)[4]\ \(\(S\_-\)[1]\^*\)] - 7\ Re[\(E\_+\)[6]\ \(\(S\_-\)[1]\^*\)] + 14\ Re[\(E\_-\)[3]\ \(\(S\_-\)[2]\^*\)] + 22\ Re[\(E\_-\)[5]\ \(\(S\_-\)[2]\^*\)] - 6\ Re[\(E\_+\)[1]\ \(\(S\_-\)[2]\^*\)] - 14\ Re[\(E\_+\)[3]\ \(\(S\_-\)[2]\^*\)] - 22\ Re[\(E\_+\)[5]\ \(\(S\_-\)[2]\^*\)] + 2\ Re[\(M\_-\)[1]\ \(\(S\_-\)[2]\^*\)] + 2\ Re[\(M\_-\)[3]\ \(\(S\_-\)[2]\^*\)] + 2\ Re[\(M\_-\)[5]\ \(\(S\_-\)[2]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(S\_-\)[2]\^*\)] - 2\ Re[\(M\_+\)[3]\ \(\(S\_-\)[2]\^*\)] - 2\ Re[\(M\_+\)[5]\ \(\(S\_-\)[2]\^*\)] - 3\ Re[\(E\_-\)[2]\ \(\(S\_-\)[3]\^*\)] + 45\ Re[\(E\_-\)[4]\ \(\(S\_-\)[3]\^*\)] + 63\ Re[\(E\_-\)[6]\ \(\(S\_-\)[3]\^*\)] - 3\ Re[\(E\_+\)[0]\ \(\(S\_-\)[3]\^*\)] - 18\ Re[\(E\_+\)[2]\ \(\(S\_-\)[3]\^*\)] - 36\ Re[\(E\_+\)[4]\ \(\(S\_-\)[3]\^*\)] - 54\ Re[\(E\_+\)[6]\ \(\(S\_-\)[3]\^*\)] + 9\ Re[\(M\_-\)[2]\ \(\(S\_-\)[3]\^*\)] + 9\ Re[\(M\_-\)[4]\ \(\(S\_-\)[3]\^*\)] + 9\ Re[\(M\_-\)[6]\ \(\(S\_-\)[3]\^*\)] - 9\ Re[\(M\_+\)[2]\ \(\(S\_-\)[3]\^*\)] - 9\ Re[\(M\_+\)[4]\ \(\(S\_-\)[3]\^*\)] - 9\ Re[\(M\_+\)[6]\ \(\(S\_-\)[3]\^*\)] - 12\ Re[\(E\_-\)[3]\ \(\(S\_-\)[4]\^*\)] + 104\ Re[\(E\_-\)[5]\ \(\(S\_-\)[4]\^*\)] - 12\ Re[\(E\_+\)[1]\ \(\(S\_-\)[4]\^*\)] - 40\ Re[\(E\_+\)[3]\ \(\(S\_-\)[4]\^*\)] - 72\ Re[\(E\_+\)[5]\ \(\(S\_-\)[4]\^*\)] + 4\ Re[\(M\_-\)[1]\ \(\(S\_-\)[4]\^*\)] + 24\ Re[\(M\_-\)[3]\ \(\(S\_-\)[4]\^*\)] + 24\ Re[\(M\_-\)[5]\ \(\(S\_-\)[4]\^*\)] - 4\ Re[\(M\_+\)[1]\ \(\(S\_-\)[4]\^*\)] - 24\ Re[\(M\_+\)[3]\ \(\(S\_-\)[4]\^*\)] - 24\ Re[\(M\_+\)[5]\ \(\(S\_-\)[4]\^*\)] - 5\ Re[\(E\_-\)[2]\ \(\(S\_-\)[5]\^*\)] - 30\ Re[\(E\_-\)[4]\ \(\(S\_-\)[5]\^*\)] + 200\ Re[\(E\_-\)[6]\ \(\(S\_-\)[5]\^*\)] - 5\ Re[\(E\_+\)[0]\ \(\(S\_-\)[5]\^*\)] - 30\ Re[\(E\_+\)[2]\ \(\(S\_-\)[5]\^*\)] - 75\ Re[\(E\_+\)[4]\ \(\(S\_-\)[5]\^*\)] - 125\ Re[\(E\_+\)[6]\ \(\(S\_-\)[5]\^*\)] + 15\ Re[\(M\_-\)[2]\ \(\(S\_-\)[5]\^*\)] + 50\ Re[\(M\_-\)[4]\ \(\(S\_-\)[5]\^*\)] + 50\ Re[\(M\_-\)[6]\ \(\(S\_-\)[5]\^*\)] - 15\ Re[\(M\_+\)[2]\ \(\(S\_-\)[5]\^*\)] - 50\ Re[\(M\_+\)[4]\ \(\(S\_-\)[5]\^*\)] - 50\ Re[\(M\_+\)[6]\ \(\(S\_-\)[5]\^*\)] - 18\ Re[\(E\_-\)[3]\ \(\(S\_-\)[6]\^*\)] - 60\ Re[\(E\_-\)[5]\ \(\(S\_-\)[6]\^*\)] - 18\ Re[\(E\_+\)[1]\ \(\(S\_-\)[6]\^*\)] - 60\ Re[\(E\_+\)[3]\ \(\(S\_-\)[6]\^*\)] - 126\ Re[\(E\_+\)[5]\ \(\(S\_-\)[6]\^*\)] + 6\ Re[\(M\_-\)[1]\ \(\(S\_-\)[6]\^*\)] + 36\ Re[\(M\_-\)[3]\ \(\(S\_-\)[6]\^*\)] + 90\ Re[\(M\_-\)[5]\ \(\(S\_-\)[6]\^*\)] - 6\ Re[\(M\_+\)[1]\ \(\(S\_-\)[6]\^*\)] - 36\ Re[\(M\_+\)[3]\ \(\(S\_-\)[6]\^*\)] - 90\ Re[\(M\_+\)[5]\ \(\(S\_-\)[6]\^*\)] + 2\ Re[\(E\_-\)[3]\ \(\(S\_+\)[0]\^*\)] + 4\ Re[\(E\_-\)[5]\ \(\(S\_+\)[0]\^*\)] - 3\ Re[\(E\_+\)[1]\ \(\(S\_+\)[0]\^*\)] - 5\ Re[\(E\_+\)[3]\ \(\(S\_+\)[0]\^*\)] - 7\ Re[\(E\_+\)[5]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_-\)[1]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_-\)[3]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_-\)[5]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[3]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[5]\ \(\(S\_+\)[0]\^*\)] + 2\ Re[\(E\_-\)[2]\ \(\(S\_+\)[1]\^*\)] + 10\ Re[\(E\_-\)[4]\ \(\(S\_+\)[1]\^*\)] + 18\ Re[\(E\_-\)[6]\ \(\(S\_+\)[1]\^*\)] + 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)] - 18\ Re[\(E\_+\)[2]\ \(\(S\_+\)[1]\^*\)] - 26\ Re[\(E\_+\)[4]\ \(\(S\_+\)[1]\^*\)] - 34\ Re[\(E\_+\)[6]\ \(\(S\_+\)[1]\^*\)] - 6\ Re[\(M\_-\)[2]\ \(\(S\_+\)[1]\^*\)] - 6\ Re[\(M\_-\)[4]\ \(\(S\_+\)[1]\^*\)] - 6\ Re[\(M\_-\)[6]\ \(\(S\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[2]\ \(\(S\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[4]\ \(\(S\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[6]\ \(\(S\_+\)[1]\^*\)] + 9\ Re[\(E\_-\)[3]\ \(\(S\_+\)[2]\^*\)] + 27\ Re[\(E\_-\)[5]\ \(\(S\_+\)[2]\^*\)] + 9\ Re[\(E\_+\)[1]\ \(\(S\_+\)[2]\^*\)] - 54\ Re[\(E\_+\)[3]\ \(\(S\_+\)[2]\^*\)] - 72\ Re[\(E\_+\)[5]\ \(\(S\_+\)[2]\^*\)] - 3\ Re[\(M\_-\)[1]\ \(\(S\_+\)[2]\^*\)] - 18\ Re[\(M\_-\)[3]\ \(\(S\_+\)[2]\^*\)] - 18\ Re[\(M\_-\)[5]\ \(\(S\_+\)[2]\^*\)] + 3\ Re[\(M\_+\)[1]\ \(\(S\_+\)[2]\^*\)] + 18\ Re[\(M\_+\)[3]\ \(\(S\_+\)[2]\^*\)] + 18\ Re[\(M\_+\)[5]\ \(\(S\_+\)[2]\^*\)] + 4\ Re[\(E\_-\)[2]\ \(\(S\_+\)[3]\^*\)] + 24\ Re[\(E\_-\)[4]\ \(\(S\_+\)[3]\^*\)] + 56\ Re[\(E\_-\)[6]\ \(\(S\_+\)[3]\^*\)] + 4\ Re[\(E\_+\)[0]\ \(\(S\_+\)[3]\^*\)] + 24\ Re[\(E\_+\)[2]\ \(\(S\_+\)[3]\^*\)] - 120\ Re[\(E\_+\)[4]\ \(\(S\_+\)[3]\^*\)] - 152\ Re[\(E\_+\)[6]\ \(\(S\_+\)[3]\^*\)] - 12\ Re[\(M\_-\)[2]\ \(\(S\_+\)[3]\^*\)] - 40\ Re[\(M\_-\)[4]\ \(\(S\_+\)[3]\^*\)] - 40\ Re[\(M\_-\)[6]\ \(\(S\_+\)[3]\^*\)] + 12\ Re[\(M\_+\)[2]\ \(\(S\_+\)[3]\^*\)] + 40\ Re[\(M\_+\)[4]\ \(\(S\_+\)[3]\^*\)] + 40\ Re[\(M\_+\)[6]\ \(\(S\_+\)[3]\^*\)] + 15\ Re[\(E\_-\)[3]\ \(\(S\_+\)[4]\^*\)] + 50\ Re[\(E\_-\)[5]\ \(\(S\_+\)[4]\^*\)] + 15\ Re[\(E\_+\)[1]\ \(\(S\_+\)[4]\^*\)] + 50\ Re[\(E\_+\)[3]\ \(\(S\_+\)[4]\^*\)] - 225\ Re[\(E\_+\)[5]\ \(\(S\_+\)[4]\^*\)] - 5\ Re[\(M\_-\)[1]\ \(\(S\_+\)[4]\^*\)] - 30\ Re[\(M\_-\)[3]\ \(\(S\_+\)[4]\^*\)] - 75\ Re[\(M\_-\)[5]\ \(\(S\_+\)[4]\^*\)] + 5\ Re[\(M\_+\)[1]\ \(\(S\_+\)[4]\^*\)] + 30\ Re[\(M\_+\)[3]\ \(\(S\_+\)[4]\^*\)] + 75\ Re[\(M\_+\)[5]\ \(\(S\_+\)[4]\^*\)] + 6\ Re[\(E\_-\)[2]\ \(\(S\_+\)[5]\^*\)] + 36\ Re[\(E\_-\)[4]\ \(\(S\_+\)[5]\^*\)] + 90\ Re[\(E\_-\)[6]\ \(\(S\_+\)[5]\^*\)] + 6\ Re[\(E\_+\)[0]\ \(\(S\_+\)[5]\^*\)] + 36\ Re[\(E\_+\)[2]\ \(\(S\_+\)[5]\^*\)] + 90\ Re[\(E\_+\)[4]\ \(\(S\_+\)[5]\^*\)] - 378\ Re[\(E\_+\)[6]\ \(\(S\_+\)[5]\^*\)] - 18\ Re[\(M\_-\)[2]\ \(\(S\_+\)[5]\^*\)] - 60\ Re[\(M\_-\)[4]\ \(\(S\_+\)[5]\^*\)] - 126\ Re[\(M\_-\)[6]\ \(\(S\_+\)[5]\^*\)] + 18\ Re[\(M\_+\)[2]\ \(\(S\_+\)[5]\^*\)] + 60\ Re[\(M\_+\)[4]\ \(\(S\_+\)[5]\^*\)] + 126\ Re[\(M\_+\)[6]\ \(\(S\_+\)[5]\^*\)] + 21\ Re[\(E\_-\)[3]\ \(\(S\_+\)[6]\^*\)] + 70\ Re[\(E\_-\)[5]\ \(\(S\_+\)[6]\^*\)] + 21\ Re[\(E\_+\)[1]\ \(\(S\_+\)[6]\^*\)] + 70\ Re[\(E\_+\)[3]\ \(\(S\_+\)[6]\^*\)] + 147\ Re[\(E\_+\)[5]\ \(\(S\_+\)[6]\^*\)] - 7\ Re[\(M\_-\)[1]\ \(\(S\_+\)[6]\^*\)] - 42\ Re[\(M\_-\)[3]\ \(\(S\_+\)[6]\^*\)] - 105\ Re[\(M\_-\)[5]\ \(\(S\_+\)[6]\^*\)] + 7\ Re[\(M\_+\)[1]\ \(\(S\_+\)[6]\^*\)] + 42\ Re[\(M\_+\)[3]\ \(\(S\_+\)[6]\^*\)] + 105\ Re[\(M\_+\)[5]\ \(\(S\_+\)[6]\^*\)] + 3\ \((3\ Re[\(E\_-\)[3]\ \(\(S\_-\)[1]\^*\)] + 5\ Re[\(E\_-\)[5]\ \(\(S\_-\)[1]\^*\)] - 2\ Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - 4\ Re[\(E\_+\)[3]\ \(\(S\_-\)[1]\^*\)] - 6\ Re[\(E\_+\)[5]\ \(\(S\_-\)[1]\^*\)] + 2\ Re[\(E\_-\)[2]\ \(\(S\_-\)[2]\^*\)] + 18\ Re[\(E\_-\)[4]\ \(\(S\_-\)[2]\^*\)] + 26\ Re[\(E\_-\)[6]\ \(\(S\_-\)[2]\^*\)] - 2\ Re[\(E\_+\)[0]\ \(\(S\_-\)[2]\^*\)] - 10\ Re[\(E\_+\)[2]\ \(\(S\_-\)[2]\^*\)] - 18\ Re[\(E\_+\)[4]\ \(\(S\_-\)[2]\^*\)] - 26\ Re[\(E\_+\)[6]\ \(\(S\_-\)[2]\^*\)] + 2\ Re[\(M\_-\)[2]\ \(\(S\_-\)[2]\^*\)] + 2\ Re[\(M\_-\)[4]\ \(\(S\_-\)[2]\^*\)] + 2\ Re[\(M\_-\)[6]\ \(\(S\_-\)[2]\^*\)] - 2\ Re[\(M\_+\)[2]\ \(\(S\_-\)[2]\^*\)] - 2\ Re[\(M\_+\)[4]\ \(\(S\_-\)[2]\^*\)] - 2\ Re[\(M\_+\)[6]\ \(\(S\_-\)[2]\^*\)] + 9\ Re[\(E\_-\)[3]\ \(\(S\_-\)[3]\^*\)] + 54\ Re[\(E\_-\)[5]\ \(\(S\_-\)[3]\^*\)] - 9\ Re[\(E\_+\)[1]\ \(\(S\_-\)[3]\^*\)] - 27\ Re[\(E\_+\)[3]\ \(\(S\_-\)[3]\^*\)] - 45\ Re[\(E\_+\)[5]\ \(\(S\_-\)[3]\^*\)] + 3\ Re[\(M\_-\)[1]\ \(\(S\_-\)[3]\^*\)] + 9\ Re[\(M\_-\)[3]\ \(\(S\_-\)[3]\^*\)] + 9\ Re[\(M\_-\)[5]\ \(\(S\_-\)[3]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(S\_-\)[3]\^*\)] - 9\ Re[\(M\_+\)[3]\ \(\(S\_-\)[3]\^*\)] - 9\ Re[\(M\_+\)[5]\ \(\(S\_-\)[3]\^*\)] - 4\ Re[\(E\_-\)[2]\ \(\(S\_-\)[4]\^*\)] + 24\ Re[\(E\_-\)[4]\ \(\(S\_-\)[4]\^*\)] + 120\ Re[\(E\_-\)[6]\ \(\(S\_-\)[4]\^*\)] - 4\ Re[\(E\_+\)[0]\ \(\(S\_-\)[4]\^*\)] - 24\ Re[\(E\_+\)[2]\ \(\(S\_-\)[4]\^*\)] - 56\ Re[\(E\_+\)[4]\ \(\(S\_-\)[4]\^*\)] - 88\ Re[\(E\_+\)[6]\ \(\(S\_-\)[4]\^*\)] + 12\ Re[\(M\_-\)[2]\ \(\(S\_-\)[4]\^*\)] + 24\ Re[\(M\_-\)[4]\ \(\(S\_-\)[4]\^*\)] + 24\ Re[\(M\_-\)[6]\ \(\(S\_-\)[4]\^*\)] - 12\ Re[\(M\_+\)[2]\ \(\(S\_-\)[4]\^*\)] - 24\ Re[\(M\_+\)[4]\ \(\(S\_-\)[4]\^*\)] - 24\ Re[\(M\_+\)[6]\ \(\(S\_-\)[4]\^*\)] - 15\ Re[\(E\_-\)[3]\ \(\(S\_-\)[5]\^*\)] + 50\ Re[\(E\_-\)[5]\ \(\(S\_-\)[5]\^*\)] - 15\ Re[\(E\_+\)[1]\ \(\(S\_-\)[5]\^*\)] - 50\ Re[\(E\_+\)[3]\ \(\(S\_-\)[5]\^*\)] - 100\ Re[\(E\_+\)[5]\ \(\(S\_-\)[5]\^*\)] + 5\ Re[\(M\_-\)[1]\ \(\(S\_-\)[5]\^*\)] + 30\ Re[\(M\_-\)[3]\ \(\(S\_-\)[5]\^*\)] + 50\ Re[\(M\_-\)[5]\ \(\(S\_-\)[5]\^*\)] - 5\ Re[\(M\_+\)[1]\ \(\(S\_-\)[5]\^*\)] - 30\ Re[\(M\_+\)[3]\ \(\(S\_-\)[5]\^*\)] - 50\ Re[\(M\_+\)[5]\ \(\(S\_-\)[5]\^*\)] - 6\ Re[\(E\_-\)[2]\ \(\(S\_-\)[6]\^*\)] - 36\ Re[\(E\_-\)[4]\ \(\(S\_-\)[6]\^*\)] + 90\ Re[\(E\_-\)[6]\ \(\(S\_-\)[6]\^*\)] - 6\ Re[\(E\_+\)[0]\ \(\(S\_-\)[6]\^*\)] - 36\ Re[\(E\_+\)[2]\ \(\(S\_-\)[6]\^*\)] - 90\ Re[\(E\_+\)[4]\ \(\(S\_-\)[6]\^*\)] - 162\ Re[\(E\_+\)[6]\ \(\(S\_-\)[6]\^*\)] + 18\ Re[\(M\_-\)[2]\ \(\(S\_-\)[6]\^*\)] + 60\ Re[\(M\_-\)[4]\ \(\(S\_-\)[6]\^*\)] + 90\ Re[\(M\_-\)[6]\ \(\(S\_-\)[6]\^*\)] - 18\ Re[\(M\_+\)[2]\ \(\(S\_-\)[6]\^*\)] - 60\ Re[\(M\_+\)[4]\ \(\(S\_-\)[6]\^*\)] - 90\ Re[\(M\_+\)[6]\ \(\(S\_-\)[6]\^*\)] + Re[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)] + 3\ Re[\(E\_-\)[4]\ \(\(S\_+\)[0]\^*\)] + 5\ Re[\(E\_-\)[6]\ \(\(S\_+\)[0]\^*\)] - 4\ Re[\(E\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 6\ Re[\(E\_+\)[4]\ \(\(S\_+\)[0]\^*\)] - 8\ Re[\(E\_+\)[6]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_-\)[2]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_-\)[4]\ \(\(S\_+\)[0]\^*\)] - Re[\(M\_-\)[6]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[4]\ \(\(S\_+\)[0]\^*\)] + Re[\(M\_+\)[6]\ \(\(S\_+\)[0]\^*\)] + 6\ Re[\(E\_-\)[3]\ \(\(S\_+\)[1]\^*\)] + 14\ Re[\(E\_-\)[5]\ \(\(S\_+\)[1]\^*\)] - 2\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 22\ Re[\(E\_+\)[3]\ \(\(S\_+\)[1]\^*\)] - 30\ Re[\(E\_+\)[5]\ \(\(S\_+\)[1]\^*\)] - 2\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 6\ Re[\(M\_-\)[3]\ \(\(S\_+\)[1]\^*\)] - 6\ Re[\(M\_-\)[5]\ \(\(S\_+\)[1]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[3]\ \(\(S\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[5]\ \(\(S\_+\)[1]\^*\)] + 3\ Re[\(E\_-\)[2]\ \(\(S\_+\)[2]\^*\)] + 18\ Re[\(E\_-\)[4]\ \(\(S\_+\)[2]\^*\)] + 36\ Re[\(E\_-\)[6]\ \(\(S\_+\)[2]\^*\)] + 3\ Re[\(E\_+\)[0]\ \(\(S\_+\)[2]\^*\)] - 9\ Re[\(E\_+\)[2]\ \(\(S\_+\)[2]\^*\)] - 63\ Re[\(E\_+\)[4]\ \(\(S\_+\)[2]\^*\)] - 81\ Re[\(E\_+\)[6]\ \(\(S\_+\)[2]\^*\)] - 9\ Re[\(M\_-\)[2]\ \(\(S\_+\)[2]\^*\)] - 18\ Re[\(M\_-\)[4]\ \(\(S\_+\)[2]\^*\)] - 18\ Re[\(M\_-\)[6]\ \(\(S\_+\)[2]\^*\)] + 9\ Re[\(M\_+\)[2]\ \(\(S\_+\)[2]\^*\)] + 18\ Re[\(M\_+\)[4]\ \(\(S\_+\)[2]\^*\)] + 18\ Re[\(M\_+\)[6]\ \(\(S\_+\)[2]\^*\)] + 12\ Re[\(E\_-\)[3]\ \(\(S\_+\)[3]\^*\)] + 40\ Re[\(E\_-\)[5]\ \(\(S\_+\)[3]\^*\)] + 12\ Re[\(E\_+\)[1]\ \(\(S\_+\)[3]\^*\)] - 24\ Re[\(E\_+\)[3]\ \(\(S\_+\)[3]\^*\)] - 136\ Re[\(E\_+\)[5]\ \(\(S\_+\)[3]\^*\)] - 4\ Re[\(M\_-\)[1]\ \(\(S\_+\)[3]\^*\)] - 24\ Re[\(M\_-\)[3]\ \(\(S\_+\)[3]\^*\)] - 40\ Re[\(M\_-\)[5]\ \(\(S\_+\)[3]\^*\)] + 4\ Re[\(M\_+\)[1]\ \(\(S\_+\)[3]\^*\)] + 24\ Re[\(M\_+\)[3]\ \(\(S\_+\)[3]\^*\)] + 40\ Re[\(M\_+\)[5]\ \(\(S\_+\)[3]\^*\)] + 5\ Re[\(E\_-\)[2]\ \(\(S\_+\)[4]\^*\)] + 30\ Re[\(E\_-\)[4]\ \(\(S\_+\)[4]\^*\)] + 75\ Re[\(E\_-\)[6]\ \(\(S\_+\)[4]\^*\)] + 5\ Re[\(E\_+\)[0]\ \(\(S\_+\)[4]\^*\)] + 30\ Re[\(E\_+\)[2]\ \(\(S\_+\)[4]\^*\)] - 50\ Re[\(E\_+\)[4]\ \(\(S\_+\)[4]\^*\)] - 250\ Re[\(E\_+\)[6]\ \(\(S\_+\)[4]\^*\)] - 15\ Re[\(M\_-\)[2]\ \(\(S\_+\)[4]\^*\)] - 50\ Re[\(M\_-\)[4]\ \(\(S\_+\)[4]\^*\)] - 75\ Re[\(M\_-\)[6]\ \(\(S\_+\)[4]\^*\)] + 15\ Re[\(M\_+\)[2]\ \(\(S\_+\)[4]\^*\)] + 50\ Re[\(M\_+\)[4]\ \(\(S\_+\)[4]\^*\)] + 75\ Re[\(M\_+\)[6]\ \(\(S\_+\)[4]\^*\)] + 18\ Re[\(E\_-\)[3]\ \(\(S\_+\)[5]\^*\)] + 60\ Re[\(E\_-\)[5]\ \(\(S\_+\)[5]\^*\)] + 18\ Re[\(E\_+\)[1]\ \(\(S\_+\)[5]\^*\)] + 60\ Re[\(E\_+\)[3]\ \(\(S\_+\)[5]\^*\)] - 90\ Re[\(E\_+\)[5]\ \(\(S\_+\)[5]\^*\)] - 6\ Re[\(M\_-\)[1]\ \(\(S\_+\)[5]\^*\)] - 36\ Re[\(M\_-\)[3]\ \(\(S\_+\)[5]\^*\)] - 90\ Re[\(M\_-\)[5]\ \(\(S\_+\)[5]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(S\_+\)[5]\^*\)] + 36\ Re[\(M\_+\)[3]\ \(\(S\_+\)[5]\^*\)] + 90\ Re[\(M\_+\)[5]\ \(\(S\_+\)[5]\^*\)] + 7\ Re[\(E\_-\)[2]\ \(\(S\_+\)[6]\^*\)] + 42\ Re[\(E\_-\)[4]\ \(\(S\_+\)[6]\^*\)] + 105\ Re[\(E\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 7\ Re[\(E\_+\)[0]\ \(\(S\_+\)[6]\^*\)] + 42\ Re[\(E\_+\)[2]\ \(\(S\_+\)[6]\^*\)] + 105\ Re[\(E\_+\)[4]\ \(\(S\_+\)[6]\^*\)] - 147\ Re[\(E\_+\)[6]\ \(\(S\_+\)[6]\^*\)] - 21\ Re[\(M\_-\)[2]\ \(\(S\_+\)[6]\^*\)] - 70\ Re[\(M\_-\)[4]\ \(\(S\_+\)[6]\^*\)] - 147\ Re[\(M\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 21\ Re[\(M\_+\)[2]\ \(\(S\_+\)[6]\^*\)] + 70\ Re[\(M\_+\)[4]\ \(\(S\_+\)[6]\^*\)] + 147\ Re[\(M\_+\)[6]\ \(\(S\_+\)[6]\^*\)])\)\ P\_1[ x] + \((20\ Re[\(E\_-\)[4]\ \(\(S\_-\)[1]\^*\)] + 30\ Re[\(E\_-\)[6]\ \(\(S\_-\)[1]\^*\)] - 15\ Re[\(E\_+\)[2]\ \(\(S\_-\)[1]\^*\)] - 25\ Re[\(E\_+\)[4]\ \(\(S\_-\)[1]\^*\)] - 35\ Re[\(E\_+\)[6]\ \(\(S\_-\)[1]\^*\)] + 16\ Re[\(E\_-\)[3]\ \(\(S\_-\)[2]\^*\)] + 110\ Re[\(E\_-\)[5]\ \(\(S\_-\)[2]\^*\)] - 24\ Re[\(E\_+\)[1]\ \(\(S\_-\)[2]\^*\)] - 70\ Re[\(E\_+\)[3]\ \(\(S\_-\)[2]\^*\)] - 110\ Re[\(E\_+\)[5]\ \(\(S\_-\)[2]\^*\)] + 10\ Re[\(M\_-\)[3]\ \(\(S\_-\)[2]\^*\)] + 10\ Re[\(M\_-\)[5]\ \(\(S\_-\)[2]\^*\)] - 10\ Re[\(M\_+\)[3]\ \(\(S\_-\)[2]\^*\)] - 10\ Re[\(M\_+\)[5]\ \(\(S\_-\)[2]\^*\)] + 12\ Re[\(E\_-\)[2]\ \(\(S\_-\)[3]\^*\)] + 495\/7\ Re[\(E\_-\)[4]\ \(\(S\_-\)[3]\^*\)] + 315\ Re[\(E\_-\)[6]\ \(\(S\_-\)[3]\^*\)] - 15\ Re[\(E\_+\)[0]\ \(\(S\_-\)[3]\^*\)] - 576\/7\ Re[\(E\_+\)[2]\ \(\(S\_-\)[3]\^*\)] - 180\ Re[\(E\_+\)[4]\ \(\(S\_-\)[3]\^*\)] - 270\ Re[\(E\_+\)[6]\ \(\(S\_-\)[3]\^*\)] + 18\ Re[\(M\_-\)[2]\ \(\(S\_-\)[3]\^*\)] + 45\ Re[\(M\_-\)[4]\ \(\(S\_-\)[3]\^*\)] + 45\ Re[\(M\_-\)[6]\ \(\(S\_-\)[3]\^*\)] - 18\ Re[\(M\_+\)[2]\ \(\(S\_-\)[3]\^*\)] - 45\ Re[\(M\_+\)[4]\ \(\(S\_-\)[3]\^*\)] - 45\ Re[\(M\_+\)[6]\ \(\(S\_-\)[3]\^*\)] + 300\/7\ Re[\(E\_-\)[3]\ \(\(S\_-\)[4]\^*\)] + 560\/3\ Re[\(E\_-\)[5]\ \(\(S\_-\)[4]\^*\)] - 60\ Re[\(E\_+\)[1]\ \(\(S\_-\)[4]\^*\)] - 4000\/21\ Re[\(E\_+\)[3]\ \(\(S\_-\)[4]\^*\)] - 360\ Re[\(E\_+\)[5]\ \(\(S\_-\)[4]\^*\)] + 20\ Re[\(M\_-\)[1]\ \(\(S\_-\)[4]\^*\)] + 480\/7\ Re[\(M\_-\)[3]\ \(\(S\_-\)[4]\^*\)] + 120\ Re[\(M\_-\)[5]\ \(\(S\_-\)[4]\^*\)] - 20\ Re[\(M\_+\)[1]\ \(\(S\_-\)[4]\^*\)] - 480\/7\ Re[\(M\_+\)[3]\ \(\(S\_-\)[4]\^*\)] - 120\ Re[\(M\_+\)[5]\ \(\(S\_-\)[4]\^*\)] - 25\ Re[\(E\_-\)[2]\ \(\(S\_-\)[5]\^*\)] + 100\ Re[\(E\_-\)[4]\ \(\(S\_-\)[5]\^*\)] + 4250\/11\ Re[\(E\_-\)[6]\ \(\(S\_-\)[5]\^*\)] - 25\ Re[\(E\_+\)[0]\ \(\(S\_-\)[5]\^*\)] - 150\ Re[\(E\_+\)[2]\ \(\(S\_-\)[5]\^*\)] - 4000\/11\ Re[\(E\_+\)[4]\ \(\(S\_-\)[5]\^*\)] - 625\ Re[\(E\_+\)[6]\ \(\(S\_-\)[5]\^*\)] + 75\ Re[\(M\_-\)[2]\ \(\(S\_-\)[5]\^*\)] + 500\/3\ Re[\(M\_-\)[4]\ \(\(S\_-\)[5]\^*\)] + 250\ Re[\(M\_-\)[6]\ \(\(S\_-\)[5]\^*\)] - 75\ Re[\(M\_+\)[2]\ \(\(S\_-\)[5]\^*\)] - 500\/3\ Re[\(M\_+\)[4]\ \(\(S\_-\)[5]\^*\)] - 250\ Re[\(M\_+\)[6]\ \(\(S\_-\)[5]\^*\)] - 90\ Re[\(E\_-\)[3]\ \(\(S\_-\)[6]\^*\)] + 2100\/11\ Re[\(E\_-\)[5]\ \(\(S\_-\)[6]\^*\)] - 90\ Re[\(E\_+\)[1]\ \(\(S\_-\)[6]\^*\)] - 300\ Re[\(E\_+\)[3]\ \(\(S\_-\)[6]\^*\)] - 88200\/143\ Re[\(E\_+\)[5]\ \(\(S\_-\)[6]\^*\)] + 30\ Re[\(M\_-\)[1]\ \(\(S\_-\)[6]\^*\)] + 180\ Re[\(M\_-\)[3]\ \(\(S\_-\)[6]\^*\)] + 3600\/11\ Re[\(M\_-\)[5]\ \(\(S\_-\)[6]\^*\)] - 30\ Re[\(M\_+\)[1]\ \(\(S\_-\)[6]\^*\)] - 180\ Re[\(M\_+\)[3]\ \(\(S\_-\)[6]\^*\)] - 3600\/11\ Re[\(M\_+\)[5]\ \(\(S\_-\)[6]\^*\)] + 10\ Re[\(E\_-\)[3]\ \(\(S\_+\)[0]\^*\)] + 20\ Re[\(E\_-\)[5]\ \(\(S\_+\)[0]\^*\)] - 25\ Re[\(E\_+\)[3]\ \(\(S\_+\)[0]\^*\)] - 35\ Re[\(E\_+\)[5]\ \(\(S\_+\)[0]\^*\)] - 5\ Re[\(M\_-\)[3]\ \(\(S\_+\)[0]\^*\)] - 5\ Re[\(M\_-\)[5]\ \(\(S\_+\)[0]\^*\)] + 5\ Re[\(M\_+\)[3]\ \(\(S\_+\)[0]\^*\)] + 5\ Re[\(M\_+\)[5]\ \(\(S\_+\)[0]\^*\)] + 12\ Re[\(E\_-\)[2]\ \(\(S\_+\)[1]\^*\)] + 50\ Re[\(E\_-\)[4]\ \(\(S\_+\)[1]\^*\)] + 90\ Re[\(E\_-\)[6]\ \(\(S\_+\)[1]\^*\)] - 18\ Re[\(E\_+\)[2]\ \(\(S\_+\)[1]\^*\)] - 130\ Re[\(E\_+\)[4]\ \(\(S\_+\)[1]\^*\)] - 170\ Re[\(E\_+\)[6]\ \(\(S\_+\)[1]\^*\)] - 12\ Re[\(M\_-\)[2]\ \(\(S\_+\)[1]\^*\)] - 30\ Re[\(M\_-\)[4]\ \(\(S\_+\)[1]\^*\)] - 30\ Re[\(M\_-\)[6]\ \(\(S\_+\)[1]\^*\)] + 12\ Re[\(M\_+\)[2]\ \(\(S\_+\)[1]\^*\)] + 30\ Re[\(M\_+\)[4]\ \(\(S\_+\)[1]\^*\)] + 30\ Re[\(M\_+\)[6]\ \(\(S\_+\)[1]\^*\)] + 342\/7\ Re[\(E\_-\)[3]\ \(\(S\_+\)[2]\^*\)] + 135\ Re[\(E\_-\)[5]\ \(\(S\_+\)[2]\^*\)] - 9\ Re[\(E\_+\)[1]\ \(\(S\_+\)[2]\^*\)] - 540\/7\ Re[\(E\_+\)[3]\ \(\(S\_+\)[2]\^*\)] - 360\ Re[\(E\_+\)[5]\ \(\(S\_+\)[2]\^*\)] - 15\ Re[\(M\_-\)[1]\ \(\(S\_+\)[2]\^*\)] - 360\/7\ Re[\(M\_-\)[3]\ \(\(S\_+\)[2]\^*\)] - 90\ Re[\(M\_-\)[5]\ \(\(S\_+\)[2]\^*\)] + 15\ Re[\(M\_+\)[1]\ \(\(S\_+\)[2]\^*\)] + 360\/7\ Re[\(M\_+\)[3]\ \(\(S\_+\)[2]\^*\)] + 90\ Re[\(M\_+\)[5]\ \(\(S\_+\)[2]\^*\)] + 20\ Re[\(E\_-\)[2]\ \(\(S\_+\)[3]\^*\)] + 880\/7\ Re[\(E\_-\)[4]\ \(\(S\_+\)[3]\^*\)] + 280\ Re[\(E\_-\)[6]\ \(\(S\_+\)[3]\^*\)] + 20\ Re[\(E\_+\)[0]\ \(\(S\_+\)[3]\^*\)] - 240\/7\ Re[\(E\_+\)[2]\ \(\(S\_+\)[3]\^*\)] - 200\ Re[\(E\_+\)[4]\ \(\(S\_+\)[3]\^*\)] - 760\ Re[\(E\_+\)[6]\ \(\(S\_+\)[3]\^*\)] - 60\ Re[\(M\_-\)[2]\ \(\(S\_+\)[3]\^*\)] - 400\/3\ Re[\(M\_-\)[4]\ \(\(S\_+\)[3]\^*\)] - 200\ Re[\(M\_-\)[6]\ \(\(S\_+\)[3]\^*\)] + 60\ Re[\(M\_+\)[2]\ \(\(S\_+\)[3]\^*\)] + 400\/3\ Re[\(M\_+\)[4]\ \(\(S\_+\)[3]\^*\)] + 200\ Re[\(M\_+\)[6]\ \(\(S\_+\)[3]\^*\)] + 75\ Re[\(E\_-\)[3]\ \(\(S\_+\)[4]\^*\)] + 8500\/33\ Re[\(E\_-\)[5]\ \(\(S\_+\)[4]\^*\)] + 75\ Re[\(E\_+\)[1]\ \(\(S\_+\)[4]\^*\)] - 250\/3\ Re[\(E\_+\)[3]\ \(\(S\_+\)[4]\^*\)] - 4500\/11\ Re[\(E\_+\)[5]\ \(\(S\_+\)[4]\^*\)] - 25\ Re[\(M\_-\)[1]\ \(\(S\_+\)[4]\^*\)] - 150\ Re[\(M\_-\)[3]\ \(\(S\_+\)[4]\^*\)] - 3000\/11\ Re[\(M\_-\)[5]\ \(\(S\_+\)[4]\^*\)] + 25\ Re[\(M\_+\)[1]\ \(\(S\_+\)[4]\^*\)] + 150\ Re[\(M\_+\)[3]\ \(\(S\_+\)[4]\^*\)] + 3000\/11\ Re[\(M\_+\)[5]\ \(\(S\_+\)[4]\^*\)] + 30\ Re[\(E\_-\)[2]\ \(\(S\_+\)[5]\^*\)] + 180\ Re[\(E\_-\)[4]\ \(\(S\_+\)[5]\^*\)] + 65700\/143\ Re[\(E\_-\)[6]\ \(\(S\_+\)[5]\^*\)] + 30\ Re[\(E\_+\)[0]\ \(\(S\_+\)[5]\^*\)] + 180\ Re[\(E\_+\)[2]\ \(\(S\_+\)[5]\^*\)] - 1800\/11\ Re[\(E\_+\)[4]\ \(\(S\_+\)[5]\^*\)] - 9450\/13\ Re[\(E\_+\)[6]\ \(\(S\_+\)[5]\^*\)] - 90\ Re[\(M\_-\)[2]\ \(\(S\_+\)[5]\^*\)] - 300\ Re[\(M\_-\)[4]\ \(\(S\_+\)[5]\^*\)] - 6300\/13\ Re[\(M\_-\)[6]\ \(\(S\_+\)[5]\^*\)] + 90\ Re[\(M\_+\)[2]\ \(\(S\_+\)[5]\^*\)] + 300\ Re[\(M\_+\)[4]\ \(\(S\_+\)[5]\^*\)] + 6300\/13\ Re[\(M\_+\)[6]\ \(\(S\_+\)[5]\^*\)] + 105\ Re[\(E\_-\)[3]\ \(\(S\_+\)[6]\^*\)] + 350\ Re[\(E\_-\)[5]\ \(\(S\_+\)[6]\^*\)] + 105\ Re[\(E\_+\)[1]\ \(\(S\_+\)[6]\^*\)] + 350\ Re[\(E\_+\)[3]\ \(\(S\_+\)[6]\^*\)] - 3675\/13\ Re[\(E\_+\)[5]\ \(\(S\_+\)[6]\^*\)] - 35\ Re[\(M\_-\)[1]\ \(\(S\_+\)[6]\^*\)] - 210\ Re[\(M\_-\)[3]\ \(\(S\_+\)[6]\^*\)] - 525\ Re[\(M\_-\)[5]\ \(\(S\_+\)[6]\^*\)] + 35\ Re[\(M\_+\)[1]\ \(\(S\_+\)[6]\^*\)] + 210\ Re[\(M\_+\)[3]\ \(\(S\_+\)[6]\^*\)] + 525\ Re[\(M\_+\)[5]\ \(\(S\_+\)[6]\^*\)])\)\ P\_2[ x] + \((35\ Re[\(E\_-\)[5]\ \(\(S\_-\)[1]\^*\)] - 28\ Re[\(E\_+\)[3]\ \(\(S\_-\)[1]\^*\)] - 42\ Re[\(E\_+\)[5]\ \(\(S\_-\)[1]\^*\)] + 30\ Re[\(E\_-\)[4]\ \(\(S\_-\)[2]\^*\)] + 182\ Re[\(E\_-\)[6]\ \(\(S\_-\)[2]\^*\)] - 54\ Re[\(E\_+\)[2]\ \(\(S\_-\)[2]\^*\)] - 126\ Re[\(E\_+\)[4]\ \(\(S\_-\)[2]\^*\)] - 182\ Re[\(E\_+\)[6]\ \(\(S\_-\)[2]\^*\)] + 14\ Re[\(M\_-\)[4]\ \(\(S\_-\)[2]\^*\)] + 14\ Re[\(M\_-\)[6]\ \(\(S\_-\)[2]\^*\)] - 14\ Re[\(M\_+\)[4]\ \(\(S\_-\)[2]\^*\)] - 14\ Re[\(M\_+\)[6]\ \(\(S\_-\)[2]\^*\)] + 27\ Re[\(E\_-\)[3]\ \(\(S\_-\)[3]\^*\)] + 128\ Re[\(E\_-\)[5]\ \(\(S\_-\)[3]\^*\)] - 54\ Re[\(E\_+\)[1]\ \(\(S\_-\)[3]\^*\)] - 169\ Re[\(E\_+\)[3]\ \(\(S\_-\)[3]\^*\)] - 315\ Re[\(E\_+\)[5]\ \(\(S\_-\)[3]\^*\)] + 27\ Re[\(M\_-\)[3]\ \(\(S\_-\)[3]\^*\)] + 63\ Re[\(M\_-\)[5]\ \(\(S\_-\)[3]\^*\)] - 27\ Re[\(M\_+\)[3]\ \(\(S\_-\)[3]\^*\)] - 63\ Re[\(M\_+\)[5]\ \(\(S\_-\)[3]\^*\)] + 20\ Re[\(E\_-\)[2]\ \(\(S\_-\)[4]\^*\)] + 304\/3\ Re[\(E\_-\)[4]\ \(\(S\_-\)[4]\^*\)] + 3640\/11\ Re[\(E\_-\)[6]\ \(\(S\_-\)[4]\^*\)] - 28\ Re[\(E\_+\)[0]\ \(\(S\_-\)[4]\^*\)] - 160\ Re[\(E\_+\)[2]\ \(\(S\_-\)[4]\^*\)] - 12136\/33\ Re[\(E\_+\)[4]\ \(\(S\_-\)[4]\^*\)] - 616\ Re[\(E\_+\)[6]\ \(\(S\_-\)[4]\^*\)] + 36\ Re[\(M\_-\)[2]\ \(\(S\_-\)[4]\^*\)] + 304\/3\ Re[\(M\_-\)[4]\ \(\(S\_-\)[4]\^*\)] + 168\ Re[\(M\_-\)[6]\ \(\(S\_-\)[4]\^*\)] - 36\ Re[\(M\_+\)[2]\ \(\(S\_-\)[4]\^*\)] - 304\/3\ Re[\(M\_+\)[4]\ \(\(S\_-\)[4]\^*\)] - 168\ Re[\(M\_+\)[6]\ \(\(S\_-\)[4]\^*\)] + 185\/3\ Re[\(E\_-\)[3]\ \(\(S\_-\)[5]\^*\)] + 8050\/33\ Re[\(E\_-\)[5]\ \(\(S\_-\)[5]\^*\)] - 105\ Re[\(E\_+\)[1]\ \(\(S\_-\)[5]\^*\)] - 11300\/33\ Re[\(E\_+\)[3]\ \(\(S\_-\)[5]\^*\)] - 288050\/429\ Re[\(E\_+\)[5]\ \(\(S\_-\)[5]\^*\)] + 35\ Re[\(M\_-\)[1]\ \(\(S\_-\)[5]\^*\)] + 380\/3\ Re[\(M\_-\)[3]\ \(\(S\_-\)[5]\^*\)] + 8050\/33\ Re[\(M\_-\)[5]\ \(\(S\_-\)[5]\^*\)] - 35\ Re[\(M\_+\)[1]\ \(\(S\_-\)[5]\^*\)] - 380\/3\ Re[\(M\_+\)[3]\ \(\(S\_-\)[5]\^*\)] - 8050\/33\ Re[\(M\_+\)[5]\ \(\(S\_-\)[5]\^*\)] - 42\ Re[\(E\_-\)[2]\ \(\(S\_-\)[6]\^*\)] + 1428\/11\ Re[\(E\_-\)[4]\ \(\(S\_-\)[6]\^*\)] + 68040\/143\ Re[\(E\_-\)[6]\ \(\(S\_-\)[6]\^*\)] - 42\ Re[\(E\_+\)[0]\ \(\(S\_-\)[6]\^*\)] - 252\ Re[\(E\_+\)[2]\ \(\(S\_-\)[6]\^*\)] - 89040\/143\ Re[\(E\_+\)[4]\ \(\(S\_-\)[6]\^*\)] - 157458\/143\ Re[\(E\_+\)[6]\ \(\(S\_-\)[6]\^*\)] + 126\ Re[\(M\_-\)[2]\ \(\(S\_-\)[6]\^*\)] + 3220\/11\ Re[\(M\_-\)[4]\ \(\(S\_-\)[6]\^*\)] + 68040\/143\ Re[\(M\_-\)[6]\ \(\(S\_-\)[6]\^*\)] - 126\ Re[\(M\_+\)[2]\ \(\(S\_-\)[6]\^*\)] - 3220\/11\ Re[\(M\_+\)[4]\ \(\(S\_-\)[6]\^*\)] - 68040\/143\ Re[\(M\_+\)[6]\ \(\(S\_-\)[6]\^*\)] + 21\ Re[\(E\_-\)[4]\ \(\(S\_+\)[0]\^*\)] + 35\ Re[\(E\_-\)[6]\ \(\(S\_+\)[0]\^*\)] - 42\ Re[\(E\_+\)[4]\ \(\(S\_+\)[0]\^*\)] - 56\ Re[\(E\_+\)[6]\ \(\(S\_+\)[0]\^*\)] - 7\ Re[\(M\_-\)[4]\ \(\(S\_+\)[0]\^*\)] - 7\ Re[\(M\_-\)[6]\ \(\(S\_+\)[0]\^*\)] + 7\ Re[\(M\_+\)[4]\ \(\(S\_+\)[0]\^*\)] + 7\ Re[\(M\_+\)[6]\ \(\(S\_+\)[0]\^*\)] + 36\ Re[\(E\_-\)[3]\ \(\(S\_+\)[1]\^*\)] + 98\ Re[\(E\_-\)[5]\ \(\(S\_+\)[1]\^*\)] - 34\ Re[\(E\_+\)[3]\ \(\(S\_+\)[1]\^*\)] - 210\ Re[\(E\_+\)[5]\ \(\(S\_+\)[1]\^*\)] - 18\ Re[\(M\_-\)[3]\ \(\(S\_+\)[1]\^*\)] - 42\ Re[\(M\_-\)[5]\ \(\(S\_+\)[1]\^*\)] + 18\ Re[\(M\_+\)[3]\ \(\(S\_+\)[1]\^*\)] + 42\ Re[\(M\_+\)[5]\ \(\(S\_+\)[1]\^*\)] + 27\ Re[\(E\_-\)[2]\ \(\(S\_+\)[2]\^*\)] + 120\ Re[\(E\_-\)[4]\ \(\(S\_+\)[2]\^*\)] + 252\ Re[\(E\_-\)[6]\ \(\(S\_+\)[2]\^*\)] - 27\ Re[\(E\_+\)[2]\ \(\(S\_+\)[2]\^*\)] - 141\ Re[\(E\_+\)[4]\ \(\(S\_+\)[2]\^*\)] - 567\ Re[\(E\_+\)[6]\ \(\(S\_+\)[2]\^*\)] - 27\ Re[\(M\_-\)[2]\ \(\(S\_+\)[2]\^*\)] - 76\ Re[\(M\_-\)[4]\ \(\(S\_+\)[2]\^*\)] - 126\ Re[\(M\_-\)[6]\ \(\(S\_+\)[2]\^*\)] + 27\ Re[\(M\_+\)[2]\ \(\(S\_+\)[2]\^*\)] + 76\ Re[\(M\_+\)[4]\ \(\(S\_+\)[2]\^*\)] + 126\ Re[\(M\_+\)[6]\ \(\(S\_+\)[2]\^*\)] + 284\/3\ Re[\(E\_-\)[3]\ \(\(S\_+\)[3]\^*\)] + 9040\/33\ Re[\(E\_-\)[5]\ \(\(S\_+\)[3]\^*\)] - 12\ Re[\(E\_+\)[1]\ \(\(S\_+\)[3]\^*\)] - 304\/3\ Re[\(E\_+\)[3]\ \(\(S\_+\)[3]\^*\)] - 11816\/33\ Re[\(E\_+\)[5]\ \(\(S\_+\)[3]\^*\)] - 28\ Re[\(M\_-\)[1]\ \(\(S\_+\)[3]\^*\)] - 304\/3\ Re[\(M\_-\)[3]\ \(\(S\_+\)[3]\^*\)] - 6440\/33\ Re[\(M\_-\)[5]\ \(\(S\_+\)[3]\^*\)] + 28\ Re[\(M\_+\)[1]\ \(\(S\_+\)[3]\^*\)] + 304\/3\ Re[\(M\_+\)[3]\ \(\(S\_+\)[3]\^*\)] + 6440\/33\ Re[\(M\_+\)[5]\ \(\(S\_+\)[3]\^*\)] + 35\ Re[\(E\_-\)[2]\ \(\(S\_+\)[4]\^*\)] + 7430\/33\ Re[\(E\_-\)[4]\ \(\(S\_+\)[4]\^*\)] + 74200\/143\ Re[\(E\_-\)[6]\ \(\(S\_+\)[4]\^*\)] + 35\ Re[\(E\_+\)[0]\ \(\(S\_+\)[4]\^*\)] - 40\ Re[\(E\_+\)[2]\ \(\(S\_+\)[4]\^*\)] - 8050\/33\ Re[\(E\_+\)[4]\ \(\(S\_+\)[4]\^*\)] - 103250\/143\ Re[\(E\_+\)[6]\ \(\(S\_+\)[4]\^*\)] - 105\ Re[\(M\_-\)[2]\ \(\(S\_+\)[4]\^*\)] - 8050\/33\ Re[\(M\_-\)[4]\ \(\(S\_+\)[4]\^*\)] - 56700\/143\ Re[\(M\_-\)[6]\ \(\(S\_+\)[4]\^*\)] + 105\ Re[\(M\_+\)[2]\ \(\(S\_+\)[4]\^*\)] + 8050\/33\ Re[\(M\_+\)[4]\ \(\(S\_+\)[4]\^*\)] + 56700\/143\ Re[\(M\_+\)[6]\ \(\(S\_+\)[4]\^*\)] + 126\ Re[\(E\_-\)[3]\ \(\(S\_+\)[5]\^*\)] + 62860\/143\ Re[\(E\_-\)[5]\ \(\(S\_+\)[5]\^*\)] + 126\ Re[\(E\_+\)[1]\ \(\(S\_+\)[5]\^*\)] - 980\/11\ Re[\(E\_+\)[3]\ \(\(S\_+\)[5]\^*\)] - 68040\/143\ Re[\(E\_+\)[5]\ \(\(S\_+\)[5]\^*\)] - 42\ Re[\(M\_-\)[1]\ \(\(S\_+\)[5]\^*\)] - 252\ Re[\(M\_-\)[3]\ \(\(S\_+\)[5]\^*\)] - 68040\/143\ Re[\(M\_-\)[5]\ \(\(S\_+\)[5]\^*\)] + 42\ Re[\(M\_+\)[1]\ \(\(S\_+\)[5]\^*\)] + 252\ Re[\(M\_+\)[3]\ \(\(S\_+\)[5]\^*\)] + 68040\/143\ Re[\(M\_+\)[5]\ \(\(S\_+\)[5]\^*\)] + 49\ Re[\(E\_-\)[2]\ \(\(S\_+\)[6]\^*\)] + 294\ Re[\(E\_-\)[4]\ \(\(S\_+\)[6]\^*\)] + 108535\/143\ Re[\(E\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 49\ Re[\(E\_+\)[0]\ \(\(S\_+\)[6]\^*\)] + 294\ Re[\(E\_+\)[2]\ \(\(S\_+\)[6]\^*\)] - 23520\/143\ Re[\(E\_+\)[4]\ \(\(S\_+\)[6]\^*\)] - 10633\/13\ Re[\(E\_+\)[6]\ \(\(S\_+\)[6]\^*\)] - 147\ Re[\(M\_-\)[2]\ \(\(S\_+\)[6]\^*\)] - 490\ Re[\(M\_-\)[4]\ \(\(S\_+\)[6]\^*\)] - 10633\/13\ Re[\(M\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 147\ Re[\(M\_+\)[2]\ \(\(S\_+\)[6]\^*\)] + 490\ Re[\(M\_+\)[4]\ \(\(S\_+\)[6]\^*\)] + 10633\/13\ Re[\(M\_+\)[6]\ \(\(S\_+\)[6]\^*\)])\)\ P\_3[ x] - \(\(1\/1001\)\((3\ \((\(-18018\)\ Re[\(E\_-\)[ 6]\ \(\(S\_-\)[1]\^*\)] + 15015\ Re[\(E\_+\)[4]\ \(\(S\_-\)[1]\^*\)] + 21021\ Re[\(E\_+\)[6]\ \(\(S\_-\)[1]\^*\)] - 16016\ Re[\(E\_-\)[5]\ \(\(S\_-\)[2]\^*\)] + 32032\ Re[\(E\_+\)[3]\ \(\(S\_-\)[2]\^*\)] + 66066\ Re[\(E\_+\)[5]\ \(\(S\_-\)[2]\^*\)] - 6006\ Re[\(M\_-\)[5]\ \(\(S\_-\)[2]\^*\)] + 6006\ Re[\(M\_+\)[5]\ \(\(S\_-\)[2]\^*\)] - 15444\ Re[\(E\_-\)[4]\ \(\(S\_-\)[3]\^*\)] - 66339\ Re[\(E\_-\)[6]\ \(\(S\_-\)[3]\^*\)] + 38610\ Re[\(E\_+\)[2]\ \(\(S\_-\)[3]\^*\)] + 95823\ Re[\(E\_+\)[4]\ \(\(S\_-\)[3]\^*\)] + 162162\ Re[\(E\_+\)[6]\ \(\(S\_-\)[3]\^*\)] - 12012\ Re[\(M\_-\)[4]\ \(\(S\_-\)[3]\^*\)] - 27027\ Re[\(M\_-\)[6]\ \(\(S\_-\)[3]\^*\)] + 12012\ Re[\(M\_+\)[4]\ \(\(S\_-\)[3]\^*\)] + 27027\ Re[\(M\_+\)[6]\ \(\(S\_-\)[3]\^*\)] - 13728\ Re[\(E\_-\)[3]\ \(\(S\_-\)[4]\^*\)] - 57512\ Re[\(E\_-\)[5]\ \(\(S\_-\)[4]\^*\)] + 32032\ Re[\(E\_+\)[1]\ \(\(S\_-\)[4]\^*\)] + 103480\ Re[\(E\_+\)[3]\ \(\(S\_-\)[4]\^*\)] + 201516\ Re[\(E\_+\)[5]\ \(\(S\_-\)[4]\^*\)] - 17160\ Re[\(M\_-\)[3]\ \(\(S\_-\)[4]\^*\)] - 44772\ Re[\(M\_-\)[5]\ \(\(S\_-\)[4]\^*\)] + 17160\ Re[\(M\_+\)[3]\ \(\(S\_-\)[4]\^*\)] + 44772\ Re[\(M\_+\)[5]\ \(\(S\_-\)[4]\^*\)] - 10010\ Re[\(E\_-\)[2]\ \(\(S\_-\)[5]\^*\)] - 46410\ Re[\(E\_-\)[4]\ \(\(S\_-\)[5]\^*\)] - 137725\ Re[\(E\_-\)[6]\ \(\(S\_-\)[5]\^*\)] + 15015\ Re[\(E\_+\)[0]\ \(\(S\_-\)[5]\^*\)] + 87815\ Re[\(E\_+\)[2]\ \(\(S\_-\)[5]\^*\)] + 206850\ Re[\(E\_+\)[4]\ \(\(S\_-\)[5]\^*\)] + 358225\ Re[\(E\_+\)[6]\ \(\(S\_-\)[5]\^*\)] - 20020\ Re[\(M\_-\)[2]\ \(\(S\_-\)[5]\^*\)] - 59150\ Re[\(M\_-\)[4]\ \(\(S\_-\)[5]\^*\)] - 107275\ Re[\(M\_-\)[6]\ \(\(S\_-\)[5]\^*\)] + 20020\ Re[\(M\_+\)[2]\ \(\(S\_-\)[5]\^*\)] + 59150\ Re[\(M\_+\)[4]\ \(\(S\_-\)[5]\^*\)] + 107275\ Re[\(M\_+\)[6]\ \(\(S\_-\)[5]\^*\)] - 27846\ Re[\(E\_-\)[3]\ \(\(S\_-\)[6]\^*\)] - 104370\ Re[\(E\_-\)[5]\ \(\(S\_-\)[6]\^*\)] + 54054\ Re[\(E\_+\)[1]\ \(\(S\_-\)[6]\^*\)] + 179130\ Re[\(E\_+\)[3]\ \(\(S\_-\)[6]\^*\)] + 358092\ Re[\(E\_+\)[5]\ \(\(S\_-\)[6]\^*\)] - 18018\ Re[\(M\_-\)[1]\ \(\(S\_-\)[6]\^*\)] - 67158\ Re[\(M\_-\)[3]\ \(\(S\_-\)[6]\^*\)] - 134820\ Re[\(M\_-\)[5]\ \(\(S\_-\)[6]\^*\)] + 18018\ Re[\(M\_+\)[1]\ \(\(S\_-\)[6]\^*\)] + 67158\ Re[\(M\_+\)[3]\ \(\(S\_-\)[6]\^*\)] + 134820\ Re[\(M\_+\)[5]\ \(\(S\_-\)[6]\^*\)] - 12012\ Re[\(E\_-\)[5]\ \(\(S\_+\)[0]\^*\)] + 21021\ Re[\(E\_+\)[5]\ \(\(S\_+\)[0]\^*\)] + 3003\ Re[\(M\_-\)[5]\ \(\(S\_+\)[0]\^*\)] - 3003\ Re[\(M\_+\)[5]\ \(\(S\_+\)[0]\^*\)] - 24024\ Re[\(E\_-\)[4]\ \(\(S\_+\)[1]\^*\)] - 54054\ Re[\(E\_-\)[6]\ \(\(S\_+\)[1]\^*\)] + 18018\ Re[\(E\_+\)[4]\ \(\(S\_+\)[1]\^*\)] + 102102\ Re[\(E\_+\)[6]\ \(\(S\_+\)[1]\^*\)] + 8008\ Re[\(M\_-\)[4]\ \(\(S\_+\)[1]\^*\)] + 18018\ Re[\(M\_-\)[6]\ \(\(S\_+\)[1]\^*\)] - 8008\ Re[\(M\_+\)[4]\ \(\(S\_+\)[1]\^*\)] - 18018\ Re[\(M\_+\)[6]\ \(\(S\_+\)[1]\^*\)] - 25740\ Re[\(E\_-\)[3]\ \(\(S\_+\)[2]\^*\)] - 74256\ Re[\(E\_-\)[5]\ \(\(S\_+\)[2]\^*\)] + 16302\ Re[\(E\_+\)[3]\ \(\(S\_+\)[2]\^*\)] + 72891\ Re[\(E\_+\)[5]\ \(\(S\_+\)[2]\^*\)] + 12870\ Re[\(M\_-\)[3]\ \(\(S\_+\)[2]\^*\)] + 33579\ Re[\(M\_-\)[5]\ \(\(S\_+\)[2]\^*\)] - 12870\ Re[\(M\_+\)[3]\ \(\(S\_+\)[2]\^*\)] - 33579\ Re[\(M\_+\)[5]\ \(\(S\_+\)[2]\^*\)] - 16016\ Re[\(E\_-\)[2]\ \(\(S\_+\)[3]\^*\)] - 73320\ Re[\(E\_-\)[4]\ \(\(S\_+\)[3]\^*\)] - 160468\ Re[\(E\_-\)[6]\ \(\(S\_+\)[3]\^*\)] + 12584\ Re[\(E\_+\)[2]\ \(\(S\_+\)[3]\^*\)] + 60060\ Re[\(E\_+\)[4]\ \(\(S\_+\)[3]\^*\)] + 182056\ Re[\(E\_+\)[6]\ \(\(S\_+\)[3]\^*\)] + 16016\ Re[\(M\_-\)[2]\ \(\(S\_+\)[3]\^*\)] + 47320\ Re[\(M\_-\)[4]\ \(\(S\_+\)[3]\^*\)] + 85820\ Re[\(M\_-\)[6]\ \(\(S\_+\)[3]\^*\)] - 16016\ Re[\(M\_+\)[2]\ \(\(S\_+\)[3]\^*\)] - 47320\ Re[\(M\_+\)[4]\ \(\(S\_+\)[3]\^*\)] - 85820\ Re[\(M\_+\)[6]\ \(\(S\_+\)[3]\^*\)] - 51870\ Re[\(E\_-\)[3]\ \(\(S\_+\)[4]\^*\)] - 153650\ Re[\(E\_-\)[5]\ \(\(S\_+\)[4]\^*\)] + 5005\ Re[\(E\_+\)[1]\ \(\(S\_+\)[4]\^*\)] + 43225\ Re[\(E\_+\)[3]\ \(\(S\_+\)[4]\^*\)] + 142800\ Re[\(E\_+\)[5]\ \(\(S\_+\)[4]\^*\)] + 15015\ Re[\(M\_-\)[1]\ \(\(S\_+\)[4]\^*\)] + 55965\ Re[\(M\_-\)[3]\ \(\(S\_+\)[4]\^*\)] + 112350\ Re[\(M\_-\)[5]\ \(\(S\_+\)[4]\^*\)] - 15015\ Re[\(M\_+\)[1]\ \(\(S\_+\)[4]\^*\)] - 55965\ Re[\(M\_+\)[3]\ \(\(S\_+\)[4]\^*\)] - 112350\ Re[\(M\_+\)[5]\ \(\(S\_+\)[4]\^*\)] - 18018\ Re[\(E\_-\)[2]\ \(\(S\_+\)[5]\^*\)] - 117558\ Re[\(E\_-\)[4]\ \(\(S\_+\)[5]\^*\)] - 275940\ Re[\(E\_-\)[6]\ \(\(S\_+\)[5]\^*\)] - 18018\ Re[\(E\_+\)[0]\ \(\(S\_+\)[5]\^*\)] + 14742\ Re[\(E\_+\)[2]\ \(\(S\_+\)[5]\^*\)] + 98280\ Re[\(E\_+\)[4]\ \(\(S\_+\)[5]\^*\)] + 276066\ Re[\(E\_+\)[6]\ \(\(S\_+\)[5]\^*\)] + 54054\ Re[\(M\_-\)[2]\ \(\(S\_+\)[5]\^*\)] + 128730\ Re[\(M\_-\)[4]\ \(\(S\_+\)[5]\^*\)] + 216972\ Re[\(M\_-\)[6]\ \(\(S\_+\)[5]\^*\)] - 54054\ Re[\(M\_+\)[2]\ \(\(S\_+\)[5]\^*\)] - 128730\ Re[\(M\_+\)[4]\ \(\(S\_+\)[5]\^*\)] - 216972\ Re[\(M\_+\)[6]\ \(\(S\_+\)[5]\^*\)] - 63063\ Re[\(E\_-\)[3]\ \(\(S\_+\)[6]\^*\)] - 222215\ Re[\(E\_-\)[5]\ \(\(S\_+\)[6]\^*\)] - 63063\ Re[\(E\_+\)[1]\ \(\(S\_+\)[6]\^*\)] + 29890\ Re[\(E\_+\)[3]\ \(\(S\_+\)[6]\^*\)] + 184191\ Re[\(E\_+\)[5]\ \(\(S\_+\)[6]\^*\)] + 21021\ Re[\(M\_-\)[1]\ \(\(S\_+\)[6]\^*\)] + 126126\ Re[\(M\_-\)[3]\ \(\(S\_+\)[6]\^*\)] + 243285\ Re[\(M\_-\)[5]\ \(\(S\_+\)[6]\^*\)] - 21021\ Re[\(M\_+\)[1]\ \(\(S\_+\)[6]\^*\)] - 126126\ Re[\(M\_+\)[3]\ \(\(S\_+\)[6]\^*\)] - 243285\ Re[\(M\_+\)[5]\ \(\(S\_+\)[6]\^*\)])\)\ P\_4[ x])\)\) + \((\(-66\)\ Re[\(E\_+\)[5]\ \(\(S\_-\)[1]\^*\)] + 70\ Re[\(E\_-\)[6]\ \(\(S\_-\)[2]\^*\)] - 150\ Re[\(E\_+\)[4]\ \(\(S\_-\)[2]\^*\)] - 286\ Re[\(E\_+\)[6]\ \(\(S\_-\)[2]\^*\)] + 22\ Re[\(M\_-\)[6]\ \(\(S\_-\)[2]\^*\)] - 22\ Re[\(M\_+\)[6]\ \(\(S\_-\)[2]\^*\)] + 70\ Re[\(E\_-\)[5]\ \(\(S\_-\)[3]\^*\)] - 200\ Re[\(E\_+\)[3]\ \(\(S\_-\)[3]\^*\)] - 5679\/13\ Re[\(E\_+\)[5]\ \(\(S\_-\)[3]\^*\)] + 45\ Re[\(M\_-\)[5]\ \(\(S\_-\)[3]\^*\)] - 45\ Re[\(M\_+\)[5]\ \(\(S\_-\)[3]\^*\)] + 200\/3\ Re[\(E\_-\)[4]\ \(\(S\_-\)[4]\^*\)] + 3340\/13\ Re[\(E\_-\)[6]\ \(\(S\_-\)[4]\^*\)] - 200\ Re[\(E\_+\)[2]\ \(\(S\_-\)[4]\^*\)] - 19900\/39\ Re[\(E\_+\)[4]\ \(\(S\_-\)[4]\^*\)] - 11688\/13\ Re[\(E\_+\)[6]\ \(\(S\_-\)[4]\^*\)] + 200\/3\ Re[\(M\_-\)[4]\ \(\(S\_-\)[4]\^*\)] + 2172\/13\ Re[\(M\_-\)[6]\ \(\(S\_-\)[4]\^*\)] - 200\/3\ Re[\(M\_+\)[4]\ \(\(S\_-\)[4]\^*\)] - 2172\/13\ Re[\(M\_+\)[6]\ \(\(S\_-\)[4]\^*\)] + 175\/3\ Re[\(E\_-\)[3]\ \(\(S\_-\)[5]\^*\)] + 8875\/39\ Re[\(E\_-\)[5]\ \(\(S\_-\)[5]\^*\)] - 150\ Re[\(E\_+\)[1]\ \(\(S\_-\)[5]\^*\)] - 19250\/39\ Re[\(E\_+\)[3]\ \(\(S\_-\)[5]\^*\)] - 38315\/39\ Re[\(E\_+\)[5]\ \(\(S\_-\)[5]\^*\)] + 250\/3\ Re[\(M\_-\)[3]\ \(\(S\_-\)[5]\^*\)] + 8875\/39\ Re[\(M\_-\)[5]\ \(\(S\_-\)[5]\^*\)] - 250\/3\ Re[\(M\_+\)[3]\ \(\(S\_-\)[5]\^*\)] - 8875\/39\ Re[\(M\_+\)[5]\ \(\(S\_-\)[5]\^*\)] + 42\ Re[\(E\_-\)[2]\ \(\(S\_-\)[6]\^*\)] + 2382\/13\ Re[\(E\_-\)[4]\ \(\(S\_-\)[6]\^*\)] + 6696\/13\ Re[\(E\_-\)[6]\ \(\(S\_-\)[6]\^*\)] - 66\ Re[\(E\_+\)[0]\ \(\(S\_-\)[6]\^*\)] - 5094\/13\ Re[\(E\_+\)[2]\ \(\(S\_-\)[6]\^*\)] - 12180\/13\ Re[\(E\_+\)[4]\ \(\(S\_-\)[6]\^*\)] - 365010\/221\ Re[\(E\_+\)[6]\ \(\(S\_-\)[6]\^*\)] + 90\ Re[\(M\_-\)[2]\ \(\(S\_-\)[6]\^*\)] + 3550\/13\ Re[\(M\_-\)[4]\ \(\(S\_-\)[6]\^*\)] + 6696\/13\ Re[\(M\_-\)[6]\ \(\(S\_-\)[6]\^*\)] - 90\ Re[\(M\_+\)[2]\ \(\(S\_-\)[6]\^*\)] - 3550\/13\ Re[\(M\_+\)[4]\ \(\(S\_-\)[6]\^*\)] - 6696\/13\ Re[\(M\_+\)[6]\ \(\(S\_-\)[6]\^*\)] + 55\ Re[\(E\_-\)[6]\ \(\(S\_+\)[0]\^*\)] - 88\ Re[\(E\_+\)[6]\ \(\(S\_+\)[0]\^*\)] - 11\ Re[\(M\_-\)[6]\ \(\(S\_+\)[0]\^*\)] + 11\ Re[\(M\_+\)[6]\ \(\(S\_+\)[0]\^*\)] + 120\ Re[\(E\_-\)[5]\ \(\(S\_+\)[1]\^*\)] - 78\ Re[\(E\_+\)[5]\ \(\(S\_+\)[1]\^*\)] - 30\ Re[\(M\_-\)[5]\ \(\(S\_+\)[1]\^*\)] + 30\ Re[\(M\_+\)[5]\ \(\(S\_+\)[1]\^*\)] + 150\ Re[\(E\_-\)[4]\ \(\(S\_+\)[2]\^*\)] + 4635\/13\ Re[\(E\_-\)[6]\ \(\(S\_+\)[2]\^*\)] - 75\ Re[\(E\_+\)[4]\ \(\(S\_+\)[2]\^*\)] - 4023\/13\ Re[\(E\_+\)[6]\ \(\(S\_+\)[2]\^*\)] - 50\ Re[\(M\_-\)[4]\ \(\(S\_+\)[2]\^*\)] - 1629\/13\ Re[\(M\_-\)[6]\ \(\(S\_+\)[2]\^*\)] + 50\ Re[\(M\_+\)[4]\ \(\(S\_+\)[2]\^*\)] + 1629\/13\ Re[\(M\_+\)[6]\ \(\(S\_+\)[2]\^*\)] + 400\/3\ Re[\(E\_-\)[3]\ \(\(S\_+\)[3]\^*\)] + 15400\/39\ Re[\(E\_-\)[5]\ \(\(S\_+\)[3]\^*\)] - 200\/3\ Re[\(E\_+\)[3]\ \(\(S\_+\)[3]\^*\)] - 10604\/39\ Re[\(E\_+\)[5]\ \(\(S\_+\)[3]\^*\)] - 200\/3\ Re[\(M\_-\)[3]\ \(\(S\_+\)[3]\^*\)] - 7100\/39\ Re[\(M\_-\)[5]\ \(\(S\_+\)[3]\^*\)] + 200\/3\ Re[\(M\_+\)[3]\ \(\(S\_+\)[3]\^*\)] + 7100\/39\ Re[\(M\_+\)[5]\ \(\(S\_+\)[3]\^*\)] + 75\ Re[\(E\_-\)[2]\ \(\(S\_+\)[4]\^*\)] + 13625\/39\ Re[\(E\_-\)[4]\ \(\(S\_+\)[4]\^*\)] + 10150\/13\ Re[\(E\_-\)[6]\ \(\(S\_+\)[4]\^*\)] - 50\ Re[\(E\_+\)[2]\ \(\(S\_+\)[4]\^*\)] - 8875\/39\ Re[\(E\_+\)[4]\ \(\(S\_+\)[4]\^*\)] - 8310\/13\ Re[\(E\_+\)[6]\ \(\(S\_+\)[4]\^*\)] - 75\ Re[\(M\_-\)[2]\ \(\(S\_+\)[4]\^*\)] - 8875\/39\ Re[\(M\_-\)[4]\ \(\(S\_+\)[4]\^*\)] - 5580\/13\ Re[\(M\_-\)[6]\ \(\(S\_+\)[4]\^*\)] + 75\ Re[\(M\_+\)[2]\ \(\(S\_+\)[4]\^*\)] + 8875\/39\ Re[\(M\_+\)[4]\ \(\(S\_+\)[4]\^*\)] + 5580\/13\ Re[\(M\_+\)[6]\ \(\(S\_+\)[4]\^*\)] + 3006\/13\ Re[\(E\_-\)[3]\ \(\(S\_+\)[5]\^*\)] + 9034\/13\ Re[\(E\_-\)[5]\ \(\(S\_+\)[5]\^*\)] - 18\ Re[\(E\_+\)[1]\ \(\(S\_+\)[5]\^*\)] - 2090\/13\ Re[\(E\_+\)[3]\ \(\(S\_+\)[5]\^*\)] - 6696\/13\ Re[\(E\_+\)[5]\ \(\(S\_+\)[5]\^*\)] - 66\ Re[\(M\_-\)[1]\ \(\(S\_+\)[5]\^*\)] - 3258\/13\ Re[\(M\_-\)[3]\ \(\(S\_+\)[5]\^*\)] - 6696\/13\ Re[\(M\_-\)[5]\ \(\(S\_+\)[5]\^*\)] + 66\ Re[\(M\_+\)[1]\ \(\(S\_+\)[5]\^*\)] + 3258\/13\ Re[\(M\_+\)[3]\ \(\(S\_+\)[5]\^*\)] + 6696\/13\ Re[\(M\_+\)[5]\ \(\(S\_+\)[5]\^*\)] + 77\ Re[\(E\_-\)[2]\ \(\(S\_+\)[6]\^*\)] + 6594\/13\ Re[\(E\_-\)[4]\ \(\(S\_+\)[6]\^*\)] + 266231\/221\ Re[\(E\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 77\ Re[\(E\_+\)[0]\ \(\(S\_+\)[6]\^*\)] - 609\/13\ Re[\(E\_+\)[2]\ \(\(S\_+\)[6]\^*\)] - 4536\/13\ Re[\(E\_+\)[4]\ \(\(S\_+\)[6]\^*\)] - 212611\/221\ Re[\(E\_+\)[6]\ \(\(S\_+\)[6]\^*\)] - 231\ Re[\(M\_-\)[2]\ \(\(S\_+\)[6]\^*\)] - 7266\/13\ Re[\(M\_-\)[4]\ \(\(S\_+\)[6]\^*\)] - 212611\/221\ Re[\(M\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 231\ Re[\(M\_+\)[2]\ \(\(S\_+\)[6]\^*\)] + 7266\/13\ Re[\(M\_+\)[4]\ \(\(S\_+\)[6]\^*\)] + 212611\/221\ Re[\(M\_+\)[6]\ \(\(S\_+\)[6]\^*\)])\)\ P\_5[x] + 1\/561\ \((\(-51051\)\ Re[\(E\_+\)[6]\ \(\(S\_-\)[1]\^*\)] - 121176\ Re[\(E\_+\)[5]\ \(\(S\_-\)[2]\^*\)] + 55080\ Re[\(E\_-\)[6]\ \(\(S\_-\)[3]\^*\)] - 172125\ Re[\(E\_+\)[4]\ \(\(S\_-\)[3]\^*\)] - 346698\ Re[\(E\_+\)[6]\ \(\(S\_-\)[3]\^*\)] + 30294\ Re[\(M\_-\)[6]\ \(\(S\_-\)[3]\^*\)] - 30294\ Re[\(M\_+\)[6]\ \(\(S\_-\)[3]\^*\)] + 54400\ Re[\(E\_-\)[5]\ \(\(S\_-\)[4]\^*\)] - 190400\ Re[\(E\_+\)[3]\ \(\(S\_-\)[4]\^*\)] - 426564\ Re[\(E\_+\)[5]\ \(\(S\_-\)[4]\^*\)] + 45900\ Re[\(M\_-\)[5]\ \(\(S\_-\)[4]\^*\)] - 45900\ Re[\(M\_+\)[5]\ \(\(S\_-\)[4]\^*\)] + 51000\ Re[\(E\_-\)[4]\ \(\(S\_-\)[5]\^*\)] + 184875\ Re[\(E\_-\)[6]\ \(\(S\_-\)[5]\^*\)] - 172125\ Re[\(E\_+\)[2]\ \(\(S\_-\)[5]\^*\)] - 446250\ Re[\(E\_+\)[4]\ \(\(S\_-\)[5]\^*\)] - 802215\ Re[\(E\_+\)[6]\ \(\(S\_-\)[5]\^*\)] + 59500\ Re[\(M\_-\)[4]\ \(\(S\_-\)[5]\^*\)] + 155805\ Re[\(M\_-\)[6]\ \(\(S\_-\)[5]\^*\)] - 59500\ Re[\(M\_+\)[4]\ \(\(S\_-\)[5]\^*\)] - 155805\ Re[\(M\_+\)[6]\ \(\(S\_-\)[5]\^*\)] + 44064\ Re[\(E\_-\)[3]\ \(\(S\_-\)[6]\^*\)] + 163710\ Re[\(E\_-\)[5]\ \(\(S\_-\)[6]\^*\)] - 121176\ Re[\(E\_+\)[1]\ \(\(S\_-\)[6]\^*\)] - 403614\ Re[\(E\_+\)[3]\ \(\(S\_-\)[6]\^*\)] - 814212\ Re[\(E\_+\)[5]\ \(\(S\_-\)[6]\^*\)] + 68850\ Re[\(M\_-\)[3]\ \(\(S\_-\)[6]\^*\)] + 192780\ Re[\(M\_-\)[5]\ \(\(S\_-\)[6]\^*\)] - 68850\ Re[\(M\_+\)[3]\ \(\(S\_-\)[6]\^*\)] - 192780\ Re[\(M\_+\)[5]\ \(\(S\_-\)[6]\^*\)] + 100980\ Re[\(E\_-\)[6]\ \(\(S\_+\)[1]\^*\)] - 59466\ Re[\(E\_+\)[6]\ \(\(S\_+\)[1]\^*\)] - 20196\ Re[\(M\_-\)[6]\ \(\(S\_+\)[1]\^*\)] + 20196\ Re[\(M\_+\)[6]\ \(\(S\_+\)[1]\^*\)] + 137700\ Re[\(E\_-\)[5]\ \(\(S\_+\)[2]\^*\)] - 59211\ Re[\(E\_+\)[5]\ \(\(S\_+\)[2]\^*\)] - 34425\ Re[\(M\_-\)[5]\ \(\(S\_+\)[2]\^*\)] + 34425\ Re[\(M\_+\)[5]\ \(\(S\_+\)[2]\^*\)] + 142800\ Re[\(E\_-\)[4]\ \(\(S\_+\)[3]\^*\)] + 347820\ Re[\(E\_-\)[6]\ \(\(S\_+\)[3]\^*\)] - 56100\ Re[\(E\_+\)[4]\ \(\(S\_+\)[3]\^*\)] - 211752\ Re[\(E\_+\)[6]\ \(\(S\_+\)[3]\^*\)] - 47600\ Re[\(M\_-\)[4]\ \(\(S\_+\)[3]\^*\)] - 124644\ Re[\(M\_-\)[6]\ \(\(S\_+\)[3]\^*\)] + 47600\ Re[\(M\_+\)[4]\ \(\(S\_+\)[3]\^*\)] + 124644\ Re[\(M\_+\)[6]\ \(\(S\_+\)[3]\^*\)] + 114750\ Re[\(E\_-\)[3]\ \(\(S\_+\)[4]\^*\)] + 345100\ Re[\(E\_-\)[5]\ \(\(S\_+\)[4]\^*\)] - 48875\ Re[\(E\_+\)[3]\ \(\(S\_+\)[4]\^*\)] - 189720\ Re[\(E\_+\)[5]\ \(\(S\_+\)[4]\^*\)] - 57375\ Re[\(M\_-\)[3]\ \(\(S\_+\)[4]\^*\)] - 160650\ Re[\(M\_-\)[5]\ \(\(S\_+\)[4]\^*\)] + 57375\ Re[\(M\_+\)[3]\ \(\(S\_+\)[4]\^*\)] + 160650\ Re[\(M\_+\)[5]\ \(\(S\_+\)[4]\^*\)] + 60588\ Re[\(E\_-\)[2]\ \(\(S\_+\)[5]\^*\)] + 285498\ Re[\(E\_-\)[4]\ \(\(S\_+\)[5]\^*\)] + 646380\ Re[\(E\_-\)[6]\ \(\(S\_+\)[5]\^*\)] - 35802\ Re[\(E\_+\)[2]\ \(\(S\_+\)[5]\^*\)] - 157896\ Re[\(E\_+\)[4]\ \(\(S\_+\)[5]\^*\)] - 426510\ Re[\(E\_+\)[6]\ \(\(S\_+\)[5]\^*\)] - 60588\ Re[\(M\_-\)[2]\ \(\(S\_+\)[5]\^*\)] - 186966\ Re[\(M\_-\)[4]\ \(\(S\_+\)[5]\^*\)] - 360612\ Re[\(M\_-\)[6]\ \(\(S\_+\)[5]\^*\)] + 60588\ Re[\(M\_+\)[2]\ \(\(S\_+\)[5]\^*\)] + 186966\ Re[\(M\_+\)[4]\ \(\(S\_+\)[5]\^*\)] + 360612\ Re[\(M\_+\)[6]\ \(\(S\_+\)[5]\^*\)] + 180642\ Re[\(E\_-\)[3]\ \(\(S\_+\)[6]\^*\)] + 548289\ Re[\(E\_-\)[5]\ \(\(S\_+\)[6]\^*\)] - 11781\ Re[\(E\_+\)[1]\ \(\(S\_+\)[6]\^*\)] - 109242\ Re[\(E\_+\)[3]\ \(\(S\_+\)[6]\^*\)] - 343833\ Re[\(E\_+\)[5]\ \(\(S\_+\)[6]\^*\)] - 51051\ Re[\(M\_-\)[1]\ \(\(S\_+\)[6]\^*\)] - 196350\ Re[\(M\_-\)[3]\ \(\(S\_+\)[6]\^*\)] - 409731\ Re[\(M\_-\)[5]\ \(\(S\_+\)[6]\^*\)] + 51051\ Re[\(M\_+\)[1]\ \(\(S\_+\)[6]\^*\)] + 196350\ Re[\(M\_+\)[3]\ \(\(S\_+\)[6]\^*\)] + 409731\ Re[\(M\_+\)[5]\ \(\(S\_+\)[6]\^*\)])\)\ P\_6[ x] - \(\(1\/46189\)\((21\ \((646646\ Re[\(E\_+\)[ 6]\ \(\(S\_-\)[2]\^*\)] + 959310\ Re[\(E\_+\)[5]\ \(\(S\_-\)[3]\^*\)] - 290700\ Re[\(E\_-\)[6]\ \(\(S\_-\)[4]\^*\)] + 1130500\ Re[\(E\_+\)[4]\ \(\(S\_-\)[4]\^*\)] + 2332440\ Re[\(E\_+\)[6]\ \(\(S\_-\)[4]\^*\)] - 213180\ Re[\(M\_-\)[6]\ \(\(S\_-\)[4]\^*\)] + 213180\ Re[\(M\_+\)[6]\ \(\(S\_-\)[4]\^*\)] - 282625\ Re[\(E\_-\)[5]\ \(\(S\_-\)[5]\^*\)] + 1130500\ Re[\(E\_+\)[3]\ \(\(S\_-\)[5]\^*\)] + 2570225\ Re[\(E\_+\)[5]\ \(\(S\_-\)[5]\^*\)] - 282625\ Re[\(M\_-\)[5]\ \(\(S\_-\)[5]\^*\)] + 282625\ Re[\(M\_+\)[5]\ \(\(S\_-\)[5]\^*\)] - 261630\ Re[\(E\_-\)[4]\ \(\(S\_-\)[6]\^*\)] - 909720\ Re[\(E\_-\)[6]\ \(\(S\_-\)[6]\^*\)] + 959310\ Re[\(E\_+\)[2]\ \(\(S\_-\)[6]\^*\)] + 2513700\ Re[\(E\_+\)[4]\ \(\(S\_-\)[6]\^*\)] + 4574850\ Re[\(E\_+\)[6]\ \(\(S\_-\)[6]\^*\)] - 339150\ Re[\(M\_-\)[4]\ \(\(S\_-\)[6]\^*\)] - 909720\ Re[\(M\_-\)[6]\ \(\(S\_-\)[6]\^*\)] + 339150\ Re[\(M\_+\)[4]\ \(\(S\_-\)[6]\^*\)] + 909720\ Re[\(M\_+\)[6]\ \(\(S\_-\)[6]\^*\)] - 799425\ Re[\(E\_-\)[6]\ \(\(S\_+\)[2]\^*\)] + 309111\ Re[\(E\_+\)[6]\ \(\(S\_+\)[2]\^*\)] + 159885\ Re[\(M\_-\)[6]\ \(\(S\_+\)[2]\^*\)] - 159885\ Re[\(M\_+\)[6]\ \(\(S\_+\)[2]\^*\)] - 904400\ Re[\(E\_-\)[5]\ \(\(S\_+\)[3]\^*\)] + 303620\ Re[\(E\_+\)[5]\ \(\(S\_+\)[3]\^*\)] + 226100\ Re[\(M\_-\)[5]\ \(\(S\_+\)[3]\^*\)] - 226100\ Re[\(M\_+\)[5]\ \(\(S\_+\)[3]\^*\)] - 847875\ Re[\(E\_-\)[4]\ \(\(S\_+\)[4]\^*\)] - 2094750\ Re[\(E\_-\)[6]\ \(\(S\_+\)[4]\^*\)] + 282625\ Re[\(E\_+\)[4]\ \(\(S\_+\)[4]\^*\)] + 1017450\ Re[\(E\_+\)[6]\ \(\(S\_+\)[4]\^*\)] + 282625\ Re[\(M\_-\)[4]\ \(\(S\_+\)[4]\^*\)] + 758100\ Re[\(M\_-\)[6]\ \(\(S\_+\)[4]\^*\)] - 282625\ Re[\(M\_+\)[4]\ \(\(S\_+\)[4]\^*\)] - 758100\ Re[\(M\_+\)[6]\ \(\(S\_+\)[4]\^*\)] - 639540\ Re[\(E\_-\)[3]\ \(\(S\_+\)[5]\^*\)] - 1943130\ Re[\(E\_-\)[5]\ \(\(S\_+\)[5]\^*\)] + 242250\ Re[\(E\_+\)[3]\ \(\(S\_+\)[5]\^*\)] + 909720\ Re[\(E\_+\)[5]\ \(\(S\_+\)[5]\^*\)] + 319770\ Re[\(M\_-\)[3]\ \(\(S\_+\)[5]\^*\)] + 909720\ Re[\(M\_-\)[5]\ \(\(S\_+\)[5]\^*\)] - 319770\ Re[\(M\_+\)[3]\ \(\(S\_+\)[5]\^*\)] - 909720\ Re[\(M\_+\)[5]\ \(\(S\_+\)[5]\^*\)] - 323323\ Re[\(E\_-\)[2]\ \(\(S\_+\)[6]\^*\)] - 1536150\ Re[\(E\_-\)[4]\ \(\(S\_+\)[6]\^*\)] - 3509135\ Re[\(E\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 174097\ Re[\(E\_+\)[2]\ \(\(S\_+\)[6]\^*\)] + 750120\ Re[\(E\_+\)[4]\ \(\(S\_+\)[6]\^*\)] + 1975435\ Re[\(E\_+\)[6]\ \(\(S\_+\)[6]\^*\)] + 323323\ Re[\(M\_-\)[2]\ \(\(S\_+\)[6]\^*\)] + 1009470\ Re[\(M\_-\)[4]\ \(\(S\_+\)[6]\^*\)] + 1975435\ Re[\(M\_-\)[6]\ \(\(S\_+\)[6]\^*\)] - 323323\ Re[\(M\_+\)[2]\ \(\(S\_+\)[6]\^*\)] - 1009470\ Re[\(M\_+\)[4]\ \(\(S\_+\)[6]\^*\)] - 1975435\ Re[\(M\_+\)[6]\ \(\(S\_+\)[6]\^*\)])\)\ P\_7[ x])\)\) - \(\(1\/2717\)\((98\ \((16302\ Re[\(E\_+\)[ 6]\ \(\(S\_-\)[3]\^*\)] + 20064\ Re[\(E\_+\)[5]\ \(\(S\_-\)[4]\^*\)] - 4750\ Re[\(E\_-\)[6]\ \(\(S\_-\)[5]\^*\)] + 21375\ Re[\(E\_+\)[4]\ \(\(S\_-\)[5]\^*\)] + 44715\ Re[\(E\_+\)[6]\ \(\(S\_-\)[5]\^*\)] - 4180\ Re[\(M\_-\)[6]\ \(\(S\_-\)[5]\^*\)] + 4180\ Re[\(M\_+\)[6]\ \(\(S\_-\)[5]\^*\)] - 4560\ Re[\(E\_-\)[5]\ \(\(S\_-\)[6]\^*\)] + 20064\ Re[\(E\_+\)[3]\ \(\(S\_-\)[6]\^*\)] + 46062\ Re[\(E\_+\)[5]\ \(\(S\_-\)[6]\^*\)] - 5130\ Re[\(M\_-\)[5]\ \(\(S\_-\)[6]\^*\)] + 5130\ Re[\(M\_+\)[5]\ \(\(S\_-\)[6]\^*\)] - 16720\ Re[\(E\_-\)[6]\ \(\(S\_+\)[3]\^*\)] + 5016\ Re[\(E\_+\)[6]\ \(\(S\_+\)[3]\^*\)] + 3344\ Re[\(M\_-\)[6]\ \(\(S\_+\)[3]\^*\)] - 3344\ Re[\(M\_+\)[6]\ \(\(S\_+\)[3]\^*\)] - 17100\ Re[\(E\_-\)[5]\ \(\(S\_+\)[4]\^*\)] + 4845\ Re[\(E\_+\)[5]\ \(\(S\_+\)[4]\^*\)] + 4275\ Re[\(M\_-\)[5]\ \(\(S\_+\)[4]\^*\)] - 4275\ Re[\(M\_+\)[5]\ \(\(S\_+\)[4]\^*\)] - 15048\ Re[\(E\_-\)[4]\ \(\(S\_+\)[5]\^*\)] - 37530\ Re[\(E\_-\)[6]\ \(\(S\_+\)[5]\^*\)] + 4446\ Re[\(E\_+\)[4]\ \(\(S\_+\)[5]\^*\)] + 15510\ Re[\(E\_+\)[6]\ \(\(S\_+\)[5]\^*\)] + 5016\ Re[\(M\_-\)[4]\ \(\(S\_+\)[5]\^*\)] + 13662\ Re[\(M\_-\)[6]\ \(\(S\_+\)[5]\^*\)] - 5016\ Re[\(M\_+\)[4]\ \(\(S\_+\)[5]\^*\)] - 13662\ Re[\(M\_+\)[6]\ \(\(S\_+\)[5]\^*\)] - 10868\ Re[\(E\_-\)[3]\ \(\(S\_+\)[6]\^*\)] - 33264\ Re[\(E\_-\)[5]\ \(\(S\_+\)[6]\^*\)] + 3762\ Re[\(E\_+\)[3]\ \(\(S\_+\)[6]\^*\)] + 13783\ Re[\(E\_+\)[5]\ \(\(S\_+\)[6]\^*\)] + 5434\ Re[\(M\_-\)[3]\ \(\(S\_+\)[6]\^*\)] + 15631\ Re[\(M\_-\)[5]\ \(\(S\_+\)[6]\^*\)] - 5434\ Re[\(M\_+\)[3]\ \(\(S\_+\)[6]\^*\)] - 15631\ Re[\(M\_+\)[5]\ \(\(S\_+\)[6]\^*\)])\)\ P\_8[x])\)\) - 294\/221\ \((728\ Re[\(E\_+\)[6]\ \(\(S\_-\)[4]\^*\)] + 810\ Re[\(E\_+\)[5]\ \(\(S\_-\)[5]\^*\)] - 162\ Re[\(E\_-\)[6]\ \(\(S\_-\)[6]\^*\)] + 810\ Re[\(E\_+\)[4]\ \(\(S\_-\)[6]\^*\)] + 1710\ Re[\(E\_+\)[6]\ \(\(S\_-\)[6]\^*\)] - 162\ Re[\(M\_-\)[6]\ \(\(S\_-\)[6]\^*\)] + 162\ Re[\(M\_+\)[6]\ \(\(S\_-\)[6]\^*\)] - 675\ Re[\(E\_-\)[6]\ \(\(S\_+\)[4]\^*\)] + 170\ Re[\(E\_+\)[6]\ \(\(S\_+\)[4]\^*\)] + 135\ Re[\(M\_-\)[6]\ \(\(S\_+\)[4]\^*\)] - 135\ Re[\(M\_+\)[6]\ \(\(S\_+\)[4]\^*\)] - 648\ Re[\(E\_-\)[5]\ \(\(S\_+\)[5]\^*\)] + 162\ Re[\(E\_+\)[5]\ \(\(S\_+\)[5]\^*\)] + 162\ Re[\(M\_-\)[5]\ \(\(S\_+\)[5]\^*\)] - 162\ Re[\(M\_+\)[5]\ \(\(S\_+\)[5]\^*\)] - 546\ Re[\(E\_-\)[4]\ \(\(S\_+\)[6]\^*\)] - 1371\ Re[\(E\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 147\ Re[\(E\_+\)[4]\ \(\(S\_+\)[6]\^*\)] + 501\ Re[\(E\_+\)[6]\ \(\(S\_+\)[6]\^*\)] + 182\ Re[\(M\_-\)[4]\ \(\(S\_+\)[6]\^*\)] + 501\ Re[\(M\_-\)[6]\ \(\(S\_+\)[6]\^*\)] - 182\ Re[\(M\_+\)[4]\ \(\(S\_+\)[6]\^*\)] - 501\ Re[\(M\_+\)[6]\ \(\(S\_+\)[6]\^*\)])\)\ P\_9[ x] - \(\(1\/4199\)\((2646\ \((2275\ Re[\(E\_+\)[ 6]\ \(\(S\_-\)[5]\^*\)] + 2376\ Re[\(E\_+\)[5]\ \(\(S\_-\)[6]\^*\)] - 1980\ Re[\(E\_-\)[6]\ \(\(S\_+\)[5]\^*\)] + 438\ Re[\(E\_+\)[6]\ \(\(S\_+\)[5]\^*\)] + 396\ Re[\(M\_-\)[6]\ \(\(S\_+\)[5]\^*\)] - 396\ Re[\(M\_+\)[6]\ \(\(S\_+\)[5]\^*\)] - 1820\ Re[\(E\_-\)[5]\ \(\(S\_+\)[6]\^*\)] + 413\ Re[\(E\_+\)[5]\ \(\(S\_+\)[6]\^*\)] + 455\ Re[\(M\_-\)[5]\ \(\(S\_+\)[6]\^*\)] - 455\ Re[\(M\_+\)[5]\ \(\(S\_+\)[6]\^*\)])\)\ P\_10[x])\)\) - 106722\/323\ \((6\ Re[\(E\_+\)[6]\ \(\(S\_-\)[6]\^*\)] - 5\ Re[\(E\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + Re[\(E\_+\)[6]\ \(\(S\_+\)[6]\^*\)] + Re[\(M\_-\)[6]\ \(\(S\_+\)[6]\^*\)] - Re[\(M\_+\)[6]\ \(\(S\_+\)[6]\^*\)])\)\ P\_11[x]\)], "Output", CellLabel->"Out[34]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(TTexp = RtoLegendre[R\_TT, 6]\)], "Input", CellLabel->"In[35]:="], Cell[BoxData[ RowBox[{\(Simplify::"time"\), \(\(:\)\(\ \)\), "\<\"Time spent on a \ transformation exceeded \\!\\(300\\) seconds, and the transformation was \ aborted. Increasing the value of TimeConstraint option may improve the result \ of simplification. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Simplify::time\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[35]:="], Cell[BoxData[ \(3\/2\ Abs[\(E\_-\)[2]]\^2 + 12\ Abs[\(E\_-\)[3]]\^2 + 45\ Abs[\(E\_-\)[4]]\^2 + 120\ Abs[\(E\_-\)[5]]\^2 + 525\/2\ Abs[\(E\_-\)[6]]\^2 + 9\/2\ Abs[\(E\_+\)[1]]\^2 + 24\ Abs[\(E\_+\)[2]]\^2 + 75\ Abs[\(E\_+\)[3]]\^2 + 180\ Abs[\(E\_+\)[4]]\^2 + 735\/2\ Abs[\(E\_+\)[5]]\^2 + 672\ Abs[\(E\_+\)[6]]\^2 - 9\/2\ Abs[\(M\_-\)[2]]\^2 - 24\ Abs[\(M\_-\)[3]]\^2 - 75\ Abs[\(M\_-\)[4]]\^2 - 180\ Abs[\(M\_-\)[5]]\^2 - 735\/2\ Abs[\(M\_-\)[6]]\^2 - 3\/2\ Abs[\(M\_+\)[1]]\^2 - 12\ Abs[\(M\_+\)[2]]\^2 - 45\ Abs[\(M\_+\)[3]]\^2 - 120\ Abs[\(M\_+\)[4]]\^2 - 525\/2\ Abs[\(M\_+\)[5]]\^2 - 504\ Abs[\(M\_+\)[6]]\^2 - 3\ Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)] - 12\ Re[\(E\_+\)[2]\ \(\(E\_-\)[2]\^*\)] - 25\ Re[\(E\_+\)[4]\ \(\(E\_-\)[2]\^*\)] - 42\ Re[\(E\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 3\ Re[\(M\_-\)[2]\ \(\(E\_-\)[2]\^*\)] - 10\ Re[\(M\_-\)[4]\ \(\(E\_-\)[2]\^*\)] - 21\ Re[\(M\_-\)[6]\ \(\(E\_-\)[2]\^*\)] + 3\ Re[\(M\_+\)[2]\ \(\(E\_-\)[2]\^*\)] + 10\ Re[\(M\_+\)[4]\ \(\(E\_-\)[2]\^*\)] + 21\ Re[\(M\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 21\ Re[\(E\_+\)[1]\ \(\(E\_-\)[3]\^*\)] - 60\ Re[\(E\_+\)[3]\ \(\(E\_-\)[3]\^*\)] - 111\ Re[\(E\_+\)[5]\ \(\(E\_-\)[3]\^*\)] + 3\ Re[\(M\_-\)[1]\ \(\(E\_-\)[3]\^*\)] - 12\ Re[\(M\_-\)[3]\ \(\(E\_-\)[3]\^*\)] - 39\ Re[\(M\_-\)[5]\ \(\(E\_-\)[3]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_-\)[3]\^*\)] + 12\ Re[\(M\_+\)[3]\ \(\(E\_-\)[3]\^*\)] + 39\ Re[\(M\_+\)[5]\ \(\(E\_-\)[3]\^*\)] + 2\ Re[\(E\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 10\ Re[\(E\_+\)[0]\ \(\(E\_-\)[4]\^*\)] - 78\ Re[\(E\_+\)[2]\ \(\(E\_-\)[4]\^*\)] - 180\ Re[\(E\_+\)[4]\ \(\(E\_-\)[4]\^*\)] - 306\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] + 12\ Re[\(M\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 30\ Re[\(M\_-\)[4]\ \(\(E\_-\)[4]\^*\)] - 96\ Re[\(M\_-\)[6]\ \(\(E\_-\)[4]\^*\)] - 12\ Re[\(M\_+\)[2]\ \(\(E\_-\)[4]\^*\)] + 30\ Re[\(M\_+\)[4]\ \(\(E\_-\)[4]\^*\)] + 96\ Re[\(M\_+\)[6]\ \(\(E\_-\)[4]\^*\)] + 21\ Re[\(E\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 54\ Re[\(E\_+\)[1]\ \(\(E\_-\)[5]\^*\)] - 210\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 420\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 10\ Re[\(M\_-\)[1]\ \(\(E\_-\)[5]\^*\)] + 30\ Re[\(M\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 60\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] - 10\ Re[\(M\_+\)[1]\ \(\(E\_-\)[5]\^*\)] - 30\ Re[\(M\_+\)[3]\ \(\(E\_-\)[5]\^*\)] + 60\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 3\ Re[\(E\_-\)[2]\ \(\(E\_-\)[6]\^*\)] + 84\ Re[\(E\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 21\ Re[\(E\_+\)[0]\ \(\(E\_-\)[6]\^*\)] - 168\ Re[\(E\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 465\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 840\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 39\ Re[\(M\_-\)[2]\ \(\(E\_-\)[6]\^*\)] + 60\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 105\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] - 39\ Re[\(M\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 60\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 105\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 3\ Re[\(M\_-\)[2]\ \(\(E\_+\)[0]\^*\)] - 10\ Re[\(M\_-\)[4]\ \(\(E\_+\)[0]\^*\)] - 21\ Re[\(M\_-\)[6]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] + 10\ Re[\(M\_+\)[4]\ \(\(E\_+\)[0]\^*\)] + 21\ Re[\(M\_+\)[6]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 12\ Re[\(M\_-\)[3]\ \(\(E\_+\)[1]\^*\)] - 39\ Re[\(M\_-\)[5]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 12\ Re[\(M\_+\)[3]\ \(\(E\_+\)[1]\^*\)] + 39\ Re[\(M\_+\)[5]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[2]\^*\)] + 12\ Re[\(M\_-\)[2]\ \(\(E\_+\)[2]\^*\)] - 30\ Re[\(M\_-\)[4]\ \(\(E\_+\)[2]\^*\)] - 96\ Re[\(M\_-\)[6]\ \(\(E\_+\)[2]\^*\)] - 12\ Re[\(M\_+\)[2]\ \(\(E\_+\)[2]\^*\)] + 30\ Re[\(M\_+\)[4]\ \(\(E\_+\)[2]\^*\)] + 96\ Re[\(M\_+\)[6]\ \(\(E\_+\)[2]\^*\)] + 10\ Re[\(M\_-\)[1]\ \(\(E\_+\)[3]\^*\)] + 30\ Re[\(M\_-\)[3]\ \(\(E\_+\)[3]\^*\)] - 60\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] - 10\ Re[\(M\_+\)[1]\ \(\(E\_+\)[3]\^*\)] - 30\ Re[\(M\_+\)[3]\ \(\(E\_+\)[3]\^*\)] + 60\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] - 10\ Re[\(E\_+\)[0]\ \(\(E\_+\)[4]\^*\)] + 30\ Re[\(E\_+\)[2]\ \(\(E\_+\)[4]\^*\)] + 39\ Re[\(M\_-\)[2]\ \(\(E\_+\)[4]\^*\)] + 60\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] - 105\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] - 39\ Re[\(M\_+\)[2]\ \(\(E\_+\)[4]\^*\)] - 60\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] + 105\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] - 21\ Re[\(E\_+\)[1]\ \(\(E\_+\)[5]\^*\)] + 120\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] + 21\ Re[\(M\_-\)[1]\ \(\(E\_+\)[5]\^*\)] + 96\ Re[\(M\_-\)[3]\ \(\(E\_+\)[5]\^*\)] + 105\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] - 21\ Re[\(M\_+\)[1]\ \(\(E\_+\)[5]\^*\)] - 96\ Re[\(M\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 105\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] - 21\ Re[\(E\_+\)[0]\ \(\(E\_+\)[6]\^*\)] - 12\ Re[\(E\_+\)[2]\ \(\(E\_+\)[6]\^*\)] + 315\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] + 78\ Re[\(M\_-\)[2]\ \(\(E\_+\)[6]\^*\)] + 190\ Re[\(M\_-\)[4]\ \(\(E\_+\)[6]\^*\)] + 168\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 78\ Re[\(M\_+\)[2]\ \(\(E\_+\)[6]\^*\)] - 190\ Re[\(M\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 168\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] - 10\ Re[\(M\_+\)[3]\ \(\(M\_-\)[1]\^*\)] - 21\ Re[\(M\_+\)[5]\ \(\(M\_-\)[1]\^*\)] - 21\ Re[\(M\_+\)[2]\ \(\(M\_-\)[2]\^*\)] - 54\ Re[\(M\_+\)[4]\ \(\(M\_-\)[2]\^*\)] - 99\ Re[\(M\_+\)[6]\ \(\(M\_-\)[2]\^*\)] + 3\ Re[\(M\_-\)[1]\ \(\(M\_-\)[3]\^*\)] - 12\ Re[\(M\_+\)[1]\ \(\(M\_-\)[3]\^*\)] - 78\ Re[\(M\_+\)[3]\ \(\(M\_-\)[3]\^*\)] - 168\ Re[\(M\_+\)[5]\ \(\(M\_-\)[3]\^*\)] - 60\ Re[\(M\_+\)[2]\ \(\(M\_-\)[4]\^*\)] - 210\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] - 400\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] + 10\ Re[\(M\_-\)[1]\ \(\(M\_-\)[5]\^*\)] - 30\ Re[\(M\_-\)[3]\ \(\(M\_-\)[5]\^*\)] - 25\ Re[\(M\_+\)[1]\ \(\(M\_-\)[5]\^*\)] - 180\ Re[\(M\_+\)[3]\ \(\(M\_-\)[5]\^*\)] - 465\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] + 21\ Re[\(M\_-\)[2]\ \(\(M\_-\)[6]\^*\)] - 120\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] - 111\ Re[\(M\_+\)[2]\ \(\(M\_-\)[6]\^*\)] - 420\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] - 903\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_+\)[3]\^*\)] - 21\ Re[\(M\_+\)[2]\ \(\(M\_+\)[4]\^*\)] + 3\ Re[\(M\_+\)[1]\ \(\(M\_+\)[5]\^*\)] - 84\ Re[\(M\_+\)[3]\ \(\(M\_+\)[5]\^*\)] - 6\ Re[\(M\_+\)[2]\ \(\(M\_+\)[6]\^*\)] - 230\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)] - 3\ \((6\ Re[\(E\_+\)[1]\ \(\(E\_-\)[2]\^*\)] + 17\ Re[\(E\_+\)[3]\ \(\(E\_-\)[2]\^*\)] + 32\ Re[\(E\_+\)[5]\ \(\(E\_-\)[2]\^*\)] + 5\ Re[\(M\_-\)[3]\ \(\(E\_-\)[2]\^*\)] + 14\ Re[\(M\_-\)[5]\ \(\(E\_-\)[2]\^*\)] - 5\ Re[\(M\_+\)[3]\ \(\(E\_-\)[2]\^*\)] - 14\ Re[\(M\_+\)[5]\ \(\(E\_-\)[2]\^*\)] - 4\ Re[\(E\_-\)[2]\ \(\(E\_-\)[3]\^*\)] + 5\ Re[\(E\_+\)[0]\ \(\(E\_-\)[3]\^*\)] + 36\ Re[\(E\_+\)[2]\ \(\(E\_-\)[3]\^*\)] + 81\ Re[\(E\_+\)[4]\ \(\(E\_-\)[3]\^*\)] + 138\ Re[\(E\_+\)[6]\ \(\(E\_-\)[3]\^*\)] + 21\ Re[\(M\_-\)[4]\ \(\(E\_-\)[3]\^*\)] + 54\ Re[\(M\_-\)[6]\ \(\(E\_-\)[3]\^*\)] - 21\ Re[\(M\_+\)[4]\ \(\(E\_-\)[3]\^*\)] - 54\ Re[\(M\_+\)[6]\ \(\(E\_-\)[3]\^*\)] - 27\ Re[\(E\_-\)[3]\ \(\(E\_-\)[4]\^*\)] + 33\ Re[\(E\_+\)[1]\ \(\(E\_-\)[4]\^*\)] + 120\ Re[\(E\_+\)[3]\ \(\(E\_-\)[4]\^*\)] + 234\ Re[\(E\_+\)[5]\ \(\(E\_-\)[4]\^*\)] - 5\ Re[\(M\_-\)[1]\ \(\(E\_-\)[4]\^*\)] + 54\ Re[\(M\_-\)[5]\ \(\(E\_-\)[4]\^*\)] + 5\ Re[\(M\_+\)[1]\ \(\(E\_-\)[4]\^*\)] - 54\ Re[\(M\_+\)[5]\ \(\(E\_-\)[4]\^*\)] - Re[\(E\_-\)[2]\ \(\(E\_-\)[5]\^*\)] - 96\ Re[\(E\_-\)[4]\ \(\(E\_-\)[5]\^*\)] + 14\ Re[\(E\_+\)[0]\ \(\(E\_-\)[5]\^*\)] + 114\ Re[\(E\_+\)[2]\ \(\(E\_-\)[5]\^*\)] + 300\ Re[\(E\_+\)[4]\ \(\(E\_-\)[5]\^*\)] + 530\ Re[\(E\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 21\ Re[\(M\_-\)[2]\ \(\(E\_-\)[5]\^*\)] + 110\ Re[\(M\_-\)[6]\ \(\(E\_-\)[5]\^*\)] + 21\ Re[\(M\_+\)[2]\ \(\(E\_-\)[5]\^*\)] - 110\ Re[\(M\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 18\ Re[\(E\_-\)[3]\ \(\(E\_-\)[6]\^*\)] - 250\ Re[\(E\_-\)[5]\ \(\(E\_-\)[6]\^*\)] + 72\ Re[\(E\_+\)[1]\ \(\(E\_-\)[6]\^*\)] + 290\ Re[\(E\_+\)[3]\ \(\(E\_-\)[6]\^*\)] + 630\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 14\ Re[\(M\_-\)[1]\ \(\(E\_-\)[6]\^*\)] - 54\ Re[\(M\_-\)[3]\ \(\(E\_-\)[6]\^*\)] + 14\ Re[\(M\_+\)[1]\ \(\(E\_-\)[6]\^*\)] + 54\ Re[\(M\_+\)[3]\ \(\(E\_-\)[6]\^*\)] + 5\ Re[\(M\_-\)[3]\ \(\(E\_+\)[0]\^*\)] + 14\ Re[\(M\_-\)[5]\ \(\(E\_+\)[0]\^*\)] - 5\ Re[\(M\_+\)[3]\ \(\(E\_+\)[0]\^*\)] - 14\ Re[\(M\_+\)[5]\ \(\(E\_+\)[0]\^*\)] + 21\ Re[\(M\_-\)[4]\ \(\(E\_+\)[1]\^*\)] + 54\ Re[\(M\_-\)[6]\ \(\(E\_+\)[1]\^*\)] - 21\ Re[\(M\_+\)[4]\ \(\(E\_+\)[1]\^*\)] - 54\ Re[\(M\_+\)[6]\ \(\(E\_+\)[1]\^*\)] - 9\ Re[\(E\_+\)[1]\ \(\(E\_+\)[2]\^*\)] - 5\ Re[\(M\_-\)[1]\ \(\(E\_+\)[2]\^*\)] + 54\ Re[\(M\_-\)[5]\ \(\(E\_+\)[2]\^*\)] + 5\ Re[\(M\_+\)[1]\ \(\(E\_+\)[2]\^*\)] - 54\ Re[\(M\_+\)[5]\ \(\(E\_+\)[2]\^*\)] + 5\ Re[\(E\_+\)[0]\ \(\(E\_+\)[3]\^*\)] - 48\ Re[\(E\_+\)[2]\ \(\(E\_+\)[3]\^*\)] - 21\ Re[\(M\_-\)[2]\ \(\(E\_+\)[3]\^*\)] + 110\ Re[\(M\_-\)[6]\ \(\(E\_+\)[3]\^*\)] + 21\ Re[\(M\_+\)[2]\ \(\(E\_+\)[3]\^*\)] - 110\ Re[\(M\_+\)[6]\ \(\(E\_+\)[3]\^*\)] + 6\ Re[\(E\_+\)[1]\ \(\(E\_+\)[4]\^*\)] - 150\ Re[\(E\_+\)[3]\ \(\(E\_+\)[4]\^*\)] - 14\ Re[\(M\_-\)[1]\ \(\(E\_+\)[4]\^*\)] - 54\ Re[\(M\_-\)[3]\ \(\(E\_+\)[4]\^*\)] + 14\ Re[\(M\_+\)[1]\ \(\(E\_+\)[4]\^*\)] + 54\ Re[\(M\_+\)[3]\ \(\(E\_+\)[4]\^*\)] + 14\ Re[\(E\_+\)[0]\ \(\(E\_+\)[5]\^*\)] - 18\ Re[\(E\_+\)[2]\ \(\(E\_+\)[5]\^*\)] - 360\ Re[\(E\_+\)[4]\ \(\(E\_+\)[5]\^*\)] - 54\ Re[\(M\_-\)[2]\ \(\(E\_+\)[5]\^*\)] - 110\ Re[\(M\_-\)[4]\ \(\(E\_+\)[5]\^*\)] + 54\ Re[\(M\_+\)[2]\ \(\(E\_+\)[5]\^*\)] + 110\ Re[\(M\_+\)[4]\ \(\(E\_+\)[5]\^*\)] + 33\ Re[\(E\_+\)[1]\ \(\(E\_+\)[6]\^*\)] - 100\ Re[\(E\_+\)[3]\ \(\(E\_+\)[6]\^*\)] - 735\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 27\ Re[\(M\_-\)[1]\ \(\(E\_+\)[6]\^*\)] - 132\ Re[\(M\_-\)[3]\ \(\(E\_+\)[6]\^*\)] - 195\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] + 27\ Re[\(M\_+\)[1]\ \(\(E\_+\)[6]\^*\)] + 132\ Re[\(M\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 195\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] + 5\ Re[\(M\_+\)[2]\ \(\(M\_-\)[1]\^*\)] + 14\ Re[\(M\_+\)[4]\ \(\(M\_-\)[1]\^*\)] + 27\ Re[\(M\_+\)[6]\ \(\(M\_-\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(M\_-\)[2]\^*\)] + 33\ Re[\(M\_+\)[3]\ \(\(M\_-\)[2]\^*\)] + 72\ Re[\(M\_+\)[5]\ \(\(M\_-\)[2]\^*\)] + 9\ Re[\(M\_-\)[2]\ \(\(M\_-\)[3]\^*\)] + 36\ Re[\(M\_+\)[2]\ \(\(M\_-\)[3]\^*\)] + 114\ Re[\(M\_+\)[4]\ \(\(M\_-\)[3]\^*\)] + 216\ Re[\(M\_+\)[6]\ \(\(M\_-\)[3]\^*\)] - 5\ Re[\(M\_-\)[1]\ \(\(M\_-\)[4]\^*\)] + 48\ Re[\(M\_-\)[3]\ \(\(M\_-\)[4]\^*\)] + 17\ Re[\(M\_+\)[1]\ \(\(M\_-\)[4]\^*\)] + 120\ Re[\(M\_+\)[3]\ \(\(M\_-\)[4]\^*\)] + 290\ Re[\(M\_+\)[5]\ \(\(M\_-\)[4]\^*\)] - 6\ Re[\(M\_-\)[2]\ \(\(M\_-\)[5]\^*\)] + 150\ Re[\(M\_-\)[4]\ \(\(M\_-\)[5]\^*\)] + 81\ Re[\(M\_+\)[2]\ \(\(M\_-\)[5]\^*\)] + 300\ Re[\(M\_+\)[4]\ \(\(M\_-\)[5]\^*\)] + 615\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] - 14\ Re[\(M\_-\)[1]\ \(\(M\_-\)[6]\^*\)] + 18\ Re[\(M\_-\)[3]\ \(\(M\_-\)[6]\^*\)] + 360\ Re[\(M\_-\)[5]\ \(\(M\_-\)[6]\^*\)] + 32\ Re[\(M\_+\)[1]\ \(\(M\_-\)[6]\^*\)] + 234\ Re[\(M\_+\)[3]\ \(\(M\_-\)[6]\^*\)] + 630\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] + 4\ Re[\(M\_+\)[1]\ \(\(M\_+\)[2]\^*\)] + 27\ Re[\(M\_+\)[2]\ \(\(M\_+\)[3]\^*\)] + Re[\(M\_+\)[1]\ \(\(M\_+\)[4]\^*\)] + 96\ Re[\(M\_+\)[3]\ \(\(M\_+\)[4]\^*\)] + 18\ Re[\(M\_+\)[2]\ \(\(M\_+\)[5]\^*\)] + 250\ Re[\(M\_+\)[4]\ \(\(M\_+\)[5]\^*\)] - 6\ Re[\(M\_+\)[1]\ \(\(M\_+\)[6]\^*\)] + 78\ Re[\(M\_+\)[3]\ \(\(M\_+\)[6]\^*\)] + 540\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_1[x] + 5\ \((3\ Abs[\(E\_-\)[3]]\^2 + 18\ Abs[\(E\_-\)[4]]\^2 + 60\ Abs[\(E\_-\)[5]]\^2 + 150\ Abs[\(E\_-\)[6]]\^2 + 6\ Abs[\(E\_+\)[2]]\^2 + 30\ Abs[\(E\_+\)[3]]\^2 + 90\ Abs[\(E\_+\)[4]]\^2 + 210\ Abs[\(E\_+\)[5]]\^2 + 420\ Abs[\(E\_+\)[6]]\^2 - 6\ Abs[\(M\_-\)[3]]\^2 - 30\ Abs[\(M\_-\)[4]]\^2 - 90\ Abs[\(M\_-\)[5]]\^2 - 210\ Abs[\(M\_-\)[6]]\^2 - 3\ Abs[\(M\_+\)[2]]\^2 - 18\ Abs[\(M\_+\)[3]]\^2 - 60\ Abs[\(M\_+\)[4]]\^2 - 150\ Abs[\(M\_+\)[5]]\^2 - 315\ Abs[\(M\_+\)[6]]\^2 - 9\ Re[\(E\_+\)[2]\ \(\(E\_-\)[2]\^*\)] - 22\ Re[\(E\_+\)[4]\ \(\(E\_-\)[2]\^*\)] - 39\ Re[\(E\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 7\ Re[\(M\_-\)[4]\ \(\(E\_-\)[2]\^*\)] - 18\ Re[\(M\_-\)[6]\ \(\(E\_-\)[2]\^*\)] + 7\ Re[\(M\_+\)[4]\ \(\(E\_-\)[2]\^*\)] + 18\ Re[\(M\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 12\ Re[\(E\_+\)[1]\ \(\(E\_-\)[3]\^*\)] - 51\ Re[\(E\_+\)[3]\ \(\(E\_-\)[3]\^*\)] - 102\ Re[\(E\_+\)[5]\ \(\(E\_-\)[3]\^*\)] - 3\ Re[\(M\_-\)[3]\ \(\(E\_-\)[3]\^*\)] - 30\ Re[\(M\_-\)[5]\ \(\(E\_-\)[3]\^*\)] + 3\ Re[\(M\_+\)[3]\ \(\(E\_-\)[3]\^*\)] + 30\ Re[\(M\_+\)[5]\ \(\(E\_-\)[3]\^*\)] + 5\ Re[\(E\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 7\ Re[\(E\_+\)[0]\ \(\(E\_-\)[4]\^*\)] - 60\ Re[\(E\_+\)[2]\ \(\(E\_-\)[4]\^*\)] - 162\ Re[\(E\_+\)[4]\ \(\(E\_-\)[4]\^*\)] - 288\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] + 3\ Re[\(M\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 12\ Re[\(M\_-\)[4]\ \(\(E\_-\)[4]\^*\)] - 78\ Re[\(M\_-\)[6]\ \(\(E\_-\)[4]\^*\)] - 3\ Re[\(M\_+\)[2]\ \(\(E\_-\)[4]\^*\)] + 12\ Re[\(M\_+\)[4]\ \(\(E\_-\)[4]\^*\)] + 78\ Re[\(M\_+\)[6]\ \(\(E\_-\)[4]\^*\)] + 30\ Re[\(E\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 45\ Re[\(E\_+\)[1]\ \(\(E\_-\)[5]\^*\)] - 180\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 390\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 7\ Re[\(M\_-\)[1]\ \(\(E\_-\)[5]\^*\)] + 12\ Re[\(M\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 30\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] - 7\ Re[\(M\_+\)[1]\ \(\(E\_-\)[5]\^*\)] - 12\ Re[\(M\_+\)[3]\ \(\(E\_-\)[5]\^*\)] + 30\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 102\ Re[\(E\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 18\ Re[\(E\_+\)[0]\ \(\(E\_-\)[6]\^*\)] - 150\ Re[\(E\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 420\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 795\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 30\ Re[\(M\_-\)[2]\ \(\(E\_-\)[6]\^*\)] + 30\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 60\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] - 30\ Re[\(M\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 30\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 60\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 7\ Re[\(M\_-\)[4]\ \(\(E\_+\)[0]\^*\)] - 18\ Re[\(M\_-\)[6]\ \(\(E\_+\)[0]\^*\)] + 7\ Re[\(M\_+\)[4]\ \(\(E\_+\)[0]\^*\)] + 18\ Re[\(M\_+\)[6]\ \(\(E\_+\)[0]\^*\)] - 3\ Re[\(M\_-\)[3]\ \(\(E\_+\)[1]\^*\)] - 30\ Re[\(M\_-\)[5]\ \(\(E\_+\)[1]\^*\)] + 3\ Re[\(M\_+\)[3]\ \(\(E\_+\)[1]\^*\)] + 30\ Re[\(M\_+\)[5]\ \(\(E\_+\)[1]\^*\)] + 3\ Re[\(M\_-\)[2]\ \(\(E\_+\)[2]\^*\)] - 12\ Re[\(M\_-\)[4]\ \(\(E\_+\)[2]\^*\)] - 78\ Re[\(M\_-\)[6]\ \(\(E\_+\)[2]\^*\)] - 3\ Re[\(M\_+\)[2]\ \(\(E\_+\)[2]\^*\)] + 12\ Re[\(M\_+\)[4]\ \(\(E\_+\)[2]\^*\)] + 78\ Re[\(M\_+\)[6]\ \(\(E\_+\)[2]\^*\)] + 9\ Re[\(E\_+\)[1]\ \(\(E\_+\)[3]\^*\)] + 7\ Re[\(M\_-\)[1]\ \(\(E\_+\)[3]\^*\)] + 12\ Re[\(M\_-\)[3]\ \(\(E\_+\)[3]\^*\)] - 30\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] - 7\ Re[\(M\_+\)[1]\ \(\(E\_+\)[3]\^*\)] - 12\ Re[\(M\_+\)[3]\ \(\(E\_+\)[3]\^*\)] + 30\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] - 7\ Re[\(E\_+\)[0]\ \(\(E\_+\)[4]\^*\)] + 48\ Re[\(E\_+\)[2]\ \(\(E\_+\)[4]\^*\)] + 30\ Re[\(M\_-\)[2]\ \(\(E\_+\)[4]\^*\)] + 30\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] - 60\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] - 30\ Re[\(M\_+\)[2]\ \(\(E\_+\)[4]\^*\)] - 30\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] + 60\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] - 12\ Re[\(E\_+\)[1]\ \(\(E\_+\)[5]\^*\)] + 150\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] + 18\ Re[\(M\_-\)[1]\ \(\(E\_+\)[5]\^*\)] + 78\ Re[\(M\_-\)[3]\ \(\(E\_+\)[5]\^*\)] + 60\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] - 18\ Re[\(M\_+\)[1]\ \(\(E\_+\)[5]\^*\)] - 78\ Re[\(M\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 60\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] - 18\ Re[\(E\_+\)[0]\ \(\(E\_+\)[6]\^*\)] + 6\ Re[\(E\_+\)[2]\ \(\(E\_+\)[6]\^*\)] + 360\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] + 69\ Re[\(M\_-\)[2]\ \(\(E\_+\)[6]\^*\)] + 160\ Re[\(M\_-\)[4]\ \(\(E\_+\)[6]\^*\)] + 105\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 69\ Re[\(M\_+\)[2]\ \(\(E\_+\)[6]\^*\)] - 160\ Re[\(M\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 105\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] - 7\ Re[\(M\_+\)[3]\ \(\(M\_-\)[1]\^*\)] - 18\ Re[\(M\_+\)[5]\ \(\(M\_-\)[1]\^*\)] - 12\ Re[\(M\_+\)[2]\ \(\(M\_-\)[2]\^*\)] - 45\ Re[\(M\_+\)[4]\ \(\(M\_-\)[2]\^*\)] - 90\ Re[\(M\_+\)[6]\ \(\(M\_-\)[2]\^*\)] - 9\ Re[\(M\_+\)[1]\ \(\(M\_-\)[3]\^*\)] - 60\ Re[\(M\_+\)[3]\ \(\(M\_-\)[3]\^*\)] - 150\ Re[\(M\_+\)[5]\ \(\(M\_-\)[3]\^*\)] - 9\ Re[\(M\_-\)[2]\ \(\(M\_-\)[4]\^*\)] - 51\ Re[\(M\_+\)[2]\ \(\(M\_-\)[4]\^*\)] - 180\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] - 370\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] + 7\ Re[\(M\_-\)[1]\ \(\(M\_-\)[5]\^*\)] - 48\ Re[\(M\_-\)[3]\ \(\(M\_-\)[5]\^*\)] - 22\ Re[\(M\_+\)[1]\ \(\(M\_-\)[5]\^*\)] - 162\ Re[\(M\_+\)[3]\ \(\(M\_-\)[5]\^*\)] - 420\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] + 12\ Re[\(M\_-\)[2]\ \(\(M\_-\)[6]\^*\)] - 150\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] - 102\ Re[\(M\_+\)[2]\ \(\(M\_-\)[6]\^*\)] - 390\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] - 840\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] - 5\ Re[\(M\_+\)[1]\ \(\(M\_+\)[3]\^*\)] - 30\ Re[\(M\_+\)[2]\ \(\(M\_+\)[4]\^*\)] - 102\ Re[\(M\_+\)[3]\ \(\(M\_+\)[5]\^*\)] - 15\ Re[\(M\_+\)[2]\ \(\(M\_+\)[6]\^*\)] - 260\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)])\)\ P\_2[ x] + \((\(-84\)\ Re[\(E\_+\)[3]\ \(\(E\_-\)[2]\^*\)] - 189\ Re[\(E\_+\)[5]\ \(\(E\_-\)[2]\^*\)] - 63\ Re[\(M\_-\)[5]\ \(\(E\_-\)[2]\^*\)] + 63\ Re[\(M\_+\)[5]\ \(\(E\_-\)[2]\^*\)] - 135\ Re[\(E\_+\)[2]\ \(\(E\_-\)[3]\^*\)] - 462\ Re[\(E\_+\)[4]\ \(\(E\_-\)[3]\^*\)] - 861\ Re[\(E\_+\)[6]\ \(\(E\_-\)[3]\^*\)] - 42\ Re[\(M\_-\)[4]\ \(\(E\_-\)[3]\^*\)] - 273\ Re[\(M\_-\)[6]\ \(\(E\_-\)[3]\^*\)] + 42\ Re[\(M\_+\)[4]\ \(\(E\_-\)[3]\^*\)] + 273\ Re[\(M\_+\)[6]\ \(\(E\_-\)[3]\^*\)] + 54\ Re[\(E\_-\)[3]\ \(\(E\_-\)[4]\^*\)] - 126\ Re[\(E\_+\)[1]\ \(\(E\_-\)[4]\^*\)] - 1840\/3\ Re[\(E\_+\)[3]\ \(\(E\_-\)[4]\^*\)] - 1428\ Re[\(E\_+\)[5]\ \(\(E\_-\)[4]\^*\)] - 168\ Re[\(M\_-\)[5]\ \(\(E\_-\)[4]\^*\)] + 168\ Re[\(M\_+\)[5]\ \(\(E\_-\)[4]\^*\)] + 42\ Re[\(E\_-\)[2]\ \(\(E\_-\)[5]\^*\)] + 896\/3\ Re[\(E\_-\)[4]\ \(\(E\_-\)[5]\^*\)] - 63\ Re[\(E\_+\)[0]\ \(\(E\_-\)[5]\^*\)] - 588\ Re[\(E\_+\)[2]\ \(\(E\_-\)[5]\^*\)] - 57050\/33\ Re[\(E\_+\)[4]\ \(\(E\_-\)[5]\^*\)] - 3360\ Re[\(E\_+\)[6]\ \(\(E\_-\)[5]\^*\)] + 42\ Re[\(M\_-\)[2]\ \(\(E\_-\)[5]\^*\)] - 420\ Re[\(M\_-\)[6]\ \(\(E\_-\)[5]\^*\)] - 42\ Re[\(M\_+\)[2]\ \(\(E\_-\)[5]\^*\)] + 420\ Re[\(M\_+\)[6]\ \(\(E\_-\)[5]\^*\)] + 231\ Re[\(E\_-\)[3]\ \(\(E\_-\)[6]\^*\)] + 10500\/11\ Re[\(E\_-\)[5]\ \(\(E\_-\)[6]\^*\)] - 399\ Re[\(E\_+\)[1]\ \(\(E\_-\)[6]\^*\)] - 1680\ Re[\(E\_+\)[3]\ \(\(E\_-\)[6]\^*\)] - 551880\/143\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] + 63\ Re[\(M\_-\)[1]\ \(\(E\_-\)[6]\^*\)] + 168\ Re[\(M\_-\)[3]\ \(\(E\_-\)[6]\^*\)] - 63\ Re[\(M\_+\)[1]\ \(\(E\_-\)[6]\^*\)] - 168\ Re[\(M\_+\)[3]\ \(\(E\_-\)[6]\^*\)] - 63\ Re[\(M\_-\)[5]\ \(\(E\_+\)[0]\^*\)] + 63\ Re[\(M\_+\)[5]\ \(\(E\_+\)[0]\^*\)] - 42\ Re[\(M\_-\)[4]\ \(\(E\_+\)[1]\^*\)] - 273\ Re[\(M\_-\)[6]\ \(\(E\_+\)[1]\^*\)] + 42\ Re[\(M\_+\)[4]\ \(\(E\_+\)[1]\^*\)] + 273\ Re[\(M\_+\)[6]\ \(\(E\_+\)[1]\^*\)] - 168\ Re[\(M\_-\)[5]\ \(\(E\_+\)[2]\^*\)] + 168\ Re[\(M\_+\)[5]\ \(\(E\_+\)[2]\^*\)] + 96\ Re[\(E\_+\)[2]\ \(\(E\_+\)[3]\^*\)] + 42\ Re[\(M\_-\)[2]\ \(\(E\_+\)[3]\^*\)] - 420\ Re[\(M\_-\)[6]\ \(\(E\_+\)[3]\^*\)] - 42\ Re[\(M\_+\)[2]\ \(\(E\_+\)[3]\^*\)] + 420\ Re[\(M\_+\)[6]\ \(\(E\_+\)[3]\^*\)] + 63\ Re[\(E\_+\)[1]\ \(\(E\_+\)[4]\^*\)] + 1400\/3\ Re[\(E\_+\)[3]\ \(\(E\_+\)[4]\^*\)] + 63\ Re[\(M\_-\)[1]\ \(\(E\_+\)[4]\^*\)] + 168\ Re[\(M\_-\)[3]\ \(\(E\_+\)[4]\^*\)] - 63\ Re[\(M\_+\)[1]\ \(\(E\_+\)[4]\^*\)] - 168\ Re[\(M\_+\)[3]\ \(\(E\_+\)[4]\^*\)] - 63\ Re[\(E\_+\)[0]\ \(\(E\_+\)[5]\^*\)] + 336\ Re[\(E\_+\)[2]\ \(\(E\_+\)[5]\^*\)] + 15120\/11\ Re[\(E\_+\)[4]\ \(\(E\_+\)[5]\^*\)] + 273\ Re[\(M\_-\)[2]\ \(\(E\_+\)[5]\^*\)] + 420\ Re[\(M\_-\)[4]\ \(\(E\_+\)[5]\^*\)] - 273\ Re[\(M\_+\)[2]\ \(\(E\_+\)[5]\^*\)] - 420\ Re[\(M\_+\)[4]\ \(\(E\_+\)[5]\^*\)] - 126\ Re[\(E\_+\)[1]\ \(\(E\_+\)[6]\^*\)] + 1050\ Re[\(E\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 41160\/13\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] + 154\ Re[\(M\_-\)[1]\ \(\(E\_+\)[6]\^*\)] + 714\ Re[\(M\_-\)[3]\ \(\(E\_+\)[6]\^*\)] + 840\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] - 154\ Re[\(M\_+\)[1]\ \(\(E\_+\)[6]\^*\)] - 714\ Re[\(M\_+\)[3]\ \(\(E\_+\)[6]\^*\)] - 840\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 63\ Re[\(M\_+\)[4]\ \(\(M\_-\)[1]\^*\)] - 154\ Re[\(M\_+\)[6]\ \(\(M\_-\)[1]\^*\)] - 126\ Re[\(M\_+\)[3]\ \(\(M\_-\)[2]\^*\)] - 399\ Re[\(M\_+\)[5]\ \(\(M\_-\)[2]\^*\)] - 135\ Re[\(M\_+\)[2]\ \(\(M\_-\)[3]\^*\)] - 588\ Re[\(M\_+\)[4]\ \(\(M\_-\)[3]\^*\)] - 1302\ Re[\(M\_+\)[6]\ \(\(M\_-\)[3]\^*\)] - 96\ Re[\(M\_-\)[3]\ \(\(M\_-\)[4]\^*\)] - 84\ Re[\(M\_+\)[1]\ \(\(M\_-\)[4]\^*\)] - 1840\/3\ Re[\(M\_+\)[3]\ \(\(M\_-\)[4]\^*\)] - 1680\ Re[\(M\_+\)[5]\ \(\(M\_-\)[4]\^*\)] - 63\ Re[\(M\_-\)[2]\ \(\(M\_-\)[5]\^*\)] - 1400\/3\ Re[\(M\_-\)[4]\ \(\(M\_-\)[5]\^*\)] - 462\ Re[\(M\_+\)[2]\ \(\(M\_-\)[5]\^*\)] - 57050\/33\ Re[\(M\_+\)[4]\ \(\(M\_-\)[5]\^*\)] - 3780\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] + 63\ Re[\(M\_-\)[1]\ \(\(M\_-\)[6]\^*\)] - 336\ Re[\(M\_-\)[3]\ \(\(M\_-\)[6]\^*\)] - 15120\/11\ Re[\(M\_-\)[5]\ \(\(M\_-\)[6]\^*\)] - 189\ Re[\(M\_+\)[1]\ \(\(M\_-\)[6]\^*\)] - 1428\ Re[\(M\_+\)[3]\ \(\(M\_-\)[6]\^*\)] - 551880\/143\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] - 54\ Re[\(M\_+\)[2]\ \(\(M\_+\)[3]\^*\)] - 42\ Re[\(M\_+\)[1]\ \(\(M\_+\)[4]\^*\)] - 896\/3\ Re[\(M\_+\)[3]\ \(\(M\_+\)[4]\^*\)] - 231\ Re[\(M\_+\)[2]\ \(\(M\_+\)[5]\^*\)] - 10500\/11\ Re[\(M\_+\)[4]\ \(\(M\_+\)[5]\^*\)] + 7\ Re[\(M\_+\)[1]\ \(\(M\_+\)[6]\^*\)] - 756\ Re[\(M\_+\)[3]\ \(\(M\_+\)[6]\^*\)] - 30240\/13\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_3[x] + 3\/143\ \((2145\ Abs[\(E\_-\)[4]]\^2 + 11180\ Abs[\(E\_-\)[5]]\^2 + 34425\ Abs[\(E\_-\)[6]]\^2 + 3575\ Abs[\(E\_+\)[3]]\^2 + 16770\ Abs[\(E\_+\)[4]]\^2 + 48195\ Abs[\(E\_+\)[5]]\^2 + 109032\ Abs[\(E\_+\)[6]]\^2 - 3575\ Abs[\(M\_-\)[4]]\^2 - 16770\ Abs[\(M\_-\)[5]]\^2 - 48195\ Abs[\(M\_-\)[6]]\^2 - 2145\ Abs[\(M\_+\)[3]]\^2 - 11180\ Abs[\(M\_+\)[4]]\^2 - 34425\ Abs[\(M\_+\)[5]]\^2 - 81774\ Abs[\(M\_+\)[6]]\^2 - 6435\ Re[\(E\_+\)[4]\ \(\(E\_-\)[2]\^*\)] - 13728\ Re[\(E\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 4719\ Re[\(M\_-\)[6]\ \(\(E\_-\)[2]\^*\)] + 4719\ Re[\(M\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 11440\ Re[\(E\_+\)[3]\ \(\(E\_-\)[3]\^*\)] - 34749\ Re[\(E\_+\)[5]\ \(\(E\_-\)[3]\^*\)] - 3861\ Re[\(M\_-\)[5]\ \(\(E\_-\)[3]\^*\)] + 3861\ Re[\(M\_+\)[5]\ \(\(E\_-\)[3]\^*\)] - 12870\ Re[\(E\_+\)[2]\ \(\(E\_-\)[4]\^*\)] - 49530\ Re[\(E\_+\)[4]\ \(\(E\_-\)[4]\^*\)] - 105534\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] - 1430\ Re[\(M\_-\)[4]\ \(\(E\_-\)[4]\^*\)] - 15444\ Re[\(M\_-\)[6]\ \(\(E\_-\)[4]\^*\)] + 1430\ Re[\(M\_+\)[4]\ \(\(E\_-\)[4]\^*\)] + 15444\ Re[\(M\_+\)[6]\ \(\(E\_-\)[4]\^*\)] + 4004\ Re[\(E\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 10296\ Re[\(E\_+\)[1]\ \(\(E\_-\)[5]\^*\)] - 53690\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 134830\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 1430\ Re[\(M\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 5590\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] - 1430\ Re[\(M\_+\)[3]\ \(\(E\_-\)[5]\^*\)] + 5590\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 3003\ Re[\(E\_-\)[2]\ \(\(E\_-\)[6]\^*\)] + 20826\ Re[\(E\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 4719\ Re[\(E\_+\)[0]\ \(\(E\_-\)[6]\^*\)] - 46332\ Re[\(E\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 143010\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 293070\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 3861\ Re[\(M\_-\)[2]\ \(\(E\_-\)[6]\^*\)] + 5590\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 13770\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] - 3861\ Re[\(M\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 5590\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 13770\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 4719\ Re[\(M\_-\)[6]\ \(\(E\_+\)[0]\^*\)] + 4719\ Re[\(M\_+\)[6]\ \(\(E\_+\)[0]\^*\)] - 3861\ Re[\(M\_-\)[5]\ \(\(E\_+\)[1]\^*\)] + 3861\ Re[\(M\_+\)[5]\ \(\(E\_+\)[1]\^*\)] - 1430\ Re[\(M\_-\)[4]\ \(\(E\_+\)[2]\^*\)] - 15444\ Re[\(M\_-\)[6]\ \(\(E\_+\)[2]\^*\)] + 1430\ Re[\(M\_+\)[4]\ \(\(E\_+\)[2]\^*\)] + 15444\ Re[\(M\_+\)[6]\ \(\(E\_+\)[2]\^*\)] + 1430\ Re[\(M\_-\)[3]\ \(\(E\_+\)[3]\^*\)] - 5590\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] - 1430\ Re[\(M\_+\)[3]\ \(\(E\_+\)[3]\^*\)] + 5590\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] + 6435\ Re[\(E\_+\)[2]\ \(\(E\_+\)[4]\^*\)] + 3861\ Re[\(M\_-\)[2]\ \(\(E\_+\)[4]\^*\)] + 5590\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] - 13770\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] - 3861\ Re[\(M\_+\)[2]\ \(\(E\_+\)[4]\^*\)] - 5590\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] + 13770\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] + 3861\ Re[\(E\_+\)[1]\ \(\(E\_+\)[5]\^*\)] + 30680\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] + 4719\ Re[\(M\_-\)[1]\ \(\(E\_+\)[5]\^*\)] + 15444\ Re[\(M\_-\)[3]\ \(\(E\_+\)[5]\^*\)] + 13770\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] - 4719\ Re[\(M\_+\)[1]\ \(\(E\_+\)[5]\^*\)] - 15444\ Re[\(M\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 13770\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] - 4719\ Re[\(E\_+\)[0]\ \(\(E\_+\)[6]\^*\)] + 20592\ Re[\(E\_+\)[2]\ \(\(E\_+\)[6]\^*\)] + 89235\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] + 20592\ Re[\(M\_-\)[2]\ \(\(E\_+\)[6]\^*\)] + 38610\ Re[\(M\_-\)[4]\ \(\(E\_+\)[6]\^*\)] + 27258\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 20592\ Re[\(M\_+\)[2]\ \(\(E\_+\)[6]\^*\)] - 38610\ Re[\(M\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 27258\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] - 4719\ Re[\(M\_+\)[5]\ \(\(M\_-\)[1]\^*\)] - 10296\ Re[\(M\_+\)[4]\ \(\(M\_-\)[2]\^*\)] - 29601\ Re[\(M\_+\)[6]\ \(\(M\_-\)[2]\^*\)] - 12870\ Re[\(M\_+\)[3]\ \(\(M\_-\)[3]\^*\)] - 46332\ Re[\(M\_+\)[5]\ \(\(M\_-\)[3]\^*\)] - 11440\ Re[\(M\_+\)[2]\ \(\(M\_-\)[4]\^*\)] - 53690\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] - 128700\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] - 6435\ Re[\(M\_-\)[3]\ \(\(M\_-\)[5]\^*\)] - 6435\ Re[\(M\_+\)[1]\ \(\(M\_-\)[5]\^*\)] - 49530\ Re[\(M\_+\)[3]\ \(\(M\_-\)[5]\^*\)] - 143010\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] - 3861\ Re[\(M\_-\)[2]\ \(\(M\_-\)[6]\^*\)] - 30680\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] - 34749\ Re[\(M\_+\)[2]\ \(\(M\_-\)[6]\^*\)] - 134830\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] - 306558\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] - 4004\ Re[\(M\_+\)[2]\ \(\(M\_+\)[4]\^*\)] - 3003\ Re[\(M\_+\)[1]\ \(\(M\_+\)[5]\^*\)] - 20826\ Re[\(M\_+\)[3]\ \(\(M\_+\)[5]\^*\)] - 15444\ Re[\(M\_+\)[2]\ \(\(M\_+\)[6]\^*\)] - 64395\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)])\)\ P\_4[x] + 1\/39\ \((\(-7722\)\ Re[\(E\_+\)[5]\ \(\(E\_-\)[2]\^*\)] - 14625\ Re[\(E\_+\)[4]\ \(\(E\_-\)[3]\^*\)] - 41184\ Re[\(E\_+\)[6]\ \(\(E\_-\)[3]\^*\)] - 5148\ Re[\(M\_-\)[6]\ \(\(E\_-\)[3]\^*\)] + 5148\ Re[\(M\_+\)[6]\ \(\(E\_-\)[3]\^*\)] - 18200\ Re[\(E\_+\)[3]\ \(\(E\_-\)[4]\^*\)] - 61515\ Re[\(E\_+\)[5]\ \(\(E\_-\)[4]\^*\)] - 2925\ Re[\(M\_-\)[5]\ \(\(E\_-\)[4]\^*\)] + 2925\ Re[\(M\_+\)[5]\ \(\(E\_-\)[4]\^*\)] + 5200\ Re[\(E\_-\)[4]\ \(\(E\_-\)[5]\^*\)] - 17550\ Re[\(E\_+\)[2]\ \(\(E\_-\)[5]\^*\)] - 71750\ Re[\(E\_+\)[4]\ \(\(E\_-\)[5]\^*\)] - 163782\ Re[\(E\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 11385\ Re[\(M\_-\)[6]\ \(\(E\_-\)[5]\^*\)] + 11385\ Re[\(M\_+\)[6]\ \(\(E\_-\)[5]\^*\)] + 4680\ Re[\(E\_-\)[3]\ \(\(E\_-\)[6]\^*\)] + 25875\ Re[\(E\_-\)[5]\ \(\(E\_-\)[6]\^*\)] - 12870\ Re[\(E\_+\)[1]\ \(\(E\_-\)[6]\^*\)] - 69975\ Re[\(E\_+\)[3]\ \(\(E\_-\)[6]\^*\)] - 183708\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] + 2925\ Re[\(M\_-\)[3]\ \(\(E\_-\)[6]\^*\)] - 2925\ Re[\(M\_+\)[3]\ \(\(E\_-\)[6]\^*\)] - 5148\ Re[\(M\_-\)[6]\ \(\(E\_+\)[1]\^*\)] + 5148\ Re[\(M\_+\)[6]\ \(\(E\_+\)[1]\^*\)] - 2925\ Re[\(M\_-\)[5]\ \(\(E\_+\)[2]\^*\)] + 2925\ Re[\(M\_+\)[5]\ \(\(E\_+\)[2]\^*\)] - 11385\ Re[\(M\_-\)[6]\ \(\(E\_+\)[3]\^*\)] + 11385\ Re[\(M\_+\)[6]\ \(\(E\_+\)[3]\^*\)] + 8125\ Re[\(E\_+\)[3]\ \(\(E\_+\)[4]\^*\)] + 2925\ Re[\(M\_-\)[3]\ \(\(E\_+\)[4]\^*\)] - 2925\ Re[\(M\_+\)[3]\ \(\(E\_+\)[4]\^*\)] + 6903\ Re[\(E\_+\)[2]\ \(\(E\_+\)[5]\^*\)] + 37260\ Re[\(E\_+\)[4]\ \(\(E\_+\)[5]\^*\)] + 5148\ Re[\(M\_-\)[2]\ \(\(E\_+\)[5]\^*\)] + 11385\ Re[\(M\_-\)[4]\ \(\(E\_+\)[5]\^*\)] - 5148\ Re[\(M\_+\)[2]\ \(\(E\_+\)[5]\^*\)] - 11385\ Re[\(M\_+\)[4]\ \(\(E\_+\)[5]\^*\)] + 3861\ Re[\(E\_+\)[1]\ \(\(E\_+\)[6]\^*\)] + 32400\ Re[\(E\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 105399\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] + 5577\ Re[\(M\_-\)[1]\ \(\(E\_+\)[6]\^*\)] + 20592\ Re[\(M\_-\)[3]\ \(\(E\_+\)[6]\^*\)] + 27963\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] - 5577\ Re[\(M\_+\)[1]\ \(\(E\_+\)[6]\^*\)] - 20592\ Re[\(M\_+\)[3]\ \(\(E\_+\)[6]\^*\)] - 27963\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 5577\ Re[\(M\_+\)[6]\ \(\(M\_-\)[1]\^*\)] - 12870\ Re[\(M\_+\)[5]\ \(\(M\_-\)[2]\^*\)] - 17550\ Re[\(M\_+\)[4]\ \(\(M\_-\)[3]\^*\)] - 56628\ Re[\(M\_+\)[6]\ \(\(M\_-\)[3]\^*\)] - 18200\ Re[\(M\_+\)[3]\ \(\(M\_-\)[4]\^*\)] - 69975\ Re[\(M\_+\)[5]\ \(\(M\_-\)[4]\^*\)] - 8125\ Re[\(M\_-\)[4]\ \(\(M\_-\)[5]\^*\)] - 14625\ Re[\(M\_+\)[2]\ \(\(M\_-\)[5]\^*\)] - 71750\ Re[\(M\_+\)[4]\ \(\(M\_-\)[5]\^*\)] - 180360\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] - 6903\ Re[\(M\_-\)[3]\ \(\(M\_-\)[6]\^*\)] - 37260\ Re[\(M\_-\)[5]\ \(\(M\_-\)[6]\^*\)] - 7722\ Re[\(M\_+\)[1]\ \(\(M\_-\)[6]\^*\)] - 61515\ Re[\(M\_+\)[3]\ \(\(M\_-\)[6]\^*\)] - 183708\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] - 5200\ Re[\(M\_+\)[3]\ \(\(M\_+\)[4]\^*\)] - 4680\ Re[\(M\_+\)[2]\ \(\(M\_+\)[5]\^*\)] - 25875\ Re[\(M\_+\)[4]\ \(\(M\_+\)[5]\^*\)] - 3432\ Re[\(M\_+\)[1]\ \(\(M\_+\)[6]\^*\)] - 23193\ Re[\(M\_+\)[3]\ \(\(M\_+\)[6]\^*\)] - 77436\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_5[x] + 1\/187\ \((17850\ Abs[\(E\_-\)[5]]\^2 + 85680\ Abs[\(E\_-\)[6]]\^2 + 26775\ Abs[\(E\_+\)[4]]\^2 + 119952\ Abs[\(E\_+\)[5]]\^2 + 333564\ Abs[\(E\_+\)[6]]\^2 - 26775\ Abs[\(M\_-\)[5]]\^2 - 119952\ Abs[\(M\_-\)[6]]\^2 - 17850\ Abs[\(M\_+\)[4]]\^2 - 85680\ Abs[\(M\_+\)[5]]\^2 - 250173\ Abs[\(M\_+\)[6]]\^2 - 51051\ Re[\(E\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 100980\ Re[\(E\_+\)[5]\ \(\(E\_-\)[3]\^*\)] - 133875\ Re[\(E\_+\)[4]\ \(\(E\_-\)[4]\^*\)] - 416636\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] - 25245\ Re[\(M\_-\)[6]\ \(\(E\_-\)[4]\^*\)] + 25245\ Re[\(M\_+\)[6]\ \(\(E\_-\)[4]\^*\)] - 142800\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 510153\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 8925\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] + 8925\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 34425\ Re[\(E\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 126225\ Re[\(E\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 535500\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 1273629\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 8925\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 34272\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] - 8925\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 34272\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 25245\ Re[\(M\_-\)[6]\ \(\(E\_+\)[2]\^*\)] + 25245\ Re[\(M\_+\)[6]\ \(\(E\_+\)[2]\^*\)] - 8925\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] + 8925\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] + 8925\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] - 34272\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] - 8925\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] + 34272\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] + 50745\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] + 25245\ Re[\(M\_-\)[3]\ \(\(E\_+\)[5]\^*\)] + 34272\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] - 25245\ Re[\(M\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 34272\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] + 41514\ Re[\(E\_+\)[2]\ \(\(E\_+\)[6]\^*\)] + 228718\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] + 36465\ Re[\(M\_-\)[2]\ \(\(E\_+\)[6]\^*\)] + 97988\ Re[\(M\_-\)[4]\ \(\(E\_+\)[6]\^*\)] + 83391\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 36465\ Re[\(M\_+\)[2]\ \(\(E\_+\)[6]\^*\)] - 97988\ Re[\(M\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 83391\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] - 87516\ Re[\(M\_+\)[6]\ \(\(M\_-\)[2]\^*\)] - 126225\ Re[\(M\_+\)[5]\ \(\(M\_-\)[3]\^*\)] - 142800\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] - 489379\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] - 133875\ Re[\(M\_+\)[3]\ \(\(M\_-\)[5]\^*\)] - 535500\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] - 50745\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] - 100980\ Re[\(M\_+\)[2]\ \(\(M\_-\)[6]\^*\)] - 510153\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] - 1322748\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] - 34425\ Re[\(M\_+\)[3]\ \(\(M\_+\)[5]\^*\)] - 30294\ Re[\(M\_+\)[2]\ \(\(M\_+\)[6]\^*\)] - 165002\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)])\)\ P\_6[ x] - \(\(1\/2431\)\((105\ \((17017\ Re[\(E\_+\)[ 6]\ \(\(E\_-\)[3]\^*\)] + 23562\ Re[\(E\_+\)[5]\ \(\(E\_-\)[4]\^*\)] + 26775\ Re[\(E\_+\)[4]\ \(\(E\_-\)[5]\^*\)] + 87780\ Re[\(E\_+\)[6]\ \(\(E\_-\)[5]\^*\)] + 2618\ Re[\(M\_-\)[6]\ \(\(E\_-\)[5]\^*\)] - 2618\ Re[\(M\_+\)[6]\ \(\(E\_-\)[5]\^*\)] - 5950\ Re[\(E\_-\)[5]\ \(\(E\_-\)[6]\^*\)] + 26180\ Re[\(E\_+\)[3]\ \(\(E\_-\)[6]\^*\)] + 96768\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] + 2618\ Re[\(M\_-\)[6]\ \(\(E\_+\)[3]\^*\)] - 2618\ Re[\(M\_+\)[6]\ \(\(E\_+\)[3]\^*\)] - 8568\ Re[\(E\_+\)[4]\ \(\(E\_+\)[5]\^*\)] - 2618\ Re[\(M\_-\)[4]\ \(\(E\_+\)[5]\^*\)] + 2618\ Re[\(M\_+\)[4]\ \(\(E\_+\)[5]\^*\)] - 7854\ Re[\(E\_+\)[3]\ \(\(E\_+\)[6]\^*\)] - 37730\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 4862\ Re[\(M\_-\)[3]\ \(\(E\_+\)[6]\^*\)] - 10010\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] + 4862\ Re[\(M\_+\)[3]\ \(\(E\_+\)[6]\^*\)] + 10010\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] + 21879\ Re[\(M\_+\)[6]\ \(\(M\_-\)[3]\^*\)] + 26180\ Re[\(M\_+\)[5]\ \(\(M\_-\)[4]\^*\)] + 26775\ Re[\(M\_+\)[4]\ \(\(M\_-\)[5]\^*\)] + 95172\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] + 8568\ Re[\(M\_-\)[5]\ \(\(M\_-\)[6]\^*\)] + 23562\ Re[\(M\_+\)[3]\ \(\(M\_-\)[6]\^*\)] + 96768\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] + 5950\ Re[\(M\_+\)[4]\ \(\(M\_+\)[5]\^*\)] + 5610\ Re[\(M\_+\)[3]\ \(\(M\_+\)[6]\^*\)] + 27720\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_7[x])\)\) + 49\/247\ \((855\ Abs[\(E\_-\)[6]]\^2 + 1197\ Abs[\(E\_+\)[5]]\^2 + 5184\ Abs[\(E\_+\)[6]]\^2 - 1197\ Abs[\(M\_-\)[6]]\^2 - 855\ Abs[\(M\_+\)[5]]\^2 - 3888\ Abs[\(M\_+\)[6]]\^2 - 6916\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] - 8208\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 8550\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 28944\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 342\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] + 342\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 342\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] + 342\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] + 342\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] - 342\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] + 2318\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] + 988\ Re[\(M\_-\)[4]\ \(\(E\_+\)[6]\^*\)] + 1296\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 988\ Re[\(M\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 1296\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] - 7904\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] - 8550\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] - 8208\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] - 29898\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] - 1672\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)])\)\ P\_8[x] - 2646\/221\ \((182\ Re[\(E\_+\)[6]\ \(\(E\_-\)[5]\^*\)] + 198\ Re[\(E\_+\)[5]\ \(\(E\_-\)[6]\^*\)] - 49\ Re[\(E\_+\)[5]\ \(\(E\_+\)[6]\^*\)] - 13\ Re[\(M\_-\)[5]\ \(\(E\_+\)[6]\^*\)] + 13\ Re[\(M\_+\)[5]\ \(\(E\_+\)[6]\^*\)] + 195\ Re[\(M\_+\)[6]\ \(\(M\_-\)[5]\^*\)] + 198\ Re[\(M\_+\)[5]\ \(\(M\_-\)[6]\^*\)] + 36\ Re[\(M\_+\)[5]\ \(\(M\_+\)[6]\^*\)])\)\ P\_9[x] + 29106\/323\ \((4\ Abs[\(E\_+\)[6]]\^2 - 3\ Abs[\(M\_+\)[6]]\^2 - 35\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] - 36\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)])\)\ P\_10[x]\)], "Output",\ CellLabel->"Out[35]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Legendre coefficients", "Subsection"], Cell["\<\ \tNext we extract the Legendre coefficients used by the traditional analysis \ of quadrupole ratios.\ \>", "Text"], Cell[BoxData[{ \(\(A\_T0 = \ Texp /. {P\_n_[x] \[Rule] 0};\)\), "\[IndentingNewLine]", \(\(A\_L0 = \ Lexp /. {P\_n_[x] \[Rule] 0};\)\), "\[IndentingNewLine]", \(\(A\_TT0 = TTexp /. {P\_n_[x] \[Rule] 0};\)\), "\[IndentingNewLine]", \(\(A\_LT1 = Coefficient[Expand[LTexp], P\_1[x]];\)\), "\[IndentingNewLine]", \(\(A\_T2 = Coefficient[Expand[Texp], P\_2[x]];\)\), "\[IndentingNewLine]", \(\(A\_L2 = Coefficient[Expand[Lexp], P\_2[x]];\)\)}], "Input", CellLabel->"In[36]:="], Cell["\<\ It will be useful to define truncation and recombination functions.\ \>", "Text"], Cell[BoxData[{ \(\(M1truncation[A_] := Select[Expand[A], Not[\((FreeQ[#, \(M\_+\)[1]])\)] &];\)\), "\[IndentingNewLine]", \(\(M1truncation[A_\/B_] := M1truncation[A]\/M1truncation[B];\)\), "\[IndentingNewLine]", \(truncate[ A_, \[ScriptL]_]\ := \ \((A /. {_[n_?\((# > \[ScriptL] &)\)] \[Rule] 0})\) /. {\(0\^*\) \[Rule] 0}\)}], "Input", CellLabel->"In[246]:="], Cell[TextData[{ "The number of terms in each expansion grows rapidly with the maximum ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL]\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[ Table[Length /@ \(\((truncate[#, \[ScriptL]] &)\) /@ {A\_L0, A\_T0, A\_LT1, A\_TT0, A\_L2, A\_T2}\), {\[ScriptL], 1, 6}], TableHeadings \[Rule] {Range[ 6], {\*"\"\<\!\(A\_0\%L\)\>\"", \*"\"\<\!\(A\_0\%T\)\>\"", \ \*"\"\<\!\(A\_1\%LT\)\>\"", \*"\"\<\!\(A\_0\%TT\)\>\"", \ \*"\"\<\!\(A\_2\%L\)\>\"", \*"\"\<\!\(A\_2\%T\)\>\""}}]\)], "Input", CellLabel->"In[45]:="], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"\\!\\(A\\_0\\%L\\)\"\>", \ "\<\"\\!\\(A\\_0\\%T\\)\"\>", "\<\"\\!\\(A\\_1\\%LT\\)\"\>", "\<\"\\!\\(A\\_0\ \\%TT\\)\"\>", "\<\"\\!\\(A\\_2\\%L\\)\"\>", "\<\"\\!\\(A\\_2\\%T\\)\"\>"}, {"1", "3", "4", "8", "9", "5", "9"}, {"2", "5", "8", "26", "27", "11", "26"}, {"3", "7", "12", "50", "52", "17", "46"}, {"4", "9", "16", "84", "85", "23", "66"}, {"5", "11", "20", "113", "116", "27", "82"}, {"6", "13", "24", "150", "153", "31", "98"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{1, 2, 3, 4, 5, 6}, {"\!\(A\_0\%L\)", "\!\(A\_0\%T\)", "\!\(A\_1\%LT\)", "\!\(A\_0\%TT\)", "\!\(A\_2\%L\)", "\!\(A\_2\%T\)"}}]]]], "Output", CellLabel->"Out[45]//TableForm="] }, Open ]], Cell[TextData[{ "but is severely limited by the assumption of ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[ Table[Length /@ \(\((truncate[#, \[ScriptL]] &)\) /@ \(M1truncation /@ \ {A\_L0, A\_T0, A\_LT1, A\_TT0, A\_L2, A\_T2}\)\), {\[ScriptL], 1, 6}], TableHeadings \[Rule] {Range[ 6], {\*"\"\<\!\(A\_0\%L\)\>\"", \*"\"\<\!\(A\_0\%T\)\>\"", \ \*"\"\<\!\(A\_1\%LT\)\>\"", \*"\"\<\!\(A\_0\%TT\)\>\"", \ \*"\"\<\!\(A\_2\%L\)\>\"", \*"\"\<\!\(A\_2\%T\)\>\""}}]\)], "Input", CellLabel->"In[46]:="], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"\\!\\(A\\_0\\%L\\)\"\>", \ "\<\"\\!\\(A\\_0\\%T\\)\"\>", "\<\"\\!\\(A\\_1\\%LT\\)\"\>", "\<\"\\!\\(A\\_0\ \\%TT\\)\"\>", "\<\"\\!\\(A\\_2\\%L\\)\"\>", "\<\"\\!\\(A\\_2\\%T\\)\"\>"}, {"1", "0", "2", "2", "4", "0", "4"}, {"2", "0", "2", "2", "4", "0", "4"}, {"3", "0", "2", "4", "8", "0", "6"}, {"4", "0", "2", "4", "8", "0", "6"}, {"5", "0", "2", "5", "11", "0", "6"}, {"6", "0", "2", "5", "11", "0", "6"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{1, 2, 3, 4, 5, 6}, {"\!\(A\_0\%L\)", "\!\(A\_0\%T\)", "\!\(A\_1\%LT\)", "\!\(A\_0\%TT\)", "\!\(A\_2\%L\)", "\!\(A\_2\%T\)"}}]]]], "Output", CellLabel->"Out[46]//TableForm="] }, Open ]], Cell[TextData[{ "Notice that ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " truncation by itself is sufficient to eliminate even multipoles." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(M1truncation /@ {A\_L0, A\_T0, A\_LT1, A\_TT0, A\_L2, A\_T2} // TableForm\)], "Input", CellLabel->"In[47]:="], Cell[BoxData[ InterpretationBox[GridBox[{ {"0"}, {\(2\ Abs[\(M\_+\)[1]]\^2\)}, {\(\(-9\)\ Re[\(M\_+\)[1]\ \(\(S\_-\)[3]\^*\)] - 15\ Re[\(M\_+\)[1]\ \(\(S\_-\)[5]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 12\ Re[\(M\_+\)[1]\ \(\(S\_+\)[3]\^*\)] + 18\ Re[\(M\_+\)[1]\ \(\(S\_+\)[5]\^*\)]\)}, {\(\(-\(3\/2\)\)\ Abs[\(M\_+\)[1]]\^2 - 3\ Re[\(M\_+\)[1]\ \(\(E\_-\)[3]\^*\)] - 10\ Re[\(M\_+\)[1]\ \(\(E\_-\)[5]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 10\ Re[\(M\_+\)[1]\ \(\(E\_+\)[3]\^*\)] - 21\ Re[\(M\_+\)[1]\ \(\(E\_+\)[5]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] - 12\ Re[\(M\_+\)[1]\ \(\(M\_-\)[3]\^*\)] - 25\ Re[\(M\_+\)[1]\ \(\(M\_-\)[5]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_+\)[3]\^*\)] + 3\ Re[\(M\_+\)[1]\ \(\(M\_+\)[5]\^*\)]\)}, {"0"}, {\(\(-Abs[\(M\_+\)[1]]\^2\) + 6\ Re[\(M\_+\)[1]\ \(\(E\_-\)[3]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] + 24\/7\ Re[\(M\_+\)[1]\ \(\(M\_-\)[3]\^*\)] + 144\/7\ Re[\(M\_+\)[1]\ \(\(M\_+\)[3]\^*\)]\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {0, Times[ 2, Power[ Abs[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1]], 2]], Plus[ Times[ -9, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ S\[UnderBracket]Subscript\[UnderBracket]Dash[ 3]]]]], Times[ -15, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ S\[UnderBracket]Subscript\[UnderBracket]Dash[ 5]]]]], Times[ 6, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ S\[UnderBracket]Subscript\[UnderBracket]Plus[ 1]]]]], Times[ 12, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ S\[UnderBracket]Subscript\[UnderBracket]Plus[ 3]]]]], Times[ 18, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ S\[UnderBracket]Subscript\[UnderBracket]Plus[ 5]]]]]], Plus[ Times[ Rational[ -3, 2], Power[ Abs[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1]], 2]], Times[ -3, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ E\[UnderBracket]Subscript\[UnderBracket]Dash[ 3]]]]], Times[ -10, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ E\[UnderBracket]Subscript\[UnderBracket]Dash[ 5]]]]], Times[ -3, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ E\[UnderBracket]Subscript\[UnderBracket]Plus[ 1]]]]], Times[ -10, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ E\[UnderBracket]Subscript\[UnderBracket]Plus[ 3]]]]], Times[ -21, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ E\[UnderBracket]Subscript\[UnderBracket]Plus[ 5]]]]], Times[ -3, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ M\[UnderBracket]Subscript\[UnderBracket]Dash[ 1]]]]], Times[ -12, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ M\[UnderBracket]Subscript\[UnderBracket]Dash[ 3]]]]], Times[ -25, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ M\[UnderBracket]Subscript\[UnderBracket]Dash[ 5]]]]], Times[ -2, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 3]]]]], Times[ 3, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 5]]]]]], 0, Plus[ Times[ -1, Power[ Abs[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1]], 2]], Times[ 6, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ E\[UnderBracket]Subscript\[UnderBracket]Dash[ 3]]]]], Times[ 6, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ E\[UnderBracket]Subscript\[UnderBracket]Plus[ 1]]]]], Times[ -2, Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ M\[UnderBracket]Subscript\[UnderBracket]Dash[ 1]]]]], Times[ Rational[ 24, 7], Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ M\[UnderBracket]Subscript\[UnderBracket]Dash[ 3]]]]], Times[ Rational[ 144, 7], Re[ Times[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 1], SuperStar[ M\[UnderBracket]Subscript\[UnderBracket]Plus[ 3]]]]]]}]]], "Output", CellLabel->"Out[47]//TableForm="] }, Open ]], Cell[TextData[{ "Thus, we verify that the traditional formulas are recovered when ", StyleBox["sp", FontSlant->"Italic"], " truncation and ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance are applied." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(R\&~\_EM = truncate[M1truncation[\(3 \((A\_T2 + \[Epsilon]\ A\_L2)\) - 2 A\_TT0\)\ \/\(12 \((A\_T0 + \[Epsilon]\ A\_L0)\)\)], 1] // MySimplify\)], "Input", CellLabel->"In[48]:="], Cell[BoxData[ \(Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)]\/Abs[\(M\_+\)[1]]\^2\)], "Output", CellLabel->"Out[48]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(R\&~\_SM = truncate[M1truncation[A\_LT1\/\(3 \((A\_T0 + \[Epsilon]\ A\_L0)\)\)], 1] // MySimplify\)], "Input", CellLabel->"In[49]:="], Cell[BoxData[ \(Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)]\/Abs[\(M\_+\)[1]]\^2\)], "Output", CellLabel->"Out[49]="] }, Open ]], Cell[TextData[{ "However, relaxation of either assumption permits additional terms to \ contribution to the traditional formulas such that it is no longer possible \ to isolate ", Cell[BoxData[ \(TraditionalForm\`Re[E\_\(\(1\)\(+\)\)/M\_\(\(1\)\(+\)\)]\)]], " or ", Cell[BoxData[ \(TraditionalForm\`Re[S\_\(\(1\)\(+\)\)/M\_\(\(1\)\(+\)\)]\)]], " if those terms are numerically significant. For example, the traditional \ combinations of Legendre coefficients take the following forms when ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance is applied without ", StyleBox["sp", FontSlant->"Italic"], " truncation. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(M1truncation[\(3 \((A\_T2 + \[Epsilon]\ A\_L2)\) - 2 A\_TT0\)\/\(12 \ \((A\_T0 + \[Epsilon]\ A\_L0)\)\)] //. recombine // MySimplify\)], "Input", CellLabel->"In[50]:="], Cell[BoxData[ \(\(\(1\/\(84\ Abs[\(M\_+\)[1]]\^2\)\)\((84\ Re[\(M\_+\)[ 1]\ \(\(E\_-\)[3]\^*\)] + 70\ Re[\(M\_+\)[1]\ \(\(E\_-\)[5]\^*\)] + 84\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 70\ Re[\(M\_+\)[1]\ \(\(E\_+\)[3]\^*\)] + 147\ Re[\(M\_+\)[1]\ \(\(E\_+\)[5]\^*\)] + 120\ Re[\(M\_+\)[1]\ \(\(M\_-\)[3]\^*\)] + 175\ Re[\(M\_+\)[1]\ \(\(M\_-\)[5]\^*\)] + 230\ Re[\(M\_+\)[1]\ \(\(M\_+\)[3]\^*\)] - 21\ Re[\(M\_+\)[1]\ \(\(M\_+\)[5]\^*\)])\)\)\)], "Output", CellLabel->"Out[50]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(M1truncation[A\_LT1\/\(3 \((A\_T0 + \[Epsilon]\ A\_L0)\)\)] //. recombine // MySimplify\)], "Input", CellLabel->"In[51]:="], Cell[BoxData[ \(\(\(-3\)\ Re[\(M\_+\)[1]\ \(\(S\_-\)[3]\^*\)] - 5\ Re[\(M\_+\)[1]\ \ \(\(S\_-\)[5]\^*\)] + 2\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 4\ \ Re[\(M\_+\)[1]\ \(\(S\_+\)[3]\^*\)] + 6\ Re[\(M\_+\)[1]\ \ \(\(S\_+\)[5]\^*\)]\)\/\(2\ Abs[\(M\_+\)[1]]\^2\)\)], "Output", CellLabel->"Out[51]="] }, Open ]], Cell[TextData[{ "Similarly, using ", StyleBox["sp", FontSlant->"Italic"], " truncation without ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance yields the following." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(truncate[\(3 \((A\_T2 + \[Epsilon]\ A\_L2)\) - 2 A\_TT0\)\/\(12 \ \((A\_T0 + \[Epsilon]\ A\_L0)\)\), 1] //. recombine // MySimplify\)], "Input",\ CellLabel->"In[52]:="], Cell[BoxData[ \(\(2\ \((\[Epsilon]\ Abs[\(S\_+\)[1]]\^2 - Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\ \^*\)] + Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + \[Epsilon]\ \ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]])\)\)\/\(Abs[\(E\_+\)[0]]\^2 + 6\ \ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_-\)[1]]\^2 + 2\ Abs[\(M\_+\)[1]]\^2 + \ \[Epsilon]\ Abs[\(S\_-\)[1]]\^2 + \[Epsilon]\ Abs[\(S\_+\)[0]]\^2 + 8\ \ \[Epsilon]\ Abs[\(S\_+\)[1]]\^2\)\)], "Output", CellLabel->"Out[52]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(truncate[A\_LT1\/\(3 \((A\_T0 + \[Epsilon]\ A\_L0)\)\), 1] //. recombine // MySimplify\)], "Input", CellLabel->"In[53]:="], Cell[BoxData[ \(\(-\(\(2\ \((Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] + Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - Re[\(M\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)])\)\)\/\(Abs[\(E\_+\)[0]]\^2 + 6\ Abs[\(E\_+\)[1]]\^2 + Abs[\(M\_-\)[1]]\^2 + 2\ Abs[\(M\_+\)[1]]\^2 + \[Epsilon]\ Abs[\(S\_-\)[1]]\^2 + \ \[Epsilon]\ Abs[\(S\_+\)[0]]\^2 + 8\ \[Epsilon]\ Abs[\(S\_+\)[1]]\^2\)\)\)\)], "Output", CellLabel->"Out[53]="] }, Open ]], Cell[TextData[{ "Notice that the formulas without ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominant become dependent upon ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\)]], ". Therefore, elimination of ", Cell[BoxData[ \(TraditionalForm\`R\_L\)]], " by Rosenbluth separation would help." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Traditional quadrupole formulas", "Subsection"], Cell[TextData[{ "The functions below form the combinations of Legendre coefficients \ traditionally used to extract quadrupole ratios from cross section data. To \ study the convergence of these formulas, the maximum ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL]\)]], " for contributing multipoles is given as an argument. In addition, there \ are versions which truncate according to ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance. However, experimentally one cannot perform such truncations \ and the question is how well ", Cell[BoxData[ \(TraditionalForm\`\(R\&~\)\_EM\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(R\&~\)\_SM\)]], " in the limit ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL] \[Rule] \[Infinity]\)]], " approximate ", Cell[BoxData[ \(TraditionalForm\`R\_EM\)]], " and ", Cell[BoxData[ \(TraditionalForm\`R\_SM\)]], ". Although most attempts to extract quadrupole amplitudes have not used \ Rosenbluth separation, one would expect better accuracy if the longitudinal \ response function were eliminated because it vanishes under the assumption of \ ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance. Rosenbluth separation can be simulated here by choosing ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon] \[Rule] 0\)]], "." }], "Text"], Cell[CellGroupData[{ Cell["M1 dominance", "Subsubsection"], Cell[BoxData[{ \(\(EMR[M1, \[ScriptL]_Integer?NonNegative] := truncate[ M1truncation[\(3 \((A\_T2 + \[Epsilon]\ A\_L2)\) - 2 \ A\_TT0\)\/\(12 \((A\_T0 + \[Epsilon]\ A\_L0)\)\)], \[ScriptL]];\)\), "\n", \(SMR[M1, \[ScriptL]_Integer?NonNegative] := truncate[M1truncation[ A\_LT1\/\(3 \((A\_T0 + \[Epsilon]\ A\_L0)\)\)], \[ScriptL]]\)}], \ "Input", CellLabel->"In[54]:="], Cell[BoxData[{ \(\(EMR[M1, \[ScriptL]_, eps_, values : {__Rule}] := \(\(EMR[M1, \[ScriptL]] /. values\) /. \[Epsilon] \[Rule] eps\) /. \(x_\^*\) \[Rule] Conjugate[x];\)\), "\[IndentingNewLine]", \(SMR[M1, \[ScriptL]_, eps_, values : {__Rule}] := \(\(SMR[M1, \[ScriptL]] /. values\) /. \[Epsilon] \[Rule] eps\) /. \(x_\^*\) \[Rule] Conjugate[x]\)}], "Input", CellLabel->"In[56]:="] }, Open ]], Cell[CellGroupData[{ Cell["truncation without M1 dominance", "Subsubsection"], Cell[BoxData[{ \(\(EMR[\[ScriptL]_Integer?NonNegative] := truncate[\(3 \((A\_T2 + \[Epsilon]\ A\_L2)\) - 2 A\_TT0\)\/\(12 \((A\ \_T0 + \[Epsilon]\ A\_L0)\)\), \[ScriptL]];\)\), "\n", \(SMR[\[ScriptL]_Integer?NonNegative] := truncate[A\_LT1\/\(3 \((A\_T0 + \[Epsilon]\ A\_L0)\)\), \ \[ScriptL]]\)}], "Input", CellLabel->"In[58]:="], Cell[BoxData[{ \(\(EMR[\[ScriptL]_, eps_, values : {__Rule}] := \(\(EMR[\[ScriptL]] /. values\) /. \[Epsilon] \[Rule] eps\) /. \(x_\^*\) \[Rule] Conjugate[x];\)\), "\[IndentingNewLine]", \(SMR[\[ScriptL]_, eps_, values : {__Rule}] := \(\(SMR[\[ScriptL]] /. values\) /. \[Epsilon] \[Rule] eps\) /. \(x_\^*\) \[Rule] Conjugate[x]\)}], "Input", CellLabel->"In[60]:="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Ranking of omitted terms", "Subsection"], Cell["\<\ Additional insight into the behavior of the traditional Legendre analysis can \ be obtained by ordering the contributions to the numerators and the \ denominator by magnitude. The following function performs that operation.\ \>", "Text"], Cell[BoxData[ \(arrange[coeff_, mpamps_, eps_] := Module[{list}, list = List @@ Expand[coeff]; list\[LeftDoubleBracket]Ordering[\((\(\(list /. mpamps\) /. \[Epsilon] \[Rule] eps\) /. \(x_\^*\) \[Rule] Conjugate[x])\), All, Abs[#1] > Abs[#2] &]\[RightDoubleBracket]]\)], "Input", CellLabel->"In[62]:="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Numerical analysis", "Section"], Cell[TextData[{ "\tTo analyze the quantitative accuracy of the traditional Legendre \ analysis, we employ the MAID2003 model. This is a realistic phenomenological \ model fitted to previous pion electro- and photo-production data. Although \ the details are clearly model dependent, other recent models give \ qualitatively similar results for the convergence and accuracy of Legendre \ coefficients and the quadrupole ratios extracted from them using the \ traditional method. The following ", Cell[BoxData[ \(TraditionalForm\`p\[VeryThinSpace]\[Pi]\^0\)]], " multipole amplitudes were obtained for ", Cell[BoxData[ \(TraditionalForm\`\((W, Q\^2)\) = \((1.23, 1.0)\)\)]], "." }], "Text"], Cell[TextData[{ "\tIdeally the quadrupole amplitudes are obtained for isospin-3/2 \ amplitudes for ", Cell[BoxData[ \(TraditionalForm\`W = M\_\[CapitalDelta]\)]], ". Although ", Cell[BoxData[ \(TraditionalForm\`M\_\[CapitalDelta] \[TildeTilde] 1.232\)]], " GeV, this parameter is model dependent. Furthermore, experimental \ analyses average over finite bins of ", Cell[BoxData[ \(TraditionalForm\`W\)]], " which are not necessarily centered upon the nominal ", Cell[BoxData[ \(TraditionalForm\`M\_\[CapitalDelta]\)]], "; the closest bin for JLab experiment e91011 was actually 1.23 GeV. \ Finally, experimental separation of isospin amplitudes is not performed often \ and would entail additional uncertainties if it were done. Therefore, it is \ useful to test the traditional Legendre analysis for both ", Cell[BoxData[ \(TraditionalForm\`p\[ThinSpace]\[Pi]\^0\)]], " and isospin-3/2 channels and for both 1.232 and 1.230 GeV. This becomes \ a little tedious, but if any of these variations were omitted, someone would \ question the conclusions." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`p\[Pi]\^0\)]], ", W=1.232" }], "Subsection"], Cell[BoxData[ \(\(maidvalues[p\[Pi]0, 1.232] = \[IndentingNewLine]Join[{\(E\_+\)[0] \[Rule] 3.894\[Times]10\^\(-1\) + \[ImaginaryI]\ \ 1.117\[Times]10\^\(-1\), \(S\_+\)[ 0] \[Rule] \(-3.516\)\[Times]10\^\(-1\) + \[ImaginaryI]\ \ 9.778\[Times]10\^\(-2\), \[IndentingNewLine]\(M\_+\)[ 1] \[Rule] \(-4.855\)\[Times]10\^\(-3\) + 1.895 \[ImaginaryI], \(E\_+\)[1] \[Rule] 4.735\[Times]10\^\(-2\) - \[ImaginaryI]\ \ 3.088\[Times]10\^\(-2\), \(S\_+\)[1] \[Rule] 1.932\[Times]10\^\(-2\) - \[ImaginaryI]\ \ 1.261\[Times]10\^\(-1\), \(M\_-\)[ 1] \[Rule] \(-4.837\)\[Times]10\^\(-1\) - \[ImaginaryI]\ \ 8.949\[Times]10\^\(-3\), \(S\_-\)[1] \[Rule] 2.638\[Times]10\^\(-1\) + \ \[ImaginaryI]\ \ 2.989\[Times]10\^\(-2\), \[IndentingNewLine]\(M\_+\)[ 2] \[Rule] \(-2.964\)\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 3.000\[Times]10\^\(-4\), \(E\_+\)[2] \[Rule] 2.258\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 9.037\[Times]10\^\(-5\), \(S\_+\)[ 2] \[Rule] \(-2.768\)\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 6.564\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_-\)[2] \[Rule] 1.854\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 1.686\[Times]10\^\(-3\), \(E\_-\)[ 2] \[Rule] \(-8.363\)\[Times]10\^\(-2\) - \[ImaginaryI]\ \ 1.176\[Times]10\^\(-3\), \(S\_-\)[2] \[Rule] 2.135\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 5.058\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_+\)[3] \[Rule] 9.687\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 3.213\[Times]10\^\(-5\), \(E\_+\)[ 3] \[Rule] \(-3.276\)\[Times]10\^\(-4\) - \[ImaginaryI]\ \ 4.398\[Times]10\^\(-6\), \(S\_+\)[3] \[Rule] 2.939\[Times]10\^\(-4\) - \[ImaginaryI]\ \ 1.771\[Times]10\^\(-6\), \[IndentingNewLine]\(M\_-\)[3] \[Rule] 3.727\[Times]10\^\(-4\) + \[ImaginaryI]\ \ 3.154\[Times]10\^\(-5\), \(E\_-\)[3] \[Rule] 1.394\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 1.823\[Times]10\^\(-5\), \(S\_-\)[ 3] \[Rule] \(-4.856\)\[Times]10\^\(-3\) - \[ImaginaryI]\ \ 6.440\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_+\)[ 4] \[Rule] \(-1.999\)\[Times]10\^\(-3\), \(E\_+\)[4] \[Rule] 1.027\[Times]10\^\(-4\), \(S\_+\)[ 4] \[Rule] \(-1.137\)\[Times]10\^\(-4\), \ \[IndentingNewLine]\(M\_-\)[4] \[Rule] 1.807\[Times]10\^\(-4\), \(E\_-\)[ 4] \[Rule] \(-3.194\)\[Times]10\^\(-3\), \(S\_-\)[4] \[Rule] 1.036\[Times]10\^\(-3\), \(M\_+\)[5] \[Rule] 5.752\[Times]10\^\(-4\), \(E\_+\)[ 5] \[Rule] \(-2.540\)\[Times]10\^\(-5\), \(S\_+\)[5] \[Rule] 2.648\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_-\)[ 5] \[Rule] \(-8.473\)\[Times]10\^\(-8\), \(E\_-\)[5] \[Rule] 8.124\[Times]10\^\(-4\), \(S\_-\)[ 5] \[Rule] \(-2.738\)\[Times]10\^\(-4\)}, Flatten[Table[{\(M\_+\)[\[ScriptL]] \[Rule] 0, \(E\_+\)[\[ScriptL]] \[Rule] 0, \(S\_+\)[\[ScriptL]] \[Rule] 0, \(M\_-\)[\[ScriptL]] \[Rule] 0, \(E\_-\)[\[ScriptL]] \[Rule] 0, \(S\_-\)[\[ScriptL]] \[Rule] 0}, {\[ScriptL], 6, 6}]]];\)\)], "Input", CellLabel->"In[63]:="], Cell[BoxData[ \(\(values = maidvalues[p\[Pi]0, 1.232];\)\)], "Input", CellLabel->"In[64]:="], Cell["The correct quadrupole ratios are", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({R\_EM, R\_SM} = {Re[\(E\_+\)[1]\/\(M\_+\)[1]], Re[\(S\_+\)[1]\/\(M\_+\)[1]]} /. values\)], "Input", CellLabel->"In[65]:="], Cell[BoxData[ \({\(-0.016359423463111665`\), \(-0.0665692189542751`\)}\)], "Output", CellLabel->"Out[65]="] }, Open ]], Cell["and would be reproduced by the traditional analysis", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\({R\&~\_EM, R\&~\_SM} /. values\) /. \(x_\^*\) \[Rule] Conjugate[x]\)], "Input", CellLabel->"In[66]:="], Cell[BoxData[ \({\(-0.016359423463111665`\), \(-0.06656921895427512`\)}\)], "Output", CellLabel->"Out[66]="] }, Open ]], Cell[TextData[{ "if its assumptions were accurate. Consider first the convergence with \ respect to ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL]\)]], " without assuming ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0.95, values], SMR[\[ScriptL], 0.95, values]}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[67]:="], Cell[BoxData[ TagBox[GridBox[{ {\(-0.00490078656187251`\), \(-0.062454466364268234`\)}, {\(-0.012038053872974583`\), \(-0.05854351486582303`\)}, {\(-0.011680915273555458`\), \(-0.05546851072372838`\)}, {\(-0.01137823529944677`\), \(-0.055133719947292825`\)}, {\(-0.011333818869828136`\), \(-0.05484031001762324`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[67]//TableForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0.95, values], SMR[\[ScriptL], 0.95, values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[68]:="], Cell[BoxData[ TagBox[GridBox[{ {"0.2995696378251431`", "0.9381883601062948`"}, {"0.73584829563944`", "0.8794382116160211`"}, {"0.7140175385700099`", "0.8332456290621113`"}, {"0.6955156656408574`", "0.8282164161361565`"}, {"0.6928006292754997`", "0.8238088245453475`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[68]//TableForm="] }, Open ]], Cell[TextData[{ "Although these values do appear to converge, their limits are incorrect. \ The relative errors are about ", Cell[BoxData[ \(TraditionalForm\`18 %\)]], " for SMR and a whopping ", Cell[BoxData[ \(TraditionalForm\`30 %\)]], " for EMR. Evidently, multipoles with ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL] > 1\)]], " and terms with ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL] \[LessEqual] 1\)]], " that do not involve ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " spoil the traditional Legendre analysis. Although the multipoles do tend \ to decrease with ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL]\)]], ", the coefficients in the multipole expansions of Legendre coefficients \ tend to increase with ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL]\)]], ". Multipole expansions for each of the required Legendre coefficients are \ given below." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(A\_L0\)], "Input", CellLabel->"In[69]:="], Cell[BoxData[ \(Abs[\(S\_-\)[1]]\^2 + 8\ Abs[\(S\_-\)[2]]\^2 + 27\ Abs[\(S\_-\)[3]]\^2 + 64\ Abs[\(S\_-\)[4]]\^2 + 125\ Abs[\(S\_-\)[5]]\^2 + 216\ Abs[\(S\_-\)[6]]\^2 + Abs[\(S\_+\)[0]]\^2 + 8\ Abs[\(S\_+\)[1]]\^2 + 27\ Abs[\(S\_+\)[2]]\^2 + 64\ Abs[\(S\_+\)[3]]\^2 + 125\ Abs[\(S\_+\)[4]]\^2 + 216\ Abs[\(S\_+\)[5]]\^2 + 343\ Abs[\(S\_+\)[6]]\^2\)], "Output", CellLabel->"Out[69]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A\_T0\)], "Input", CellLabel->"In[70]:="], Cell[BoxData[ \(2\ Abs[\(E\_-\)[2]]\^2 + 9\ Abs[\(E\_-\)[3]]\^2 + 24\ Abs[\(E\_-\)[4]]\^2 + 50\ Abs[\(E\_-\)[5]]\^2 + 90\ Abs[\(E\_-\)[6]]\^2 + Abs[\(E\_+\)[0]]\^2 + 6\ Abs[\(E\_+\)[1]]\^2 + 18\ Abs[\(E\_+\)[2]]\^2 + 40\ Abs[\(E\_+\)[3]]\^2 + 75\ Abs[\(E\_+\)[4]]\^2 + 126\ Abs[\(E\_+\)[5]]\^2 + 196\ Abs[\(E\_+\)[6]]\^2 + Abs[\(M\_-\)[1]]\^2 + 6\ Abs[\(M\_-\)[2]]\^2 + 18\ Abs[\(M\_-\)[3]]\^2 + 40\ Abs[\(M\_-\)[4]]\^2 + 75\ Abs[\(M\_-\)[5]]\^2 + 126\ Abs[\(M\_-\)[6]]\^2 + 2\ Abs[\(M\_+\)[1]]\^2 + 9\ Abs[\(M\_+\)[2]]\^2 + 24\ Abs[\(M\_+\)[3]]\^2 + 50\ Abs[\(M\_+\)[4]]\^2 + 90\ Abs[\(M\_+\)[5]]\^2 + 147\ Abs[\(M\_+\)[6]]\^2\)], "Output", CellLabel->"Out[70]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A\_LT1 //. recombine\)], "Input", CellLabel->"In[73]:="], Cell[BoxData[ \(9\ Re[\(E\_-\)[3]\ \(\(S\_-\)[1]\^*\)] + 15\ Re[\(E\_-\)[5]\ \(\(S\_-\)[1]\^*\)] - 6\ Re[\(E\_+\)[1]\ \(\(S\_-\)[1]\^*\)] - 12\ Re[\(E\_+\)[3]\ \(\(S\_-\)[1]\^*\)] - 18\ Re[\(E\_+\)[5]\ \(\(S\_-\)[1]\^*\)] + 6\ Re[\(E\_-\)[2]\ \(\(S\_-\)[2]\^*\)] + 54\ Re[\(E\_-\)[4]\ \(\(S\_-\)[2]\^*\)] + 78\ Re[\(E\_-\)[6]\ \(\(S\_-\)[2]\^*\)] - 6\ Re[\(E\_+\)[0]\ \(\(S\_-\)[2]\^*\)] - 30\ Re[\(E\_+\)[2]\ \(\(S\_-\)[2]\^*\)] - 54\ Re[\(E\_+\)[4]\ \(\(S\_-\)[2]\^*\)] - 78\ Re[\(E\_+\)[6]\ \(\(S\_-\)[2]\^*\)] + 6\ Re[\(M\_-\)[2]\ \(\(S\_-\)[2]\^*\)] + 6\ Re[\(M\_-\)[4]\ \(\(S\_-\)[2]\^*\)] + 6\ Re[\(M\_-\)[6]\ \(\(S\_-\)[2]\^*\)] - 6\ Re[\(M\_+\)[2]\ \(\(S\_-\)[2]\^*\)] - 6\ Re[\(M\_+\)[4]\ \(\(S\_-\)[2]\^*\)] - 6\ Re[\(M\_+\)[6]\ \(\(S\_-\)[2]\^*\)] + 27\ Re[\(E\_-\)[3]\ \(\(S\_-\)[3]\^*\)] + 162\ Re[\(E\_-\)[5]\ \(\(S\_-\)[3]\^*\)] - 27\ Re[\(E\_+\)[1]\ \(\(S\_-\)[3]\^*\)] - 81\ Re[\(E\_+\)[3]\ \(\(S\_-\)[3]\^*\)] - 135\ Re[\(E\_+\)[5]\ \(\(S\_-\)[3]\^*\)] + 9\ Re[\(M\_-\)[1]\ \(\(S\_-\)[3]\^*\)] + 27\ Re[\(M\_-\)[3]\ \(\(S\_-\)[3]\^*\)] + 27\ Re[\(M\_-\)[5]\ \(\(S\_-\)[3]\^*\)] - 9\ Re[\(M\_+\)[1]\ \(\(S\_-\)[3]\^*\)] - 27\ Re[\(M\_+\)[3]\ \(\(S\_-\)[3]\^*\)] - 27\ Re[\(M\_+\)[5]\ \(\(S\_-\)[3]\^*\)] - 12\ Re[\(E\_-\)[2]\ \(\(S\_-\)[4]\^*\)] + 72\ Re[\(E\_-\)[4]\ \(\(S\_-\)[4]\^*\)] + 360\ Re[\(E\_-\)[6]\ \(\(S\_-\)[4]\^*\)] - 12\ Re[\(E\_+\)[0]\ \(\(S\_-\)[4]\^*\)] - 72\ Re[\(E\_+\)[2]\ \(\(S\_-\)[4]\^*\)] - 168\ Re[\(E\_+\)[4]\ \(\(S\_-\)[4]\^*\)] - 264\ Re[\(E\_+\)[6]\ \(\(S\_-\)[4]\^*\)] + 36\ Re[\(M\_-\)[2]\ \(\(S\_-\)[4]\^*\)] + 72\ Re[\(M\_-\)[4]\ \(\(S\_-\)[4]\^*\)] + 72\ Re[\(M\_-\)[6]\ \(\(S\_-\)[4]\^*\)] - 36\ Re[\(M\_+\)[2]\ \(\(S\_-\)[4]\^*\)] - 72\ Re[\(M\_+\)[4]\ \(\(S\_-\)[4]\^*\)] - 72\ Re[\(M\_+\)[6]\ \(\(S\_-\)[4]\^*\)] - 45\ Re[\(E\_-\)[3]\ \(\(S\_-\)[5]\^*\)] + 150\ Re[\(E\_-\)[5]\ \(\(S\_-\)[5]\^*\)] - 45\ Re[\(E\_+\)[1]\ \(\(S\_-\)[5]\^*\)] - 150\ Re[\(E\_+\)[3]\ \(\(S\_-\)[5]\^*\)] - 300\ Re[\(E\_+\)[5]\ \(\(S\_-\)[5]\^*\)] + 15\ Re[\(M\_-\)[1]\ \(\(S\_-\)[5]\^*\)] + 90\ Re[\(M\_-\)[3]\ \(\(S\_-\)[5]\^*\)] + 150\ Re[\(M\_-\)[5]\ \(\(S\_-\)[5]\^*\)] - 15\ Re[\(M\_+\)[1]\ \(\(S\_-\)[5]\^*\)] - 90\ Re[\(M\_+\)[3]\ \(\(S\_-\)[5]\^*\)] - 150\ Re[\(M\_+\)[5]\ \(\(S\_-\)[5]\^*\)] - 18\ Re[\(E\_-\)[2]\ \(\(S\_-\)[6]\^*\)] - 108\ Re[\(E\_-\)[4]\ \(\(S\_-\)[6]\^*\)] + 270\ Re[\(E\_-\)[6]\ \(\(S\_-\)[6]\^*\)] - 18\ Re[\(E\_+\)[0]\ \(\(S\_-\)[6]\^*\)] - 108\ Re[\(E\_+\)[2]\ \(\(S\_-\)[6]\^*\)] - 270\ Re[\(E\_+\)[4]\ \(\(S\_-\)[6]\^*\)] - 486\ Re[\(E\_+\)[6]\ \(\(S\_-\)[6]\^*\)] + 54\ Re[\(M\_-\)[2]\ \(\(S\_-\)[6]\^*\)] + 180\ Re[\(M\_-\)[4]\ \(\(S\_-\)[6]\^*\)] + 270\ Re[\(M\_-\)[6]\ \(\(S\_-\)[6]\^*\)] - 54\ Re[\(M\_+\)[2]\ \(\(S\_-\)[6]\^*\)] - 180\ Re[\(M\_+\)[4]\ \(\(S\_-\)[6]\^*\)] - 270\ Re[\(M\_+\)[6]\ \(\(S\_-\)[6]\^*\)] + 3\ Re[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)] + 9\ Re[\(E\_-\)[4]\ \(\(S\_+\)[0]\^*\)] + 15\ Re[\(E\_-\)[6]\ \(\(S\_+\)[0]\^*\)] - 12\ Re[\(E\_+\)[2]\ \(\(S\_+\)[0]\^*\)] - 18\ Re[\(E\_+\)[4]\ \(\(S\_+\)[0]\^*\)] - 24\ Re[\(E\_+\)[6]\ \(\(S\_+\)[0]\^*\)] - 3\ Re[\(M\_-\)[2]\ \(\(S\_+\)[0]\^*\)] - 3\ Re[\(M\_-\)[4]\ \(\(S\_+\)[0]\^*\)] - 3\ Re[\(M\_-\)[6]\ \(\(S\_+\)[0]\^*\)] + 3\ Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)] + 3\ Re[\(M\_+\)[4]\ \(\(S\_+\)[0]\^*\)] + 3\ Re[\(M\_+\)[6]\ \(\(S\_+\)[0]\^*\)] + 18\ Re[\(E\_-\)[3]\ \(\(S\_+\)[1]\^*\)] + 42\ Re[\(E\_-\)[5]\ \(\(S\_+\)[1]\^*\)] - 6\ Re[\(E\_+\)[1]\ \(\(S\_+\)[1]\^*\)] - 66\ Re[\(E\_+\)[3]\ \(\(S\_+\)[1]\^*\)] - 90\ Re[\(E\_+\)[5]\ \(\(S\_+\)[1]\^*\)] - 6\ Re[\(M\_-\)[1]\ \(\(S\_+\)[1]\^*\)] - 18\ Re[\(M\_-\)[3]\ \(\(S\_+\)[1]\^*\)] - 18\ Re[\(M\_-\)[5]\ \(\(S\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)] + 18\ Re[\(M\_+\)[3]\ \(\(S\_+\)[1]\^*\)] + 18\ Re[\(M\_+\)[5]\ \(\(S\_+\)[1]\^*\)] + 9\ Re[\(E\_-\)[2]\ \(\(S\_+\)[2]\^*\)] + 54\ Re[\(E\_-\)[4]\ \(\(S\_+\)[2]\^*\)] + 108\ Re[\(E\_-\)[6]\ \(\(S\_+\)[2]\^*\)] + 9\ Re[\(E\_+\)[0]\ \(\(S\_+\)[2]\^*\)] - 27\ Re[\(E\_+\)[2]\ \(\(S\_+\)[2]\^*\)] - 189\ Re[\(E\_+\)[4]\ \(\(S\_+\)[2]\^*\)] - 243\ Re[\(E\_+\)[6]\ \(\(S\_+\)[2]\^*\)] - 27\ Re[\(M\_-\)[2]\ \(\(S\_+\)[2]\^*\)] - 54\ Re[\(M\_-\)[4]\ \(\(S\_+\)[2]\^*\)] - 54\ Re[\(M\_-\)[6]\ \(\(S\_+\)[2]\^*\)] + 27\ Re[\(M\_+\)[2]\ \(\(S\_+\)[2]\^*\)] + 54\ Re[\(M\_+\)[4]\ \(\(S\_+\)[2]\^*\)] + 54\ Re[\(M\_+\)[6]\ \(\(S\_+\)[2]\^*\)] + 36\ Re[\(E\_-\)[3]\ \(\(S\_+\)[3]\^*\)] + 120\ Re[\(E\_-\)[5]\ \(\(S\_+\)[3]\^*\)] + 36\ Re[\(E\_+\)[1]\ \(\(S\_+\)[3]\^*\)] - 72\ Re[\(E\_+\)[3]\ \(\(S\_+\)[3]\^*\)] - 408\ Re[\(E\_+\)[5]\ \(\(S\_+\)[3]\^*\)] - 12\ Re[\(M\_-\)[1]\ \(\(S\_+\)[3]\^*\)] - 72\ Re[\(M\_-\)[3]\ \(\(S\_+\)[3]\^*\)] - 120\ Re[\(M\_-\)[5]\ \(\(S\_+\)[3]\^*\)] + 12\ Re[\(M\_+\)[1]\ \(\(S\_+\)[3]\^*\)] + 72\ Re[\(M\_+\)[3]\ \(\(S\_+\)[3]\^*\)] + 120\ Re[\(M\_+\)[5]\ \(\(S\_+\)[3]\^*\)] + 15\ Re[\(E\_-\)[2]\ \(\(S\_+\)[4]\^*\)] + 90\ Re[\(E\_-\)[4]\ \(\(S\_+\)[4]\^*\)] + 225\ Re[\(E\_-\)[6]\ \(\(S\_+\)[4]\^*\)] + 15\ Re[\(E\_+\)[0]\ \(\(S\_+\)[4]\^*\)] + 90\ Re[\(E\_+\)[2]\ \(\(S\_+\)[4]\^*\)] - 150\ Re[\(E\_+\)[4]\ \(\(S\_+\)[4]\^*\)] - 750\ Re[\(E\_+\)[6]\ \(\(S\_+\)[4]\^*\)] - 45\ Re[\(M\_-\)[2]\ \(\(S\_+\)[4]\^*\)] - 150\ Re[\(M\_-\)[4]\ \(\(S\_+\)[4]\^*\)] - 225\ Re[\(M\_-\)[6]\ \(\(S\_+\)[4]\^*\)] + 45\ Re[\(M\_+\)[2]\ \(\(S\_+\)[4]\^*\)] + 150\ Re[\(M\_+\)[4]\ \(\(S\_+\)[4]\^*\)] + 225\ Re[\(M\_+\)[6]\ \(\(S\_+\)[4]\^*\)] + 54\ Re[\(E\_-\)[3]\ \(\(S\_+\)[5]\^*\)] + 180\ Re[\(E\_-\)[5]\ \(\(S\_+\)[5]\^*\)] + 54\ Re[\(E\_+\)[1]\ \(\(S\_+\)[5]\^*\)] + 180\ Re[\(E\_+\)[3]\ \(\(S\_+\)[5]\^*\)] - 270\ Re[\(E\_+\)[5]\ \(\(S\_+\)[5]\^*\)] - 18\ Re[\(M\_-\)[1]\ \(\(S\_+\)[5]\^*\)] - 108\ Re[\(M\_-\)[3]\ \(\(S\_+\)[5]\^*\)] - 270\ Re[\(M\_-\)[5]\ \(\(S\_+\)[5]\^*\)] + 18\ Re[\(M\_+\)[1]\ \(\(S\_+\)[5]\^*\)] + 108\ Re[\(M\_+\)[3]\ \(\(S\_+\)[5]\^*\)] + 270\ Re[\(M\_+\)[5]\ \(\(S\_+\)[5]\^*\)] + 21\ Re[\(E\_-\)[2]\ \(\(S\_+\)[6]\^*\)] + 126\ Re[\(E\_-\)[4]\ \(\(S\_+\)[6]\^*\)] + 315\ Re[\(E\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 21\ Re[\(E\_+\)[0]\ \(\(S\_+\)[6]\^*\)] + 126\ Re[\(E\_+\)[2]\ \(\(S\_+\)[6]\^*\)] + 315\ Re[\(E\_+\)[4]\ \(\(S\_+\)[6]\^*\)] - 441\ Re[\(E\_+\)[6]\ \(\(S\_+\)[6]\^*\)] - 63\ Re[\(M\_-\)[2]\ \(\(S\_+\)[6]\^*\)] - 210\ Re[\(M\_-\)[4]\ \(\(S\_+\)[6]\^*\)] - 441\ Re[\(M\_-\)[6]\ \(\(S\_+\)[6]\^*\)] + 63\ Re[\(M\_+\)[2]\ \(\(S\_+\)[6]\^*\)] + 210\ Re[\(M\_+\)[4]\ \(\(S\_+\)[6]\^*\)] + 441\ Re[\(M\_+\)[6]\ \(\(S\_+\)[6]\^*\)]\)], "Output", CellLabel->"Out[73]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A\_TT0 //. recombine\)], "Input", CellLabel->"In[74]:="], Cell[BoxData[ \(3\/2\ Abs[\(E\_-\)[2]]\^2 + 12\ Abs[\(E\_-\)[3]]\^2 + 45\ Abs[\(E\_-\)[4]]\^2 + 120\ Abs[\(E\_-\)[5]]\^2 + 525\/2\ Abs[\(E\_-\)[6]]\^2 + 9\/2\ Abs[\(E\_+\)[1]]\^2 + 24\ Abs[\(E\_+\)[2]]\^2 + 75\ Abs[\(E\_+\)[3]]\^2 + 180\ Abs[\(E\_+\)[4]]\^2 + 735\/2\ Abs[\(E\_+\)[5]]\^2 + 672\ Abs[\(E\_+\)[6]]\^2 - 9\/2\ Abs[\(M\_-\)[2]]\^2 - 24\ Abs[\(M\_-\)[3]]\^2 - 75\ Abs[\(M\_-\)[4]]\^2 - 180\ Abs[\(M\_-\)[5]]\^2 - 735\/2\ Abs[\(M\_-\)[6]]\^2 - 3\/2\ Abs[\(M\_+\)[1]]\^2 - 12\ Abs[\(M\_+\)[2]]\^2 - 45\ Abs[\(M\_+\)[3]]\^2 - 120\ Abs[\(M\_+\)[4]]\^2 - 525\/2\ Abs[\(M\_+\)[5]]\^2 - 504\ Abs[\(M\_+\)[6]]\^2 - 3\ Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)] - 12\ Re[\(E\_+\)[2]\ \(\(E\_-\)[2]\^*\)] - 25\ Re[\(E\_+\)[4]\ \(\(E\_-\)[2]\^*\)] - 42\ Re[\(E\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 3\ Re[\(M\_-\)[2]\ \(\(E\_-\)[2]\^*\)] - 10\ Re[\(M\_-\)[4]\ \(\(E\_-\)[2]\^*\)] - 21\ Re[\(M\_-\)[6]\ \(\(E\_-\)[2]\^*\)] + 3\ Re[\(M\_+\)[2]\ \(\(E\_-\)[2]\^*\)] + 10\ Re[\(M\_+\)[4]\ \(\(E\_-\)[2]\^*\)] + 21\ Re[\(M\_+\)[6]\ \(\(E\_-\)[2]\^*\)] - 21\ Re[\(E\_+\)[1]\ \(\(E\_-\)[3]\^*\)] - 60\ Re[\(E\_+\)[3]\ \(\(E\_-\)[3]\^*\)] - 111\ Re[\(E\_+\)[5]\ \(\(E\_-\)[3]\^*\)] + 3\ Re[\(M\_-\)[1]\ \(\(E\_-\)[3]\^*\)] - 12\ Re[\(M\_-\)[3]\ \(\(E\_-\)[3]\^*\)] - 39\ Re[\(M\_-\)[5]\ \(\(E\_-\)[3]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_-\)[3]\^*\)] + 12\ Re[\(M\_+\)[3]\ \(\(E\_-\)[3]\^*\)] + 39\ Re[\(M\_+\)[5]\ \(\(E\_-\)[3]\^*\)] + 2\ Re[\(E\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 10\ Re[\(E\_+\)[0]\ \(\(E\_-\)[4]\^*\)] - 78\ Re[\(E\_+\)[2]\ \(\(E\_-\)[4]\^*\)] - 180\ Re[\(E\_+\)[4]\ \(\(E\_-\)[4]\^*\)] - 306\ Re[\(E\_+\)[6]\ \(\(E\_-\)[4]\^*\)] + 12\ Re[\(M\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 30\ Re[\(M\_-\)[4]\ \(\(E\_-\)[4]\^*\)] - 96\ Re[\(M\_-\)[6]\ \(\(E\_-\)[4]\^*\)] - 12\ Re[\(M\_+\)[2]\ \(\(E\_-\)[4]\^*\)] + 30\ Re[\(M\_+\)[4]\ \(\(E\_-\)[4]\^*\)] + 96\ Re[\(M\_+\)[6]\ \(\(E\_-\)[4]\^*\)] + 21\ Re[\(E\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 54\ Re[\(E\_+\)[1]\ \(\(E\_-\)[5]\^*\)] - 210\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 420\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 10\ Re[\(M\_-\)[1]\ \(\(E\_-\)[5]\^*\)] + 30\ Re[\(M\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 60\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] - 10\ Re[\(M\_+\)[1]\ \(\(E\_-\)[5]\^*\)] - 30\ Re[\(M\_+\)[3]\ \(\(E\_-\)[5]\^*\)] + 60\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 3\ Re[\(E\_-\)[2]\ \(\(E\_-\)[6]\^*\)] + 84\ Re[\(E\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 21\ Re[\(E\_+\)[0]\ \(\(E\_-\)[6]\^*\)] - 168\ Re[\(E\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 465\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 840\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] + 39\ Re[\(M\_-\)[2]\ \(\(E\_-\)[6]\^*\)] + 60\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 105\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] - 39\ Re[\(M\_+\)[2]\ \(\(E\_-\)[6]\^*\)] - 60\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 105\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 3\ Re[\(M\_-\)[2]\ \(\(E\_+\)[0]\^*\)] - 10\ Re[\(M\_-\)[4]\ \(\(E\_+\)[0]\^*\)] - 21\ Re[\(M\_-\)[6]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] + 10\ Re[\(M\_+\)[4]\ \(\(E\_+\)[0]\^*\)] + 21\ Re[\(M\_+\)[6]\ \(\(E\_+\)[0]\^*\)] + 3\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 12\ Re[\(M\_-\)[3]\ \(\(E\_+\)[1]\^*\)] - 39\ Re[\(M\_-\)[5]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 12\ Re[\(M\_+\)[3]\ \(\(E\_+\)[1]\^*\)] + 39\ Re[\(M\_+\)[5]\ \(\(E\_+\)[1]\^*\)] - 3\ Re[\(E\_+\)[0]\ \(\(E\_+\)[2]\^*\)] + 12\ Re[\(M\_-\)[2]\ \(\(E\_+\)[2]\^*\)] - 30\ Re[\(M\_-\)[4]\ \(\(E\_+\)[2]\^*\)] - 96\ Re[\(M\_-\)[6]\ \(\(E\_+\)[2]\^*\)] - 12\ Re[\(M\_+\)[2]\ \(\(E\_+\)[2]\^*\)] + 30\ Re[\(M\_+\)[4]\ \(\(E\_+\)[2]\^*\)] + 96\ Re[\(M\_+\)[6]\ \(\(E\_+\)[2]\^*\)] + 10\ Re[\(M\_-\)[1]\ \(\(E\_+\)[3]\^*\)] + 30\ Re[\(M\_-\)[3]\ \(\(E\_+\)[3]\^*\)] - 60\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] - 10\ Re[\(M\_+\)[1]\ \(\(E\_+\)[3]\^*\)] - 30\ Re[\(M\_+\)[3]\ \(\(E\_+\)[3]\^*\)] + 60\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] - 10\ Re[\(E\_+\)[0]\ \(\(E\_+\)[4]\^*\)] + 30\ Re[\(E\_+\)[2]\ \(\(E\_+\)[4]\^*\)] + 39\ Re[\(M\_-\)[2]\ \(\(E\_+\)[4]\^*\)] + 60\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] - 105\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] - 39\ Re[\(M\_+\)[2]\ \(\(E\_+\)[4]\^*\)] - 60\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] + 105\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] - 21\ Re[\(E\_+\)[1]\ \(\(E\_+\)[5]\^*\)] + 120\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] + 21\ Re[\(M\_-\)[1]\ \(\(E\_+\)[5]\^*\)] + 96\ Re[\(M\_-\)[3]\ \(\(E\_+\)[5]\^*\)] + 105\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] - 21\ Re[\(M\_+\)[1]\ \(\(E\_+\)[5]\^*\)] - 96\ Re[\(M\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 105\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] - 21\ Re[\(E\_+\)[0]\ \(\(E\_+\)[6]\^*\)] - 12\ Re[\(E\_+\)[2]\ \(\(E\_+\)[6]\^*\)] + 315\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] + 78\ Re[\(M\_-\)[2]\ \(\(E\_+\)[6]\^*\)] + 190\ Re[\(M\_-\)[4]\ \(\(E\_+\)[6]\^*\)] + 168\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] - 78\ Re[\(M\_+\)[2]\ \(\(E\_+\)[6]\^*\)] - 190\ Re[\(M\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 168\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] - 3\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] - 10\ Re[\(M\_+\)[3]\ \(\(M\_-\)[1]\^*\)] - 21\ Re[\(M\_+\)[5]\ \(\(M\_-\)[1]\^*\)] - 21\ Re[\(M\_+\)[2]\ \(\(M\_-\)[2]\^*\)] - 54\ Re[\(M\_+\)[4]\ \(\(M\_-\)[2]\^*\)] - 99\ Re[\(M\_+\)[6]\ \(\(M\_-\)[2]\^*\)] + 3\ Re[\(M\_-\)[1]\ \(\(M\_-\)[3]\^*\)] - 12\ Re[\(M\_+\)[1]\ \(\(M\_-\)[3]\^*\)] - 78\ Re[\(M\_+\)[3]\ \(\(M\_-\)[3]\^*\)] - 168\ Re[\(M\_+\)[5]\ \(\(M\_-\)[3]\^*\)] - 60\ Re[\(M\_+\)[2]\ \(\(M\_-\)[4]\^*\)] - 210\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] - 400\ Re[\(M\_+\)[6]\ \(\(M\_-\)[4]\^*\)] + 10\ Re[\(M\_-\)[1]\ \(\(M\_-\)[5]\^*\)] - 30\ Re[\(M\_-\)[3]\ \(\(M\_-\)[5]\^*\)] - 25\ Re[\(M\_+\)[1]\ \(\(M\_-\)[5]\^*\)] - 180\ Re[\(M\_+\)[3]\ \(\(M\_-\)[5]\^*\)] - 465\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] + 21\ Re[\(M\_-\)[2]\ \(\(M\_-\)[6]\^*\)] - 120\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] - 111\ Re[\(M\_+\)[2]\ \(\(M\_-\)[6]\^*\)] - 420\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] - 903\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_+\)[3]\^*\)] - 21\ Re[\(M\_+\)[2]\ \(\(M\_+\)[4]\^*\)] + 3\ Re[\(M\_+\)[1]\ \(\(M\_+\)[5]\^*\)] - 84\ Re[\(M\_+\)[3]\ \(\(M\_+\)[5]\^*\)] - 6\ Re[\(M\_+\)[2]\ \(\(M\_+\)[6]\^*\)] - 230\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)]\)], "Output", CellLabel->"Out[74]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A\_L2 //. recombine\)], "Input", CellLabel->"In[75]:="], Cell[BoxData[ \(8\ Abs[\(S\_-\)[2]]\^2 + 216\/7\ Abs[\(S\_-\)[3]]\^2 + 1600\/21\ Abs[\(S\_-\)[4]]\^2 + 5000\/33\ Abs[\(S\_-\)[5]]\^2 + 37800\/143\ Abs[\(S\_-\)[6]]\^2 + 8\ Abs[\(S\_+\)[1]]\^2 + 216\/7\ Abs[\(S\_+\)[2]]\^2 + 1600\/21\ Abs[\(S\_+\)[3]]\^2 + 5000\/33\ Abs[\(S\_+\)[4]]\^2 + 37800\/143\ Abs[\(S\_+\)[5]]\^2 + 5488\/13\ Abs[\(S\_+\)[6]]\^2 + 18\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_-\)[1]] + 576\/7\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_-\)[2]] + 1500\/7\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_-\)[3]] + 4800\/11\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_-\)[4]] + 8\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[0]] + 18\ Re[\(\(S\_+\)[2]\^*\)\ \(S\_+\)[0]] + 8\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]] + 72\/7\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[1]] + 576\/7\ Re[\(\(S\_+\)[3]\^*\)\ \(S\_+\)[1]] + 72\/7\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[2]] + 96\/7\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[2]] + 1500\/7\ Re[\(\(S\_+\)[4]\^*\)\ \(S\_+\)[2]] + 96\/7\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_+\)[3]] + 4000\/231\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[3]] + 4800\/11\ Re[\(\(S\_+\)[5]\^*\)\ \(S\_+\)[3]] + 4000\/231\ Re[\(\(S\_-\)[4]\^*\)\ \(S\_+\)[4]] + 3000\/143\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[4]] + 110250\/143\ Re[\(\(S\_+\)[6]\^*\)\ \(S\_+\)[4]] + 3000\/143\ Re[\(\(S\_-\)[5]\^*\)\ \(S\_+\)[5]] + 3528\/143\ Re[\(\(S\_-\)[6]\^*\)\ \(S\_+\)[6]]\)], "Output", CellLabel->"Out[75]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(A\_T2 //. recombine\)], "Input", CellLabel->"In[76]:="], Cell[BoxData[ \(\(-Abs[\(E\_-\)[2]]\^2\) + 36\/7\ Abs[\(E\_-\)[3]]\^2 + 150\/7\ Abs[\(E\_-\)[4]]\^2 + 1700\/33\ Abs[\(E\_-\)[5]]\^2 + 14175\/143\ Abs[\(E\_-\)[6]]\^2 + 3\ Abs[\(E\_+\)[1]]\^2 + 108\/7\ Abs[\(E\_+\)[2]]\^2 + 850\/21\ Abs[\(E\_+\)[3]]\^2 + 900\/11\ Abs[\(E\_+\)[4]]\^2 + 1575\/11\ Abs[\(E\_+\)[5]]\^2 + 2968\/13\ Abs[\(E\_+\)[6]]\^2 + 3\ Abs[\(M\_-\)[2]]\^2 + 108\/7\ Abs[\(M\_-\)[3]]\^2 + 850\/21\ Abs[\(M\_-\)[4]]\^2 + 900\/11\ Abs[\(M\_-\)[5]]\^2 + 1575\/11\ Abs[\(M\_-\)[6]]\^2 - Abs[\(M\_+\)[1]]\^2 + 36\/7\ Abs[\(M\_+\)[2]]\^2 + 150\/7\ Abs[\(M\_+\)[3]]\^2 + 1700\/33\ Abs[\(M\_+\)[4]]\^2 + 14175\/143\ Abs[\(M\_+\)[5]]\^2 + 168\ Abs[\(M\_+\)[6]]\^2 + 2\ Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)] - 24\/7\ Re[\(E\_+\)[2]\ \(\(E\_-\)[2]\^*\)] - 6\ Re[\(M\_-\)[2]\ \(\(E\_-\)[2]\^*\)] + 6\ Re[\(M\_+\)[2]\ \(\(E\_-\)[2]\^*\)] - 18\/7\ Re[\(E\_+\)[1]\ \(\(E\_-\)[3]\^*\)] - 40\/7\ Re[\(E\_+\)[3]\ \(\(E\_-\)[3]\^*\)] - 6\ Re[\(M\_-\)[1]\ \(\(E\_-\)[3]\^*\)] - 72\/7\ Re[\(M\_-\)[3]\ \(\(E\_-\)[3]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_-\)[3]\^*\)] + 72\/7\ Re[\(M\_+\)[3]\ \(\(E\_-\)[3]\^*\)] + 144\/7\ Re[\(E\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 36\/7\ Re[\(E\_+\)[2]\ \(\(E\_-\)[4]\^*\)] - 600\/77\ Re[\(E\_+\)[4]\ \(\(E\_-\)[4]\^*\)] - 72\/7\ Re[\(M\_-\)[2]\ \(\(E\_-\)[4]\^*\)] - 100\/7\ Re[\(M\_-\)[4]\ \(\(E\_-\)[4]\^*\)] + 72\/7\ Re[\(M\_+\)[2]\ \(\(E\_-\)[4]\^*\)] + 100\/7\ Re[\(M\_+\)[4]\ \(\(E\_-\)[4]\^*\)] + 500\/7\ Re[\(E\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 1700\/231\ Re[\(E\_+\)[3]\ \(\(E\_-\)[5]\^*\)] - 1400\/143\ Re[\(E\_+\)[5]\ \(\(E\_-\)[5]\^*\)] - 100\/7\ Re[\(M\_-\)[3]\ \(\(E\_-\)[5]\^*\)] - 200\/11\ Re[\(M\_-\)[5]\ \(\(E\_-\)[5]\^*\)] + 100\/7\ Re[\(M\_+\)[3]\ \(\(E\_-\)[5]\^*\)] + 200\/11\ Re[\(M\_+\)[5]\ \(\(E\_-\)[5]\^*\)] + 1800\/11\ Re[\(E\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 1350\/143\ Re[\(E\_+\)[4]\ \(\(E\_-\)[6]\^*\)] - 1680\/143\ Re[\(E\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 200\/11\ Re[\(M\_-\)[4]\ \(\(E\_-\)[6]\^*\)] - 3150\/143\ Re[\(M\_-\)[6]\ \(\(E\_-\)[6]\^*\)] + 200\/11\ Re[\(M\_+\)[4]\ \(\(E\_-\)[6]\^*\)] + 3150\/143\ Re[\(M\_+\)[6]\ \(\(E\_-\)[6]\^*\)] - 6\ Re[\(M\_-\)[2]\ \(\(E\_+\)[0]\^*\)] + 6\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)] - 6\ Re[\(M\_-\)[1]\ \(\(E\_+\)[1]\^*\)] - 72\/7\ Re[\(M\_-\)[3]\ \(\(E\_+\)[1]\^*\)] + 6\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)] + 72\/7\ Re[\(M\_+\)[3]\ \(\(E\_+\)[1]\^*\)] + 12\ Re[\(E\_+\)[0]\ \(\(E\_+\)[2]\^*\)] - 72\/7\ Re[\(M\_-\)[2]\ \(\(E\_+\)[2]\^*\)] - 100\/7\ Re[\(M\_-\)[4]\ \(\(E\_+\)[2]\^*\)] + 72\/7\ Re[\(M\_+\)[2]\ \(\(E\_+\)[2]\^*\)] + 100\/7\ Re[\(M\_+\)[4]\ \(\(E\_+\)[2]\^*\)] + 360\/7\ Re[\(E\_+\)[1]\ \(\(E\_+\)[3]\^*\)] - 100\/7\ Re[\(M\_-\)[3]\ \(\(E\_+\)[3]\^*\)] - 200\/11\ Re[\(M\_-\)[5]\ \(\(E\_+\)[3]\^*\)] + 100\/7\ Re[\(M\_+\)[3]\ \(\(E\_+\)[3]\^*\)] + 200\/11\ Re[\(M\_+\)[5]\ \(\(E\_+\)[3]\^*\)] + 900\/7\ Re[\(E\_+\)[2]\ \(\(E\_+\)[4]\^*\)] - 200\/11\ Re[\(M\_-\)[4]\ \(\(E\_+\)[4]\^*\)] - 3150\/143\ Re[\(M\_-\)[6]\ \(\(E\_+\)[4]\^*\)] + 200\/11\ Re[\(M\_+\)[4]\ \(\(E\_+\)[4]\^*\)] + 3150\/143\ Re[\(M\_+\)[6]\ \(\(E\_+\)[4]\^*\)] + 2800\/11\ Re[\(E\_+\)[3]\ \(\(E\_+\)[5]\^*\)] - 3150\/143\ Re[\(M\_-\)[5]\ \(\(E\_+\)[5]\^*\)] + 3150\/143\ Re[\(M\_+\)[5]\ \(\(E\_+\)[5]\^*\)] + 63000\/143\ Re[\(E\_+\)[4]\ \(\(E\_+\)[6]\^*\)] - 336\/13\ Re[\(M\_-\)[6]\ \(\(E\_+\)[6]\^*\)] + 336\/13\ Re[\(M\_+\)[6]\ \(\(E\_+\)[6]\^*\)] - 2\ Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)] + 18\/7\ Re[\(M\_+\)[2]\ \(\(M\_-\)[2]\^*\)] + 12\ Re[\(M\_-\)[1]\ \(\(M\_-\)[3]\^*\)] + 24\/7\ Re[\(M\_+\)[1]\ \(\(M\_-\)[3]\^*\)] + 36\/7\ Re[\(M\_+\)[3]\ \(\(M\_-\)[3]\^*\)] + 360\/7\ Re[\(M\_-\)[2]\ \(\(M\_-\)[4]\^*\)] + 40\/7\ Re[\(M\_+\)[2]\ \(\(M\_-\)[4]\^*\)] + 1700\/231\ Re[\(M\_+\)[4]\ \(\(M\_-\)[4]\^*\)] + 900\/7\ Re[\(M\_-\)[3]\ \(\(M\_-\)[5]\^*\)] + 600\/77\ Re[\(M\_+\)[3]\ \(\(M\_-\)[5]\^*\)] + 1350\/143\ Re[\(M\_+\)[5]\ \(\(M\_-\)[5]\^*\)] + 2800\/11\ Re[\(M\_-\)[4]\ \(\(M\_-\)[6]\^*\)] + 1400\/143\ Re[\(M\_+\)[4]\ \(\(M\_-\)[6]\^*\)] + 126\/11\ Re[\(M\_+\)[6]\ \(\(M\_-\)[6]\^*\)] + 144\/7\ Re[\(M\_+\)[1]\ \(\(M\_+\)[3]\^*\)] + 500\/7\ Re[\(M\_+\)[2]\ \(\(M\_+\)[4]\^*\)] + 1800\/11\ Re[\(M\_+\)[3]\ \(\(M\_+\)[5]\^*\)] + 44100\/143\ Re[\(M\_+\)[4]\ \(\(M\_+\)[6]\^*\)]\)], "Output", CellLabel->"Out[76]="] }, Open ]], Cell[TextData[{ "Therefore, it becomes a detailed quantitative issue whether or not higher \ multipoles decline rapidly enough to obtain rapid convergence. This is \ especially important for ", Cell[BoxData[ \(TraditionalForm\`A\_0\%T\)]], " and ", Cell[BoxData[ \(TraditionalForm\`A\_0\%L\)]], " because all contributions are positive and there is no possible benefit \ from fortuitous cancellations. The table below evaluates the convergence of \ each term in ", Cell[BoxData[ \(TraditionalForm\`R\&~\_EM\)]], " and ", Cell[BoxData[ \(TraditionalForm\`R\&~\_SM\)]], " separately. Notice that terms with ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL] = 2\)]], " produce a ", Cell[BoxData[ \(TraditionalForm\`6 %\)]], " change in ", Cell[BoxData[ \(TraditionalForm\`A\_1\%LT\)]], " and hence in ", Cell[BoxData[ \(TraditionalForm\`R\&~\_SM\)]], ". Even terms with ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL] = 3\)]], " make a perceptible contribution. Similarly, the longitudinal \ contribution to the denominator of ", Cell[BoxData[ \(TraditionalForm\`R\&~\_SM\)]], ", namely ", Cell[BoxData[ \(TraditionalForm\`A\_0\%L\)]], ", results in another ", Cell[BoxData[ \(TraditionalForm\`4 %\)]], " reduction when ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon] \[Tilde] 1\)]], " even at the level of ", StyleBox["sp", FontSlant->"Italic"], " truncation due to errors in ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance and increases with ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL]\)]], ". Consequently, the relative accuracy of ", Cell[BoxData[ \(TraditionalForm\`R\&~\_SM\)]], " is no better than ", Cell[BoxData[ \(TraditionalForm\`10 %\)]], ". The accuracy of ", Cell[BoxData[ \(TraditionalForm\`R\&~\_EM\)]], " is even worse because the cancellation between ", Cell[BoxData[ \(TraditionalForm\`A\_2\%T\)]], " and ", Cell[BoxData[ \(TraditionalForm\`A\_0\%TT\)]], " amplifies numerical errors." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[ Table[\(\((\(truncate[#, \[ScriptL]] &\) /@ {A\_L0, A\_T0, A\_LT1, A\_TT0, A\_L2, A\_T2})\) /. values\) /. \(x_\^*\) \[Rule] Conjugate[x], {\[ScriptL], 1, 5}], TableHeadings \[Rule] {Range[ 5], {\*"\"\<\!\(A\_0\%L\)\>\"", \*"\"\<\!\(A\_0\%T\)\>\"", \ \*"\"\<\!\(A\_1\%LT\)\>\"", \*"\"\<\!\(A\_0\%TT\)\>\"", \ \*"\"\<\!\(A\_2\%L\)\>\"", \*"\"\<\!\(A\_2\%T\)\>\""}}]\)], "Input", CellLabel->"In[77]:="], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"\\!\\(A\\_0\\%L\\)\"\>", \ "\<\"\\!\\(A\\_0\\%T\\)\"\>", "\<\"\\!\\(A\\_1\\%LT\\)\"\>", "\<\"\\!\\(A\\_0\ \\%TT\\)\"\>", "\<\"\\!\\(A\\_2\\%L\\)\"\>", "\<\"\\!\\(A\\_2\\%T\\)\"\>"}, {"1", "0.33386311970000004`", "7.599425748051`", \(-1.4832802817999997`\), \ \(-5.2200007928774985`\), "0.14081567519999996`", \(-3.7689656035450003`\)}, {"2", "0.3377167057471504`", "7.623495410532264`", \(-1.3952663511906214`\), \ \(-5.155617482291534`\), "0.10171094508989116`", \(-3.916240630383367`\)}, {"3", "0.3383590259400426`", "7.627503273800551`", \(-1.3227483145018906`\), \ \(-5.137301935418979`\), "0.07893698959333294`", \(-3.87126187885316`\)}, {"4", "0.3384293328452926`", "7.627950010260901`", \(-1.3148495528812902`\), \ \(-5.130997545316128`\), "0.08086684095734158`", \(-3.8592917380060148`\)}, {"5", "0.338438855107419`", "7.628012868193199`", \(-1.3078640329782893`\), \ \(-5.129829046002605`\), "0.08116514378401364`", \(-3.8573870397952343`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{1, 2, 3, 4, 5}, {"\!\(A\_0\%L\)", "\!\(A\_0\%T\)", "\!\(A\_1\%LT\)", "\!\(A\_0\%TT\)", "\!\(A\_2\%L\)", "\!\(A\_2\%T\)"}}]]]], "Output", CellLabel->"Out[77]//TableForm="] }, Open ]], Cell["\<\ \tNotice that elimination of the longitudinal contribution by means of \ Rosenbluth separation, such that\ \>", "Text"], Cell[BoxData[{ \(R\&~\_EM\ \[LongRightArrow]\ \(\(\(\ \)\(3 A\_2\%T - 2 A\_0\%TT\)\)\/\(12 A\_0\%T\)\)\), "\[IndentingNewLine]", \(R\&~\_SM\ \[LongRightArrow]\ \(\(\(\ \)\(A\_1\%LT\)\)\/\(3 A\_0\%T\)\)\)}], "DisplayFormula", FontFamily->"Times New Roman"], Cell[TextData[{ "can be simulated in the present analysis by taking ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon] \[Rule] 0\)]], ". This improves the accuracy of the extracted quadrupole ratios, as shown \ in the table below, " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0. , values], SMR[\[ScriptL], 0. , values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[78]:="], Cell[BoxData[ TagBox[GridBox[{ {"0.5810808918974375`", "0.9773446303296159`"}, {"0.9605069687671194`", "0.9164489149422888`"}, {"0.8943507681278122`", "0.8683605757009851`"}, {"0.8787376782336803`", "0.8631246276933986`"}, {"0.876475249088706`", "0.8585319528636834`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[78]//TableForm="] }, Open ]], Cell[TextData[{ "but results with approximately ", Cell[BoxData[ \(TraditionalForm\`15 %\)]], " relative errors still remain unsatisfactory. If ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance were valid, one would obtain much faster convergence" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[M1, \[ScriptL], 0. , values], SMR[M1, \[ScriptL], 0. , values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[79]:="], Cell[BoxData[ TagBox[GridBox[{ {"1.`", "1.0000000000000002`"}, {"1.`", "1.0000000000000002`"}, {"0.9986044562106383`", "0.9994221902738474`"}, {"0.9986044562106383`", "0.9994221902738474`"}, {"0.9986448328203202`", "0.9994377053440084`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[79]//TableForm="] }, Open ]], Cell[TextData[{ "and more accurate results, but in none of the models considered does ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance apply at better than the ", Cell[BoxData[ \(TraditionalForm\`10 %\)]], " level. " }], "Text"], Cell[TextData[{ "\tIt is also instructive to examine the relative sizes of the various \ contributions to both the numerators and the common denominator of these \ formulas. Thus, we find that the most important neglected term is ", Cell[BoxData[ \(TraditionalForm\`Re[\(M\_\(\(1\)\(-\)\)\) \ \(E\_\(\(1\)\(+\)\)\^*\)]\)]], ". That term alone would explain an error of approximately ", Cell[BoxData[ \(TraditionalForm\`30 %\)]], ". However, the numerical values of neglected terms do not decrease \ especially rapidly and there are very important cancellations. For example, \ the third term involves ", Cell[BoxData[ \(TraditionalForm\`E\_\(\(2\)\(-\)\)\)]], " but is cancelled almost complete by ", Cell[BoxData[ \(TraditionalForm\`\(\(\[VerticalSeparator]\)\(S\_\(\(1\)\(+\)\)\)\( \ \[VerticalSeparator] \^2\)\)\)]], " when ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon] \[Rule] 1\)]], ". Thus, final value depends upon ", Cell[BoxData[ \(TraditionalForm\`\[Epsilon]\)]], ", contrary to the original assumptions, and depends upon delicate \ cancellations. Therefore, one cannot expect good accuracy from this method." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(EMRterms = arrange[3 \((A\_T2 + \[Epsilon]\ A\_L2)\) - 2 A\_TT0, values, 0.95];\)\), "\[IndentingNewLine]", \(Take[EMRterms, 10]\)}], "Input", CellLabel->"In[80]:="], Cell[BoxData[ \({24\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)], \(-24\)\ Re[\(M\_-\)[ 1]\ \(\(E\_+\)[1]\^*\)], 12\ Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)], 24\ \[Epsilon]\ Abs[\(S\_+\)[1]]\^2, 24\ \[Epsilon]\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[ 0]], \(-24\)\ Re[\(M\_-\)[1]\ \(\(E\_-\)[3]\^*\)], 12\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)], 20\ Re[\(M\_+\)[3]\ \(\(M\_-\)[1]\^*\)], \(-12\)\ Re[\(M\_-\)[ 2]\ \(\(E\_+\)[0]\^*\)], 54\ \[Epsilon]\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_-\)[1]]}\)], "Output", CellLabel->"Out[81]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Take[\(\(EMRterms /. values\) /. \[Epsilon] \[Rule] 0.95\) /. \(x_\^*\) \[Rule] Conjugate[x], 10]\)], "Input", CellLabel->"In[82]:="], Cell[BoxData[ \({\(-1.409939622`\), 0.5430443971200001`, \(-0.3923625744`\), 0.3710579707199999`, \(-0.17103908575727997`\), 0.16183058736648`, \(-0.138099672`\), \(-0.0937177886274`\), \ \(-0.08889362640000001`\), \(-0.0658147048308`\)}\)], "Output", CellLabel->"Out[82]="] }, Open ]], Cell[TextData[{ "\tSimilarly, the most important neglected term in the numerator of ", Cell[BoxData[ \(TraditionalForm\`R\&~\_SM\)]], " is ", Cell[BoxData[ \(TraditionalForm\`Re[\(E\_\(\(2\)\(-\)\)\) \ \(S\_\(\(0\)\(+\)\)\^*\)]\)]], ", but its magnitude relative to the leading term is smaller than found for \ ", Cell[BoxData[ \(TraditionalForm\`R\&~\_EM\)]], ". However, convergence is far from monotonic because there are many other \ contributions of similar size and variable sign. Nevertheless, one expects \ the convergence of ", Cell[BoxData[ \(TraditionalForm\`R\&~\_SM\)]], " to be superior to that of ", Cell[BoxData[ \(TraditionalForm\`R\&~\_EM\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(ALT1terms = arrange[A\_LT1, values, 0.95];\)\), "\[IndentingNewLine]", \(Take[ALT1terms, 10]\)}], "Input", CellLabel->"In[83]:="], Cell[BoxData[ \({6\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)], 3\ Re[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)], \(-6\)\ Re[\(E\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)], \(-6\)\ Re[\(E\_+\)[ 0]\ \(\(S\_-\)[2]\^*\)], \(-6\)\ Re[\(M\_-\)[ 1]\ \(\(S\_+\)[1]\^*\)], 9\ Re[\(E\_-\)[3]\ \(\(S\_-\)[1]\^*\)], 3\ Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)], \(-6\)\ Re[\(E\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)], 9\ Re[\(M\_-\)[1]\ \(\(S\_-\)[3]\^*\)], \(-3\)\ Re[\(M\_-\)[ 2]\ \(\(S\_+\)[0]\^*\)]}\)], "Output", CellLabel->"Out[84]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(ALT1values = \(\(ALT1terms /. values\) /. \[Epsilon] \[Rule] 0.95\) /. \(x_\^*\) \[Rule] Conjugate[x];\)\), "\[IndentingNewLine]", \(Take[ALT1values, 10]\)}], "Input", CellLabel->"In[85]:="], Cell[BoxData[ \({\(-1.4343197916`\), 0.08786795616000001`, \(-0.06940756079999999`\), \(-0.049916038716`\), 0.04929969060000001`, 0.033101252052299995`, 0.031352274`, \(-0.028852619999999995`\), 0.0211448116404`, 0.019061420760000003`}\)], "Output", CellLabel->"Out[86]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FoldList[Plus, 0, ALT1values] // Rest\)], "Input", CellLabel->"In[87]:="], Cell[BoxData[ \({\(-1.4343197916`\), \(-1.34645183544`\), \(-1.41585939624`\), \ \(-1.465775434956`\), \(-1.4164757443559999`\), \(-1.3833744923037`\), \ \(-1.3520222183036998`\), \(-1.3808748383036997`\), \(-1.3597300266632997`\), \ \(-1.3406686059032997`\), \(-1.3513819657957797`\), \(-1.3412748721957797`\), \ \(-1.3509096171037798`\), \(-1.3414887000469797`\), \(-1.3353342409909796`\), \ \(-1.3401752617909797`\), \(-1.3353688658449796`\), \(-1.3315720728889795`\), \ \(-1.3352544354889795`\), \(-1.3319586130429795`\), \(-1.3287439462429795`\), \ \(-1.3263684605756996`\), \(-1.3241527538516995`\), \(-1.3220442086516995`\), \ \(-1.3199615128254594`\), \(-1.3179749569254593`\), \(-1.3198026899037834`\), \ \(-1.3180969669279314`\), \(-1.3195433530553695`\), \(-1.3181607356794496`\), \ \(-1.3168905978678056`\), \(-1.3157851444278057`\), \(-1.3147454562678056`\), \ \(-1.3137068282331656`\), \(-1.3128206691531656`\), \(-1.3121292013131656`\), \ \(-1.3127933230131656`\), \(-1.3121341091571657`\), \(-1.3114841413971656`\), \ \(-1.3121232337299655`\), \(-1.3115398343799656`\), \(-1.3110368836546855`\), \ \(-1.3105594700866856`\), \(-1.3101783435694856`\), \(-1.3098795490414856`\), \ \(-1.3096234771414856`\), \(-1.3093847700874857`\), \(-1.3096230169354857`\), \ \(-1.3093924661674856`\), \(-1.3091874855948462`\), \(-1.308987454042846`\), \ \(-1.308796851682846`\), \(-1.3086250969428461`\), \(-1.3084565032157898`\), \ \(-1.3086249319517897`\), \(-1.3084732788917897`\), \(-1.3083241694837897`\), \ \(-1.3081766798700616`\), \(-1.3080340489050617`\), \(-1.3081629287203889`\), \ \(-1.308042319360389`\), \(-1.308160722190389`\), \(-1.3080658622803891`\), \ \(-1.307990446657989`\), \(-1.307922739945989`\), \(-1.307980760605989`\), \ \(-1.308038155759989`\), \(-1.307984428049589`\), \(-1.308033348333741`\), \ \(-1.3079891828137409`\), \(-1.3079550898687409`\), \(-1.307988455136741`\), \ \(-1.307955770934741`\), \(-1.307927119211541`\), \(-1.307899415941461`\), \ \(-1.307872406351061`\), \(-1.3078487828870609`\), \(-1.307825635217061`\), \ \(-1.3078487413310609`\), \(-1.3078284551774608`\), \(-1.3078483946624608`\), \ \(-1.3078284615776608`\), \(-1.3078463363072608`\), \(-1.3078629875312608`\), \ \(-1.307849508756861`\), \(-1.307862963288861`\), \(-1.307872147362261`\), \ \(-1.3078800299706925`\), \(-1.3078730982534104`\), \(-1.3078689858034904`\), \ \(-1.3078651135801305`\), \(-1.3078620317416305`\), \(-1.3078589859971506`\), \ \(-1.3078613000843506`\), \(-1.3078633864403506`\), \(-1.3078616348918506`\), \ \(-1.3078631963644907`\), \(-1.3078642622268586`\), \(-1.3078640806270185`\), \ \(-1.3078640511613138`\), \(-1.307864040052194`\), \(-1.3078640365723329`\), \ \(-1.3078640335840752`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\), \ \(-1.3078640329782896`\), \(-1.3078640329782896`\), \(-1.3078640329782896`\)}\ \)], "Output", CellLabel->"Out[87]="] }, Open ]], Cell[TextData[{ "\tFinally, the most important neglected term in the common denominator is \ ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]M\_\(\(1\)\(-\)\)\ \[RightBracketingBar]\^2\)]], ", but its convergence is faster and more monotonic because all terms are \ positive." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(DenomTerms = arrange[A\_T0 + \[Epsilon]\ A\_L0, values, 0.95];\)\), "\[IndentingNewLine]", \(Take[DenomTerms, 10]\)}], "Input", CellLabel->"In[88]:="], Cell[BoxData[ \({2\ Abs[\(M\_+\)[1]]\^2, Abs[\(M\_-\)[1]]\^2, Abs[\(E\_+\)[0]]\^2, \[Epsilon]\ Abs[\(S\_+\)[0]]\^2, 8\ \[Epsilon]\ Abs[\(S\_+\)[1]]\^2, \[Epsilon]\ Abs[\(S\_-\)[1]]\^2, 6\ Abs[\(E\_+\)[1]]\^2, 2\ Abs[\(E\_-\)[2]]\^2, 9\ Abs[\(M\_+\)[2]]\^2, 8\ \[Epsilon]\ Abs[\(S\_-\)[2]]\^2}\)], "Output", CellLabel->"Out[89]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(DenomValues = \(\(DenomTerms /. values\) /. \[Epsilon] \[Rule] 0.95\) /. \(x_\^*\) \[Rule] Conjugate[x];\)\), "\[IndentingNewLine]", \(Take[DenomValues, 10]\)}], "Input", CellLabel->"In[90]:="], Cell[BoxData[ \({7.18209714205`, 0.23404577460100007`, 0.16410924999999998`, 0.12652431398000005`, 0.12368599023999997`, 0.06695965949499999`, 0.019173581400000007`, 0.013990719751999997`, 0.007907576400000001`, 0.0034642704433566394`}\)], "Output", CellLabel->"Out[91]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`p\[Pi]\^0\)]], ", W=1.230" }], "Subsection"], Cell[BoxData[ \(\(maidvalues[p\[Pi]0, 1.230] = \[IndentingNewLine]Join[{\(E\_+\)[0] \[Rule] 3.906\[Times]10\^\(-1\) + \[ImaginaryI]\ \ 1.111\[Times]10\^\(-1\), \(S\_+\)[ 0] \[Rule] \(-3.516\)\[Times]10\^\(-1\) + \[ImaginaryI]\ \ 9.654\[Times]10\^\(-2\), \[IndentingNewLine]\(M\_+\)[1] \[Rule] 6.332\[Times]10\^\(-2\) + 1.923 \[ImaginaryI], \(E\_+\)[ 1] \[Rule] 4.596\[Times]10\^\(-2\) - \[ImaginaryI]\ \ 3.327\[Times]10\^\(-2\), \(S\_+\)[1] \[Rule] 1.461\[Times]10\^\(-2\) - \[ImaginaryI]\ \ 1.300\[Times]10\^\(-1\), \(M\_-\)[ 1] \[Rule] \(-4.847\)\[Times]10\^\(-1\) - \[ImaginaryI]\ \ 7.778\[Times]10\^\(-3\), \(S\_-\)[1] \[Rule] 2.619\[Times]10\^\(-1\) + \ \[ImaginaryI]\ \ 2.822\[Times]10\^\(-2\), \[IndentingNewLine]\(M\_+\)[ 2] \[Rule] \(-2.965\)\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 2.918\[Times]10\^\(-4\), \(E\_+\)[2] \[Rule] 2.260\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 8.675\[Times]10\^\(-5\), \(S\_+\)[ 2] \[Rule] \(-2.765\)\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 6.300\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_-\)[2] \[Rule] 1.825\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 1.613\[Times]10\^\(-3\), \(E\_-\)[ 2] \[Rule] \(-8.314\)\[Times]10\^\(-2\) - \[ImaginaryI]\ \ 1.138\[Times]10\^\(-3\), \(S\_-\)[2] \[Rule] 2.138\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 5.008\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_+\)[3] \[Rule] 9.608\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 3.070\[Times]10\^\(-5\), \(E\_+\)[ 3] \[Rule] \(-3.301\)\[Times]10\^\(-4\) - \[ImaginaryI]\ \ 4.180\[Times]10\^\(-6\), \(S\_+\)[3] \[Rule] 2.974\[Times]10\^\(-4\) - \[ImaginaryI]\ \ 1.648\[Times]10\^\(-6\), \[IndentingNewLine]\(M\_-\)[3] \[Rule] 3.544\[Times]10\^\(-4\) + \[ImaginaryI]\ \ 2.999\[Times]10\^\(-5\), \(E\_-\)[3] \[Rule] 1.384\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 1.730\[Times]10\^\(-5\), \(S\_-\)[ 3] \[Rule] \(-4.830\)\[Times]10\^\(-3\) - \[ImaginaryI]\ \ 6.157\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_+\)[ 4] \[Rule] \(-1.984\)\[Times]10\^\(-3\), \(E\_+\)[4] \[Rule] 1.028\[Times]10\^\(-4\), \(S\_+\)[ 4] \[Rule] \(-1.135\)\[Times]10\^\(-4\), \ \[IndentingNewLine]\(M\_-\)[4] \[Rule] 1.759\[Times]10\^\(-4\), \(E\_-\)[ 4] \[Rule] \(-3.163\)\[Times]10\^\(-3\), \(S\_-\)[4] \[Rule] 1.031\[Times]10\^\(-3\), \(M\_+\)[5] \[Rule] 5.685\[Times]10\^\(-4\), \(E\_+\)[ 5] \[Rule] \(-2.543\)\[Times]10\^\(-5\), \(S\_+\)[5] \[Rule] 2.651\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_-\)[ 5] \[Rule] \(-3.043\)\[Times]10\^\(-9\), \(E\_-\)[5] \[Rule] 8.028\[Times]10\^\(-4\), \(S\_-\)[ 5] \[Rule] \(-2.718\)\[Times]10\^\(-4\)}, Flatten[Table[{\(M\_+\)[\[ScriptL]] \[Rule] 0, \(E\_+\)[\[ScriptL]] \[Rule] 0, \(S\_+\)[\[ScriptL]] \[Rule] 0, \(M\_-\)[\[ScriptL]] \[Rule] 0, \(E\_-\)[\[ScriptL]] \[Rule] 0, \(S\_-\)[\[ScriptL]] \[Rule] 0}, {\[ScriptL], 6, 6}]]];\)\)], "Input", CellLabel->"In[93]:="], Cell[BoxData[ \(\(values = maidvalues[p\[Pi]0, 1.230];\)\)], "Input", CellLabel->"In[94]:="], Cell["The correct quadrupole ratios are", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({R\_EM, R\_SM} = {Re[\(E\_+\)[1]\/\(M\_+\)[1]], Re[\(S\_+\)[1]\/\(M\_+\)[1]]} /. values\)], "Input", CellLabel->"In[95]:="], Cell[BoxData[ \({\(-0.016496228686702207`\), \(-0.06727958879405914`\)}\)], "Output", CellLabel->"Out[95]="] }, Open ]], Cell[TextData[{ "while the traditional estimators give the following convergence patterns. \ These results are very similar, of course, to those for the nominal ", Cell[BoxData[ \(TraditionalForm\`M\_\[CapitalDelta]\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0.95, values], SMR[\[ScriptL], 0.95, values]}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[96]:="], Cell[BoxData[ TagBox[GridBox[{ {\(-0.0055595563564253304`\), \(-0.06361627392243858`\)}, {\(-0.012475557456890668`\), \(-0.059854291068148734`\)}, {\(-0.011456744376607873`\), \(-0.056851544505272855`\)}, {\(-0.01116545624158025`\), \(-0.056531088805787534`\)}, {\(-0.011116029264854757`\), \(-0.056246192268194946`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[96]//TableForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0.95, values], SMR[\[ScriptL], 0.95, values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[97]:="], Cell[BoxData[ TagBox[GridBox[{ {"0.33701984023214654`", "0.9455508730465958`"}, {"0.7562672471282695`", "0.8896352094445906`"}, {"0.6945068836153613`", "0.8450043397157758`"}, {"0.6768490212905964`", "0.8402412948573088`"}, {"0.6738527621052872`", "0.836006778227538`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[97]//TableForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(EMRterms = arrange[3 \((A\_T2 + \[Epsilon]\ A\_L2)\) - 2 A\_TT0, values, 0.95];\)\), "\[IndentingNewLine]", \(Take[EMRterms, 10]\)}], "Input", CellLabel->"In[98]:="], Cell[BoxData[ \({24\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)], \(-24\)\ Re[\(M\_-\)[ 1]\ \(\(E\_+\)[1]\^*\)], 12\ Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)], 24\ \[Epsilon]\ Abs[\(S\_+\)[1]]\^2, 24\ \[Epsilon]\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[ 0]], \(-24\)\ Re[\(M\_-\)[1]\ \(\(E\_-\)[3]\^*\)], 12\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)], 20\ Re[\(M\_+\)[3]\ \(\(M\_-\)[1]\^*\)], \(-12\)\ Re[\(M\_-\)[ 2]\ \(\(E\_+\)[0]\^*\)], 54\ \[Epsilon]\ Re[\(\(S\_-\)[3]\^*\)\ \(S\_-\)[1]]}\)], "Output", CellLabel->"Out[99]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(ALT1terms = arrange[A\_LT1, values, 0.95];\)\), "\[IndentingNewLine]", \(Take[ALT1terms, 10]\)}], "Input", CellLabel->"In[100]:="], Cell[BoxData[ \({6\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)], 3\ Re[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)], \(-6\)\ Re[\(E\_+\)[ 1]\ \(\(S\_-\)[1]\^*\)], \(-6\)\ Re[\(E\_+\)[ 0]\ \(\(S\_-\)[2]\^*\)], \(-6\)\ Re[\(M\_-\)[ 1]\ \(\(S\_+\)[1]\^*\)], 9\ Re[\(E\_-\)[3]\ \(\(S\_-\)[1]\^*\)], 3\ Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)], \(-6\)\ Re[\(E\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)], 9\ Re[\(M\_-\)[1]\ \(\(S\_-\)[3]\^*\)], \(-3\)\ Re[\(M\_-\)[ 2]\ \(\(S\_+\)[0]\^*\)]}\)], "Output", CellLabel->"Out[101]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Isospin-3/2, W=1.232", "Subsection"], Cell[TextData[{ "Values below are for ", Cell[BoxData[ \(TraditionalForm\`W = 1.232\)]], "." }], "Text"], Cell[BoxData[ \(\(maidvalues[1.232] = Join[{\(E\_+\)[ 0] \[Rule] \(-1.389\)\[Times]10\^\(-1\) + \[ImaginaryI]\ \ 3.689\[Times]10\^\(-2\), \(S\_+\)[ 0] \[Rule] \(-5.420\)\[Times]10\^\(-1\) + \[ImaginaryI]\ \ 1.440\[Times]10\^\(-1\), \[IndentingNewLine]\(M\_+\)[1] \[Rule] 2.675\[Times]10\^\(-2\) + 2.840 \[ImaginaryI], \(E\_+\)[ 1] \[Rule] \(-4.286\)\[Times]10\^\(-4\) - \[ImaginaryI]\ \ 4.314\[Times]10\^\(-2\), \(S\_+\)[ 1] \[Rule] \(-1.762\)\[Times]10\^\(-3\) - \[ImaginaryI]\ \ 1.876\[Times]10\^\(-1\), \(M\_-\)[ 1] \[Rule] \(-1.042\)\[Times]10\^\(-1\) + \[ImaginaryI]\ \ 1.275\[Times]10\^\(-2\), \(S\_-\)[ 1] \[Rule] \(-2.074\)\[Times]10\^\(-1\) + \ \[ImaginaryI]\ \ 1.772\[Times]10\^\(-2\), \[IndentingNewLine]\(M\_+\)[ 2] \[Rule] \(-5.386\)\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 3.182\[Times]10\^\(-4\), \(E\_+\)[ 2] \[Rule] \(-4.579\)\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 2.705\[Times]10\^\(-5\), \(S\_+\)[ 2] \[Rule] \(-7.989\)\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 4.720\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_-\)[2] \[Rule] 1.217\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 1.225\[Times]10\^\(-4\), \(E\_-\)[ 2] \[Rule] \(-1.338\)\[Times]10\^\(-1\) - \[ImaginaryI]\ \ 1.878\[Times]10\^\(-3\), \(S\_-\)[2] \[Rule] 6.870\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 2.739\[Times]10\^\(-5\), \[IndentingNewLine]\(M\_+\)[3] \[Rule] 1.620\[Times]10\^\(-2\) + \[ImaginaryI]\ \ 4.758\[Times]10\^\(-5\), \(E\_+\)[ 3] \[Rule] \(-2.076\)\[Times]10\^\(-3\) - \[ImaginaryI]\ \ 6.017\[Times]10\^\(-6\), \(S\_+\)[ 3] \[Rule] \(-7.631\)\[Times]10\^\(-4\) - \[ImaginaryI]\ \ 2.216\[Times]10\^\(-6\), \[IndentingNewLine]\(M\_-\)[ 3] \[Rule] \(-1.249\)\[Times]10\^\(-3\) + \[ImaginaryI]\ \ 1.252\[Times]10\^\(-6\), \(E\_-\)[3] \[Rule] 1.969\[Times]10\^\(-2\) - \[ImaginaryI]\ \ 1.823\[Times]10\^\(-6\), \(S\_-\)[ 3] \[Rule] \(-3.956\)\[Times]10\^\(-4\) - \[ImaginaryI]\ \ 1.107\[Times]10\^\(-7\), \[IndentingNewLine]\(M\_+\)[ 4] \[Rule] \(-3.388\)\[Times]10\^\(-3\), \(E\_+\)[ 4] \[Rule] \(-7.337\)\[Times]10\^\(-5\), \(S\_+\)[ 4] \[Rule] \(-3.104\)\[Times]10\^\(-4\), \ \[IndentingNewLine]\(M\_-\)[4] \[Rule] 1.498\[Times]10\^\(-4\), \(E\_-\)[ 4] \[Rule] \(-4.892\)\[Times]10\^\(-3\), \(S\_-\)[4] \[Rule] 2.098\[Times]10\^\(-4\), \(M\_+\)[5] \[Rule] 9.758\[Times]10\^\(-4\), \(E\_+\)[ 5] \[Rule] \(-1.024\)\[Times]10\^\(-5\), \(S\_+\)[ 5] \[Rule] \(-1.822\)\[Times]10\^\(-5\), \ \[IndentingNewLine]\(M\_-\)[5] \[Rule] \(-2.805\)\[Times]10\^\(-5\), \(E\_-\)[ 5] \[Rule] 1.216\[Times]10\^\(-3\), \(S\_-\)[ 5] \[Rule] \(-3.496\)\[Times]10\^\(-5\)}, Flatten[Table[{\(M\_+\)[\[ScriptL]] \[Rule] 0, \(E\_+\)[\[ScriptL]] \[Rule] 0, \(S\_+\)[\[ScriptL]] \[Rule] 0, \(M\_-\)[\[ScriptL]] \[Rule] 0, \(E\_-\)[\[ScriptL]] \[Rule] 0, \(S\_-\)[\[ScriptL]] \[Rule] 0}, {\[ScriptL], 6, 6}]]];\)\)], "Input", CellLabel->"In[102]:="], Cell[BoxData[ \(\(values = maidvalues[1.232];\)\)], "Input", CellLabel->"In[103]:="], Cell[CellGroupData[{ Cell[BoxData[ \({R\_EM, R\_SM} = {Re[\(E\_+\)[1]\/\(M\_+\)[1]], Re[\(S\_+\)[1]\/\(M\_+\)[1]]} /. values\)], "Input", CellLabel->"In[104]:="], Cell[BoxData[ \({\(-0.015190214677396587`\), \(-0.06605632142351485`\)}\)], "Output", CellLabel->"Out[104]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0.95, values], SMR[\[ScriptL], 0.95, values]}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[105]:="], Cell[BoxData[ TagBox[GridBox[{ {\(-0.010891744055690605`\), \(-0.06411775080605792`\)}, {\(-0.007833271313889`\), \(-0.05731869373517307`\)}, {\(-0.007357821080731974`\), \(-0.058195379058770993`\)}, {\(-0.006859408904716004`\), \(-0.057506509270084896`\)}, {\(-0.0067959456290393`\), \(-0.05758851908592395`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[105]//TableForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0.95, values], SMR[\[ScriptL], 0.95, values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[106]:="], Cell[BoxData[ TagBox[GridBox[{ {"0.7170237081571854`", "0.9706527615271225`"}, {"0.5156787761232302`", "0.8677245795701948`"}, {"0.4843790056292353`", "0.8809963649906563`"}, {"0.451567607857641`", "0.8705678431801627`"}, {"0.4473897027375021`", "0.871809356695777`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[106]//TableForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[M1, \[ScriptL], 0.95, values], SMR[M1, \[ScriptL], 0.95, values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[107]:="], Cell[BoxData[ TagBox[GridBox[{ {"1.`", "0.9999999999999998`"}, {"1.`", "0.9999999999999998`"}, {"0.9838821339235186`", "1.000069567440169`"}, {"0.9838821339235186`", "1.000069567440169`"}, {"0.9837308356088363`", "1.0000679237768297`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[107]//TableForm="] }, Open ]], Cell[TextData[{ "Interestingly, the relative importance of various neglected terms is \ different for isospin-3/2 and ", Cell[BoxData[ \(TraditionalForm\`p\[ThinSpace]\[Pi]\^0\)]], " channels. For ", Cell[BoxData[ \(TraditionalForm\`\(R\&~\)\_EM\)]], " we now find that longitudinal contamination is most import and that the \ ", Cell[BoxData[ \(TraditionalForm\`Re[\(E\_\(\(0\)\(+\)\)\) \ \(E\_\(\(2\)\(-\)\)\^*\)]\)]], " contribution becomes the most important transverse term. Therefore, \ significant improvement in ", Cell[BoxData[ \(TraditionalForm\`\(R\&~\)\_EM\)]], " can be made, in principle, by performing Rosenbluth separation, but the \ results would still not be satisfactory." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(EMRterms = arrange[3 \((A\_T2 + \[Epsilon]\ A\_L2)\) - 2 A\_TT0, values, 0.95];\)\), "\[IndentingNewLine]", \(Take[EMRterms, 10]\)}], "Input", CellLabel->"In[108]:="], Cell[BoxData[ \({24\ Re[\(M\_+\)[1]\ \(\(E\_+\)[1]\^*\)], 24\ \[Epsilon]\ Abs[\(S\_+\)[1]]\^2, 54\ \[Epsilon]\ Re[\(\(S\_+\)[2]\^*\)\ \(S\_+\)[0]], 12\ Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)], 276\/7\ Abs[\(M\_+\)[2]]\^2, \(-6\)\ Abs[\(E\_-\)[2]]\^2, 12\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)], 12\ Re[\(M\_+\)[2]\ \(\(E\_-\)[2]\^*\)], 24\ \[Epsilon]\ Re[\(\(S\_-\)[2]\^*\)\ \(S\_+\)[0]], 24\ \[Epsilon]\ Re[\(\(S\_-\)[1]\^*\)\ \(S\_+\)[1]]}\)], "Output", CellLabel->"Out[109]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Take[\(\(EMRterms /. values\) /. \[Epsilon] \[Rule] 0.95\) /. \(x_\^*\) \[Rule] Conjugate[x], 10]\)], "Input", CellLabel->"In[110]:="], Cell[BoxData[ \({\(-2.9406975612000004`\), 0.8024885138832`, 0.22247962523999998`, 0.22218648695999998`, 0.1143823192774629`, \(-0.107435801304`\), 0.089914708776`, 0.0864704450448`, \(-0.084806785152`\), \(-0.06746139695999999`\)}\)], \ "Output", CellLabel->"Out[110]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(ALT1terms = arrange[A\_LT1, values, 0.95];\)\), "\[IndentingNewLine]", \(Take[ALT1terms, 10]\)}], "Input", CellLabel->"In[111]:="], Cell[BoxData[ \({6\ Re[\(M\_+\)[1]\ \(\(S\_+\)[1]\^*\)], 3\ Re[\(E\_-\)[2]\ \(\(S\_+\)[0]\^*\)], 3\ Re[\(M\_+\)[2]\ \(\(S\_+\)[0]\^*\)], \(-6\)\ Re[\(E\_+\)[ 1]\ \(\(S\_+\)[1]\^*\)], 9\ Re[\(E\_-\)[3]\ \(\(S\_-\)[1]\^*\)], \(-12\)\ Re[\(E\_+\)[ 2]\ \(\(S\_+\)[0]\^*\)], 9\ Re[\(E\_-\)[4]\ \(\(S\_+\)[0]\^*\)], \(-3\)\ Re[\(M\_-\)[ 2]\ \(\(S\_+\)[0]\^*\)], \(-6\)\ Re[\(M\_-\)[ 1]\ \(\(S\_+\)[1]\^*\)], 27\ Re[\(M\_+\)[2]\ \(\(S\_+\)[2]\^*\)]}\)], "Output", CellLabel->"Out[112]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(ALT1values = \(\(ALT1terms /. values\) /. \[Epsilon] \[Rule] 0.95\) /. \(x_\^*\) \[Rule] Conjugate[x];\)\), "\[IndentingNewLine]", \(Take[ALT1values, 10]\)}], "Input", CellLabel->"In[113]:="], Cell[BoxData[ \({\(-3.1969868009999995`\), 0.216747504`, 0.08771382240000002`, \(-0.0485629151592`\), \(-0.03675364473204001`\), \ \(-0.029828558400000003`\), 0.023863176000000003`, 0.019735500000000003`, 0.013249797599999998`, 0.01161816909408`}\)], "Output", CellLabel->"Out[114]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0. , values], SMR[\[ScriptL], 0. , values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[115]:="], Cell[BoxData[ TagBox[GridBox[{ {"0.9932378313609053`", "1.007103442196293`"}, {"0.8315768684842327`", "0.9002898785756792`"}, {"0.8005289970875151`", "0.9140415605502203`"}, {"0.767084458696917`", "0.9032203226948657`"}, {"0.7627561679099859`", "0.9045080918690038`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[115]//TableForm="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Isospin-3/2, W=1.230", "Subsection"], Cell[TextData[{ "Values below are for ", Cell[BoxData[ \(TraditionalForm\`W = 1.230\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(maidvalues[1.230] = Join[{\(E\_+\)[ 0] \[Rule] \(-1.380\)\[Times]10\^\(-1\) + \[ImaginaryI]\ 3.623\ \[Times]10\^\(-2\), \(S\_+\)[ 0] \[Rule] \(-5.416\)\[Times]10\^\(-1\) + \[ImaginaryI]\ 1.423\ \[Times]10\^\(-1\), \[IndentingNewLine]\(M\_+\)[1] \[Rule] 1.291\[Times]10\^\(-1\) + 2.883 \[ImaginaryI], \(E\_+\)[ 1] \[Rule] \(-2.110\)\[Times]10\^\(-3\) - \[ImaginaryI]\ 4.678\ \[Times]10\^\(-2\), \(S\_+\)[ 1] \[Rule] \(-8.658\)\[Times]10\^\(-3\) - \[ImaginaryI]\ 1.934\ \[Times]10\^\(-1\), \(M\_-\)[ 1] \[Rule] \(-1.047\)\[Times]10\^\(-1\) + \[ImaginaryI]\ 1.257\ \[Times]10\^\(-2\), \(S\_-\)[ 1] \[Rule] \(-2.077\)\[Times]10\^\(-1\) + \ \[ImaginaryI]\ \ 1.743\[Times]10\^\(-2\), \[IndentingNewLine]\(M\_+\)[ 2] \[Rule] \(-5.382\)\[Times]10\^\(-2\) + \[ImaginaryI]\ 3.100\ \[Times]10\^\(-4\), \(E\_+\)[ 2] \[Rule] \(-4.462\)\[Times]10\^\(-3\) + \[ImaginaryI]\ 2.570\ \[Times]10\^\(-5\), \(S\_+\)[ 2] \[Rule] \(-7.902\)\[Times]10\^\(-3\) + \[ImaginaryI]\ 4.552\ \[Times]10\^\(-5\), \[IndentingNewLine]\(M\_-\)[2] \[Rule] 1.207\[Times]10\^\(-2\) + \[ImaginaryI]\ 1.179\[Times]10\^\(-4\), \ \(E\_-\)[2] \[Rule] \(-1.330\)\[Times]10\^\(-1\) - \[ImaginaryI]\ 1.816\ \[Times]10\^\(-3\), \(S\_-\)[2] \[Rule] 6.870\[Times]10\^\(-3\) + \[ImaginaryI]\ 2.739\[Times]10\^\(-5\), \ \[IndentingNewLine]\(M\_+\)[3] \[Rule] 1.607\[Times]10\^\(-2\) + \[ImaginaryI]\ 4.547\[Times]10\^\(-5\), \ \(E\_+\)[3] \[Rule] \(-2.051\)\[Times]10\^\(-3\) - \[ImaginaryI]\ 5.727\ \[Times]10\^\(-6\), \(S\_+\)[ 3] \[Rule] \(-7.358\)\[Times]10\^\(-4\) - \[ImaginaryI]\ 2.059\ \[Times]10\^\(-6\), \[IndentingNewLine]\(M\_-\)[ 3] \[Rule] \(-1.232\)\[Times]10\^\(-3\) + \[ImaginaryI]\ 1.199\ \[Times]10\^\(-6\), \(E\_-\)[3] \[Rule] 1.957\[Times]10\^\(-2\) - \[ImaginaryI]\ 1.915\[Times]10\^\(-6\), \ \(S\_-\)[3] \[Rule] \(-4.245\)\[Times]10\^\(-4\) - \[ImaginaryI]\ 9.674\ \[Times]10\^\(-8\), \[IndentingNewLine]\(M\_+\)[ 4] \[Rule] \(-3.362\)\[Times]10\^\(-3\), \(E\_+\)[ 4] \[Rule] \(-6.589\)\[Times]10\^\(-5\), \(S\_+\)[ 4] \[Rule] \(-3.040\)\[Times]10\^\(-4\), \[IndentingNewLine]\(M\ \_-\)[4] \[Rule] 1.466\[Times]10\^\(-4\), \(E\_-\)[ 4] \[Rule] \(-4.845\)\[Times]10\^\(-3\), \(S\_-\)[4] \[Rule] 2.130\[Times]10\^\(-4\), \(M\_+\)[5] \[Rule] 9.644\[Times]10\^\(-4\), \(E\_+\)[ 5] \[Rule] \(-1.000\)\[Times]10\^\(-5\), \(S\_+\)[ 5] \[Rule] \(-1.653\)\[Times]10\^\(-5\), \[IndentingNewLine]\(M\ \_-\)[5] \[Rule] \(-2.697\)\[Times]10\^\(-5\), \(E\_-\)[5] \[Rule] 1.202\[Times]10\^\(-3\), \(S\_-\)[ 5] \[Rule] \(-3.635\)\[Times]10\^\(-5\)}, Flatten[Table[{\(M\_+\)[\[ScriptL]] \[Rule] 0, \(E\_+\)[\[ScriptL]] \[Rule] 0, \(S\_+\)[\[ScriptL]] \[Rule] 0, \(M\_-\)[\[ScriptL]] \[Rule] 0, \(E\_-\)[\[ScriptL]] \[Rule] 0, \(S\_-\)[\[ScriptL]] \[Rule] 0}, {\[ScriptL], 6, 6}]]]\)], "Input", CellLabel->"In[116]:="], Cell[BoxData[ \({\(E\_+\)[0] \[Rule] \(-0.13799999999999998`\) + 0.036230000000000005`\ \[ImaginaryI], \(S\_+\)[ 0] \[Rule] \(-0.5416000000000001`\) + 0.1423`\ \[ImaginaryI], \(M\_+\)[ 1] \[Rule] \(\(0.1291`\)\(\[InvisibleSpace]\)\) + 2.883`\ \[ImaginaryI], \(E\_+\)[1] \[Rule] \(-0.00211`\) - 0.04678`\ \[ImaginaryI], \(S\_+\)[ 1] \[Rule] \(-0.008657999999999999`\) - 0.19340000000000002`\ \[ImaginaryI], \(M\_-\)[ 1] \[Rule] \(-0.1047`\) + 0.01257`\ \[ImaginaryI], \(S\_-\)[ 1] \[Rule] \(-0.2077`\) + 0.01743`\ \[ImaginaryI], \(M\_+\)[ 2] \[Rule] \(-0.05382`\) + 0.00031`\ \[ImaginaryI], \(E\_+\)[ 2] \[Rule] \(-0.004462`\) + 0.0000257`\ \[ImaginaryI], \(S\_+\)[ 2] \[Rule] \(-0.007902000000000001`\) + 0.00004552`\ \[ImaginaryI], \(M\_-\)[ 2] \[Rule] \(\(0.01207`\)\(\[InvisibleSpace]\)\) + 0.0001179`\ \[ImaginaryI], \(E\_-\)[2] \[Rule] \(-0.133`\) - 0.0018160000000000001`\ \[ImaginaryI], \(S\_-\)[ 2] \[Rule] \(\(0.00687`\)\(\[InvisibleSpace]\)\) + 0.00002739`\ \[ImaginaryI], \(M\_+\)[ 3] \[Rule] \(\(0.01607`\)\(\[InvisibleSpace]\)\) + 0.000045470000000000003`\ \[ImaginaryI], \(E\_+\)[ 3] \[Rule] \(-0.0020510000000000003`\) - 5.727`*^-6\ \[ImaginaryI], \(S\_+\)[3] \[Rule] \(-0.0007358`\) - 2.059`*^-6\ \[ImaginaryI], \(M\_-\)[3] \[Rule] \(-0.001232`\) + 1.199`*^-6\ \[ImaginaryI], \(E\_-\)[ 3] \[Rule] \(\(0.01957`\)\(\[InvisibleSpace]\)\) - 1.915`*^-6\ \[ImaginaryI], \(S\_-\)[3] \[Rule] \(-0.0004245`\) - 9.673999999999999`*^-8\ \[ImaginaryI], \(M\_+\)[ 4] \[Rule] \(-0.003362`\), \(E\_+\)[ 4] \[Rule] \(-0.00006589000000000001`\), \(S\_+\)[ 4] \[Rule] \(-0.000304`\), \(M\_-\)[4] \[Rule] 0.0001466`, \(E\_-\)[4] \[Rule] \(-0.0048449999999999995`\), \(S\_-\)[ 4] \[Rule] 0.000213`, \(M\_+\)[5] \[Rule] 0.0009644`, \(E\_+\)[5] \[Rule] \(-0.00001`\), \(S\_+\)[ 5] \[Rule] \(-0.000016530000000000003`\), \(M\_-\)[ 5] \[Rule] \(-0.000026970000000000004`\), \(E\_-\)[5] \[Rule] 0.001202`, \(S\_-\)[5] \[Rule] \(-0.00003635`\), \(M\_+\)[6] \[Rule] 0, \(E\_+\)[6] \[Rule] 0, \(S\_+\)[6] \[Rule] 0, \(M\_-\)[6] \[Rule] 0, \(E\_-\)[6] \[Rule] 0, \(S\_-\)[6] \[Rule] 0}\)], "Output", CellLabel->"Out[116]="] }, Open ]], Cell[BoxData[ \(\(values = maidvalues[1.230];\)\)], "Input", CellLabel->"In[117]:="], Cell[CellGroupData[{ Cell[BoxData[ \({R\_EM, R\_SM} = {Re[\(E\_+\)[1]\/\(M\_+\)[1]], Re[\(S\_+\)[1]\/\(M\_+\)[1]]} /. values\)], "Input", CellLabel->"In[118]:="], Cell[BoxData[ \({\(-0.01622638898757617`\), \(-0.06708286251761379`\)}\)], "Output", CellLabel->"Out[118]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0.95, values], SMR[\[ScriptL], 0.95, values]}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[119]:="], Cell[BoxData[ TagBox[GridBox[{ {\(-0.011620137773924077`\), \(-0.06532089337487075`\)}, {\(-0.00869000069212172`\), \(-0.058759325068835674`\)}, {\(-0.007522542559337834`\), \(-0.05971594990941732`\)}, {\(-0.007046235910268961`\), \(-0.059054790948966894`\)}, {\(-0.006978298970712489`\), \(-0.0591417628031033`\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[119]//TableForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0.95, values], SMR[\[ScriptL], 0.95, values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[120]:="], Cell[BoxData[ TagBox[GridBox[{ {"0.7161259219670564`", "0.9737344371331739`"}, {"0.5355474159269367`", "0.8759215522952285`"}, {"0.46359929896278906`", "0.8901818984504103`"}, {"0.4342454698740399`", "0.8803260435325194`"}, {"0.4300586517465755`", "0.8816225274759941`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[120]//TableForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[M1, \[ScriptL], 0.95, values], SMR[M1, \[ScriptL], 0.95, values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[121]:="], Cell[BoxData[ TagBox[GridBox[{ {"0.9999999999999998`", "0.9999999999999998`"}, {"0.9999999999999998`", "0.9999999999999998`"}, {"0.9400339069004793`", "1.0002134152893558`"}, {"0.9400339069004793`", "1.0002134152893558`"}, {"0.9393777232089114`", "1.0002038753047526`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[121]//TableForm="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{EMR[\[ScriptL], 0. , values], SMR[\[ScriptL], 0. , values]}/{R\_EM, R\_SM}, {\[ScriptL], 1, 5}] // TableForm\)], "Input", CellLabel->"In[122]:="], Cell[BoxData[ TagBox[GridBox[{ {"0.9946182868327733`", "1.0101158550294038`"}, {"0.8492630401883748`", "0.9086284610447106`"}, {"0.7763954977768631`", "0.9234037808446091`"}, {"0.7464738547481162`", "0.9131786102849891`"}, {"0.742136062016442`", "0.9145231824456698`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[122]//TableForm="] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Other traditional formulas", "Section"], Cell["\<\ One sometimes sees attempts to extract other multipole products from \ truncated Legendre expansions, but extreme care must be exercised. For \ example, the following truncations appear to give simple estimators.\ \>", "Text"], Cell[BoxData[{ \(\(A\_LT0 = \ LTexp /. {P\_n_[x] \[Rule] 0};\)\), "\[IndentingNewLine]", \(\(A\_T1 = Coefficient[Expand[Texp], P\_1[x]];\)\)}], "Input", CellLabel->"In[153]:="], Cell[CellGroupData[{ Cell[BoxData[ \(M1truncation[truncate[A\_T1, 1]] // MySimplify\)], "Input", CellLabel->"In[130]:="], Cell[BoxData[ \(2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)]\)], "Output", CellLabel->"Out[130]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(M1truncation[truncate[A\_LT0, 1]] // MySimplify\)], "Input", CellLabel->"In[132]:="], Cell[BoxData[ \(Re[\(M\_+\)[1]\ \(\(S\_+\)[0]\^*\)]\)], "Output", CellLabel->"Out[132]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(M1truncation[ truncate[\(-\(\(A\_T2 + 2 A\_T0 + 2 A\_TT0\)\/8\)\), 1]] // MySimplify\)], "Input", CellLabel->"In[161]:="], Cell[BoxData[ \(Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)]\)], "Output", CellLabel->"Out[161]="] }, Open ]], Cell["\<\ Thus, it is useful to define the following accuracy parameters.\ \>", "Text"], Cell[BoxData[{ \(\(f\_E0p[lmax_, values_] := \(truncate[A\_T1, lmax]\/\(2 Re[\(M\_+\)[1] \ \(\(E\_+\)[0]\^*\)]\) /. values\) /. \(x_\^*\) \[Rule] Conjugate[x];\)\), "\[IndentingNewLine]", \(\(f\_S0p[lmax_, values_] := \(truncate[A\_LT0, lmax]\/Re[\(M\_+\)[1] \ \(\(S\_+\)[0]\^*\)] /. values\) /. \(x_\^*\) \[Rule] Conjugate[x];\)\), "\[IndentingNewLine]", \(\(f\_M1m[lmax_, values_] := \(truncate[\(-\((A\_T2 + 2 A\_T0 + 2 A\_TT0)\)\)/8, \ lmax]\/Re[\(M\_+\)[1] \(\(M\_-\)[1]\^*\)] /. values\) /. \(x_\^*\) \[Rule] Conjugate[x];\)\)}], "Input", CellLabel->"In[173]:="], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[ Table[{f\_E0p[\[ScriptL], maidvalues[p\[Pi]0, 1.232]], f\_S0p[\[ScriptL], maidvalues[p\[Pi]0, 1.232]], f\_M1m[\[ScriptL], maidvalues[p\[Pi]0, 1.232]]}, {\[ScriptL], 1, 5}], TableHeadings \[Rule] {Range[ 5], {\*"\"\<\!\(f\_\(\(E0\)\(+\)\)\)\>\"", \*"\"\<\!\(f\_\(\(S0\)\ \(+\)\)\)\>\"", \*"\"\<\!\(f\_\(\(M1\)\(-\)\)\)\>\""}}]\)], "Input", CellLabel->"In[176]:="], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"\\!\\(f\\_\\(\\(E0\\)\\(+\\)\\)\\)\"\>", "\<\"\\!\ \\(f\\_\\(\\(S0\\)\\(+\\)\\)\\)\"\>", "\<\"\\!\\(f\\_\\(\\(M1\\)\\(-\\)\\)\\)\ \"\>"}, {"1", "2.116968679851088`", \(-0.22661700448766553`\), "8.469240954058762`"}, {"2", "1.8133217018903665`", \(-0.6840211321260236`\), "8.722756838172117`"}, {"3", "1.7497855293356381`", \(-0.7261042041743792`\), "9.48957429159`"}, {"4", "1.7404550695287178`", \(-0.7647630514848391`\), "9.707510711792121`"}, {"5", "1.7392689789949185`", \(-0.7693677363347617`\), "9.744877371985124`"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {{1, 2, 3, 4, 5}, { "\!\(f\_\(\(E0\)\(+\)\)\)", "\!\(f\_\(\(S0\)\(+\)\)\)", "\!\(f\_\(\(M1\)\(-\)\)\)"}}]]]], "Output", CellLabel->"Out[176]//TableForm="] }, Open ]], Cell[TextData[{ "However, these estimators are hopelessly inaccurate at ", Cell[BoxData[ \(TraditionalForm\`Q\^2 = 1\)]], ". The estimator for ", Cell[BoxData[ \(TraditionalForm\`Re\ \(M\_\(\(1\)\(-\)\)\) \ \(M\_\(\(1\)\(+\)\)\^*\)\)]], " isn't even the numerically leading term; for MAID2003 it is only fifth! \ Therefore, these formulas are worthless under these conditions." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Take[arrange[A\_T1, maidvalues[p\[Pi]0, 1.232], 0. ], 10]\)], "Input", CellLabel->"In[142]:="], Cell[BoxData[ \({2\ Re[\(M\_+\)[1]\ \(\(E\_+\)[0]\^*\)], \(-2\)\ Re[\(M\_-\)[ 1]\ \(\(E\_+\)[0]\^*\)], 6\ Re[\(E\_+\)[0]\ \(\(E\_+\)[1]\^*\)], \(-2\)\ Re[\(M\_-\)[ 1]\ \(\(E\_-\)[2]\^*\)], 6\ Re[\(M\_-\)[1]\ \(\(M\_-\)[2]\^*\)], 54\/5\ Re[\(E\_-\)[2]\ \(\(E\_-\)[3]\^*\)], 288\/7\ Re[\(M\_+\)[2]\ \(\(M\_+\)[3]\^*\)], 54\/5\ Re[\(M\_+\)[1]\ \(\(M\_+\)[2]\^*\)], 18\/5\ Re[\(M\_+\)[2]\ \(\(E\_+\)[1]\^*\)], \(-\(6\/5\)\)\ Re[\(E\_+\)[ 1]\ \(\(E\_-\)[2]\^*\)]}\)], "Output", CellLabel->"Out[142]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(% /. maidvalues[p\[Pi]0, 1.232]\) /. \(x_\^*\) \[Rule] Conjugate[x]\)], "Input", CellLabel->"In[143]:="], Cell[BoxData[ \({0.41956192600000003`, 0.3787047666`, 0.08993276400000001`, \(-0.080924710048`\), \(-0.05389731608400001`\), \ \(-0.012590895295584`\), \(-0.011812650829714286`\), 0.007693943760000001`, \(-0.005085784800000001`\), 0.004708278744`}\)], "Output", CellLabel->"Out[143]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Take[arrange[A\_LT0, maidvalues[p\[Pi]0, 1.232], 0. ], 10]\)], "Input", CellLabel->"In[144]:="], Cell[BoxData[ \({Re[\(M\_+\)[ 1]\ \(\(S\_+\)[0]\^*\)], \(-Re[\(M\_-\)[ 1]\ \(\(S\_+\)[0]\^*\)]\), \(-Re[\(E\_+\)[ 0]\ \(\(S\_-\)[1]\^*\)]\), \(-3\)\ Re[\(E\_+\)[ 1]\ \(\(S\_+\)[0]\^*\)], 2\ Re[\(E\_-\)[2]\ \(\(S\_-\)[1]\^*\)], 2\ Re[\(M\_-\)[1]\ \(\(S\_-\)[2]\^*\)], 2\ Re[\(E\_+\)[0]\ \(\(S\_+\)[1]\^*\)], 2\ Re[\(E\_-\)[3]\ \(\(S\_+\)[0]\^*\)], \(-6\)\ Re[\(E\_+\)[ 1]\ \(\(S\_-\)[2]\^*\)], \(-3\)\ Re[\(E\_+\)[ 0]\ \(\(S\_-\)[3]\^*\)]}\)], "Output", CellLabel->"Out[144]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(% /. maidvalues[p\[Pi]0, 1.232]\) /. \(x_\^*\) \[Rule] Conjugate[x]\)], "Input", CellLabel->"In[145]:="], Cell[BoxData[ \({0.18700011800000002`, \(-0.16919388678000002`\), \(-0.106062433`\), 0.05900311920000001`, \(-0.04419348927999999`\), \ \(-0.02065489528084`\), \(-0.013124323999999998`\), \(-0.0097990429412`\), \ \(-0.006056163537599999`\), 0.0056943596400000004`}\)], "Output", CellLabel->"Out[145]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Take[ arrange[\(-\(\(A\_T2 + 2 A\_T0 + 2 A\_TT0\)\/8\)\), maidvalues[p\[Pi]0, 1.232], 0. ], 10]\)], "Input", CellLabel->"In[164]:="], Cell[BoxData[ \({\(-\(1\/4\)\)\ Abs[\(M\_-\)[1]]\^2, \(-\(1\/4\)\)\ \ Abs[\(E\_+\)[0]]\^2, \(-\(3\/2\)\)\ Re[\(M\_+\)[2]\ \(\(E\_+\)[0]\^*\)], 1\/2\ Re[\(E\_+\)[0]\ \(\(E\_-\)[2]\^*\)], Re[\(M\_+\)[1]\ \(\(M\_-\)[1]\^*\)], 5\/2\ Re[\(M\_+\)[3]\ \(\(M\_-\)[1]\^*\)], 3\/2\ Re[\(M\_-\)[ 2]\ \(\(E\_+\)[ 0]\^*\)], \(-3\)\ Abs[\(E\_+\)[1]]\^2, \(-\(3\/4\)\)\ \ Abs[\(E\_-\)[2]]\^2, \(-\(3\/2\)\)\ Re[\(M\_+\)[ 2]\ \(\(E\_-\)[2]\^*\)]}\)], "Output", CellLabel->"Out[164]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(% /. maidvalues[p\[Pi]0, 1.232]\) /. \(x_\^*\) \[Rule] Conjugate[x]\)], "Input", CellLabel->"In[165]:="], Cell[BoxData[ \({\(-0.05851144365025002`\), \(-0.041027312499999996`\), 0.017262459`, \(-0.0163484406`\), \(-0.0146099915`\), \ \(-0.011714723578425`\), 0.011111703300000001`, \(-0.009586790700000004`\), \ \(-0.005246519906999999`\), \(-0.0037176605999999996`\)}\)], "Output", CellLabel->"Out[165]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Conclusions", "Section"], Cell[TextData[{ "\tDespite the intuitive appeal of the traditional Legendre analysis for \ quadrupole amplitudes, neither of its truncation hypotheses are sufficiently \ accurate at the level of experimental precision that is now possible. Even \ if truncation to ", Cell[BoxData[ \(TraditionalForm\`\[ScriptL] \[LessEqual] 1\)]], " were accurate, contributions to the Legendre coefficients from terms that \ do not involve ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " are not negligible. Rosenbluth separation would help for ", Cell[BoxData[ \(TraditionalForm\`\(R\&~\)\_EM\)]], ", but errors would remain about ", Cell[BoxData[ \(TraditionalForm\`15 %\)]], " for both ", Cell[BoxData[ \(TraditionalForm\`p\[ThinSpace]\[Pi]\^0\)]], " quadrupole ratios. The accuracy expected for ", Cell[BoxData[ \(TraditionalForm\`\(R\&~\)\_SM\%\((p\[VeryThinSpace]\[Pi]\^0)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(R\&~\)\_SM\%\((3/2)\)\)]], " are similar but, surprisingly, the accuracy is even worse for ", Cell[BoxData[ \(TraditionalForm\`\(R\&~\)\_EM\%\((3/2)\)\)]], " than for ", Cell[BoxData[ \(TraditionalForm\`\(R\&~\)\_EM\%\((p\[VeryThinSpace]\[Pi]\^0)\)\)]], ": errors of about ", Cell[BoxData[ \(TraditionalForm\`25 %\)]], " are expected in ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(\(R\&~\)\_EM\%\((3/2)\)\), "TraditionalForm"]}], TraditionalForm]]], " with and almost ", Cell[BoxData[ \(TraditionalForm\`60 %\)]], " without Rosenbluth separation. The cancellation between ", Cell[BoxData[ \(TraditionalForm\`A\_2\%T\)]], " and ", Cell[BoxData[ \(TraditionalForm\`A\_0\%TT\)]], " amplifies the relative error. The present estimates were made using the \ MAID2003 model. Qualitatively similar results have been obtained with other \ models also, but the details are model dependent because these estimators \ involve rather delicate cancellations among many nonnegligible terms. \ Therefore, the traditional Legendre analysis does not offer reliable \ estimates of ", Cell[BoxData[ \(TraditionalForm\`N \[Rule] \[CapitalDelta]\)]], "quadrupole ratios, with truncation errors that are especially severe for \ EMR. These errors probably increase with ", Cell[BoxData[ \(TraditionalForm\`Q\^2\)]], " as ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(+\)\)\)]], " dominance deteriorates, but for that analysis I will return to numerical \ evaluation methods. The behavior of truncated Legendre estimates for ", Cell[BoxData[ \(TraditionalForm\`E\_\(\(0\)\(+\)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`S\_\(\(0\)\(+\)\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`M\_\(\(1\)\(-\)\)\)]], " is even worse." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowToolbars->"EditBar", WindowSize->{1016, 668}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, Visible->True, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, ShowSelection->True, InputAliases->{"intt"->RowBox[ {"\[Integral]", RowBox[ {"\[SelectionPlaceholder]", RowBox[ {"\[DifferentialD]", "\[Placeholder]"}]}]}], "dintt"->RowBox[ { SubsuperscriptBox[ "\[Integral]", "\[SelectionPlaceholder]", "\[Placeholder]"], RowBox[ {"\[Placeholder]", RowBox[ {"\[DifferentialD]", "\[Placeholder]"}]}]}], "sumt"->RowBox[ { UnderoverscriptBox[ "\[Sum]", RowBox[ {"\[SelectionPlaceholder]", "=", "\[Placeholder]"}], "\[Placeholder]"], "\[Placeholder]"}], "prodt"->RowBox[ { UnderoverscriptBox[ "\[Product]", RowBox[ {"\[SelectionPlaceholder]", "=", "\[Placeholder]"}], "\[Placeholder]"], "\[Placeholder]"}], "dt"->RowBox[ { SubscriptBox[ "\[PartialD]", "\[Placeholder]"], " ", "\[SelectionPlaceholder]"}], "notation"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation>"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation<"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "symb"->RowBox[ {"Symbolize", "[", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], "]"}], "infixnotation"->RowBox[ {"InfixNotation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], ",", "\[Placeholder]"}], "]"}], "addia"->RowBox[ {"AddInputAlias", "[", RowBox[ {"\"\[Placeholder]\"", "\[Rule]", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "pattwraper"->TagBox[ "\[Placeholder]", NotationPatternTag, TagStyle -> "NotationPatternWrapperStyle"], "madeboxeswraper"->TagBox[ "\[Placeholder]", NotationMadeBoxesTag, TagStyle -> "NotationMadeBoxesWrapperStyle"]}, Magnification->1.25, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Title"], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. Make modifications to any definition using commands in the \ Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], ScriptMinSize->9], Cell[StyleData[All, "Presentation"], ScriptMinSize->9], Cell[StyleData[All, "SlideShow"], PageWidth->WindowWidth, ScrollingOptions->{"PagewiseDisplay"->True, "VerticalScrollRange"->Fit}, ShowCellBracket->False, ScriptMinSize->9], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ShowCellLabel->False, ImageSize->{200, 200}, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the Notebook level.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{True, True}, PrintingOptions->{"FirstPageHeader"->False, "FacingPages"->True}, CellLabelAutoDelete->False, CellFrameLabelMargins->6, StyleMenuListing->None], Cell[StyleData["Notebook", "Presentation"], FontSize->12] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellFrame->{{0, 0}, {0, 0.25}}, CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CellFrameMargins->9, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing->{0.95, 11}, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->36], Cell[StyleData["Title", "Presentation"], CellMargins->{{25, 30}, {40, 30}}, CellFrameMargins->{{6, 10}, {10, 14}}, FontSize->54], Cell[StyleData["Title", "Printout"], CellMargins->{{18, 30}, {4, 0}}, CellFrameMargins->4, FontSize->30] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{18, 30}, {0, 10}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, LineSpacing->{1, 0}, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->24, FontSlant->"Italic"], Cell[StyleData["Subtitle", "Presentation"], CellMargins->{{25, 40}, {55, 10}}, FontSize->36], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{18, 30}, {0, 10}}, FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SectionFirst"], CellFrame->{{0, 0}, {0, 3}}, CellMargins->{{18, 30}, {4, 30}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CellFrameMargins->3, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontSize->18, FontWeight->"Bold"], Cell[StyleData["SectionFirst", "Presentation"], CellFrame->{{0, 0}, {0, 8}}, CellMargins->{{25, 30}, {65, 10}}, FontSize->27], Cell[StyleData["SectionFirst", "Printout"], FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellMargins->{{18, 30}, {4, 30}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontSize->18, FontWeight->"Bold"], Cell[StyleData["Section", "Presentation"], CellMargins->{{25, 30}, {30, 15}}, FontSize->27], Cell[StyleData["Section", "Printout"], FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{25, 30}, {30, 20}}, FontSize->21], Cell[StyleData["Subsection", "Printout"], FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{18, 30}, {4, 12}}, CellGroupingRules->{"SectionGrouping", 60}, PageBreakBelow->False, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"Subsubsection", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{25, 30}, {10, 10}}, FontSize->18], Cell[StyleData["Subsubsection", "Printout"], FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{18, 10}, {Inherited, 6}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, TextJustification->1, LineSpacing->{1, 2}, CounterIncrements->"Text"], Cell[StyleData["Text", "Presentation"], CellMargins->{{25, 10}, {6, 20}}, FontSize->18], Cell[StyleData["Text", "Printout"], CellMargins->{{18, 30}, {Inherited, 4}}, Hyphenation->True, LineSpacing->{1, 3}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Caption"], CellMargins->{{55, 50}, {5, 5}}, PageBreakAbove->False, FontSize->10], Cell[StyleData["Caption", "Presentation"], CellMargins->{{80, 50}, {8, 14}}, FontSize->15], Cell[StyleData["Caption", "Printout"], CellMargins->{{55, 55}, {5, 2}}, Hyphenation->True, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Inline Formatting", "Section"], Cell["\<\ These styles are for modifying individual words or letters in a cell \ exclusive of the cell tag.\ \>", "Text"], Cell[StyleData["RM"], StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["BF"], StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["IT"], StyleMenuListing->None, FontSlant->"Italic"], Cell[StyleData["TR"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["TI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["TB"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["TBI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["MR"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["MO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["MB"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["MBO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["SR"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["SO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SB"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Inert"], StyleMenuListing->None, Background->RGBColor[0.870588, 0.905882, 0.972549]], Cell[StyleData["Inert", "Printout"], StyleMenuListing->None, Background->GrayLevel[1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output to the \ kernel. Be careful when modifying, renaming, or removing these styles, \ because the front end associates special meanings with these style names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{55, 10}, {5, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, LinebreakAdjustments->{0.85, 2, 10, 0, 1}, CounterIncrements->"Input", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellMargins->{{80, 10}, {8, 15}}, FontSize->18], Cell[StyleData["Input", "Printout"], CellMargins->{{55, 55}, {0, 10}}, ShowCellLabel->False, LinebreakAdjustments->{0.85, 2, 10, 1, 1}, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{55, 10}, {8, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelPositioning->Left, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->"Formula", FormatType->InputForm, CounterIncrements->"Output"], Cell[StyleData["Output", "Presentation"], CellMargins->{{80, 10}, {30, 12}}, FontSize->16], Cell[StyleData["Output", "Printout"], CellMargins->{{55, 55}, {10, 10}}, ShowCellLabel->False, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellDingbat->"\[LongDash]", CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, AutoStyleOptions->{"UnmatchedBracketStyle"->None}, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Message", "Presentation"], CellMargins->{{80, 10}, {15, 8}}, FontSize->15], Cell[StyleData["Message", "Printout"], CellMargins->{{55, 55}, {0, 3}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, TextAlignment->Left, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{80, 10}, {12, 8}}, FontSize->16], Cell[StyleData["Print", "Printout"], CellMargins->{{54, 72}, {2, 10}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", StyleMenuListing->None], Cell[StyleData["Graphics", "Presentation"], CellMargins->{{80, 10}, {10, 10}}, FontSize->16], Cell[StyleData["Graphics", "Printout"], CellMargins->{{55, 55}, {0, 15}}, ImageSize->{0.0625, 0.0625}, ImageMargins->{{35, Inherited}, {Inherited, 0}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], CellMargins->{{9, Inherited}, {Inherited, Inherited}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontSlant->"Oblique"], Cell[StyleData["CellLabel", "Presentation"], FontSize->14], Cell[StyleData["CellLabel", "Printout"], CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Unique Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{45, Inherited}, {2, 20}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Author", "Presentation"], CellMargins->{{65, 30}, {4, 30}}, FontSize->21], Cell[StyleData["Author", "Printout"], CellMargins->{{36, Inherited}, {2, 30}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Address"], CellMargins->{{45, Inherited}, {2, 2}}, CellGroupingRules->{"TitleGrouping", 30}, PageBreakBelow->False, LineSpacing->{1, 1}, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->12, FontSlant->"Italic"], Cell[StyleData["Address", "Presentation"], CellMargins->{{65, 30}, {40, 2}}, FontSize->18], Cell[StyleData["Address", "Printout"], CellMargins->{{36, Inherited}, {2, 2}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Abstract"], CellMargins->{{45, 75}, {Inherited, 30}}, LineSpacing->{1, 0}], Cell[StyleData["Abstract", "Presentation"], CellMargins->{{65, 30}, {8, 25}}, FontSize->18], Cell[StyleData["Abstract", "Printout"], CellMargins->{{36, 67}, {Inherited, 50}}, Hyphenation->True, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Reference"], CellMargins->{{18, 40}, {2, 2}}, TextJustification->1, LineSpacing->{1, 0}], Cell[StyleData["Reference", "Presentation"], CellMargins->{{25, 40}, {2, 2}}, FontSize->18], Cell[StyleData["Reference", "Printout"], CellMargins->{{18, 40}, {Inherited, 0}}, Hyphenation->True, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext ButtonBoxes. The \ \"Hyperlink\" style is for links within the same Notebook, or between \ Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "Condensed"], FontSize->11], Cell[StyleData["Hyperlink", "SlideShow"]], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line help \ system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "Condensed"], FontSize->11], Cell[StyleData["MainBookLink", "SlideShow"]], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "Condensed"], FontSize->11], Cell[StyleData["AddOnsLink", "SlideShow"]], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "Condensed"], FontSize->11], Cell[StyleData["RefGuideLink", "SlideShow"]], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLinkText"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLinkText", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLinkText", "Condensed"], FontSize->11], Cell[StyleData["RefGuideLinkText", "SlideShow"]], Cell[StyleData["RefGuideLinkText", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "Condensed"], FontSize->11], Cell[StyleData["GettingStartedLink", "SlideShow"]], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DemosLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Demos", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["DemosLink", "SlideShow"]], Cell[StyleData["DemosLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TourLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Tour", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["TourLink", "SlideShow"]], Cell[StyleData["TourLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "Condensed"], FontSize->11], Cell[StyleData["OtherInformationLink", "SlideShow"]], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MasterIndexLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MasterIndex", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MasterIndexLink", "SlideShow"]], Cell[StyleData["MasterIndexLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard ButtonFunctions, for use \ in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Slide Show Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["SlideShowNavigationBar"], Editable->False, CellFrame->True, CellMargins->{{0, 0}, {3, 3}}, CellElementSpacings->{"CellMinHeight"->0.8125}, CellGroupingRules->{"SectionGrouping", 30}, CellFrameMargins->False, CellFrameColor->GrayLevel[1], CellFrameLabelMargins->False, TextAlignment->Center, CounterIncrements->"SlideShowNavigationBar", StyleMenuListing->None, FontSize->10, Background->GrayLevel[0.8], Magnification->1, GridBoxOptions->{GridBaseline->Center, RowSpacings->0, ColumnSpacings->0, ColumnWidths->{3.5, 3.5, 3.5, 3.5, 13, 5, 4}, RowAlignments->Baseline, ColumnAlignments->{ Center, Center, Center, Center, Center, Center, Right, Center}}], Cell[StyleData["SlideShowNavigationBar", "Presentation"]], Cell[StyleData["SlideShowNavigationBar", "SlideShow"], Deletable->False, ShowCellBracket->False, CellMargins->{{-1, -1}, {-1, -1}}, PageBreakAbove->True, CellFrameMargins->{{1, 1}, {0, 0}}], Cell[StyleData["SlideShowNavigationBar", "Printout"], CellMargins->{{18, 4}, {4, 4}}, LineSpacing->{1, 3}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SlideShowSection"], CellFrame->{{0, 0}, {0, 0.5}}, CellMargins->{{0, 0}, {10, 0}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CellFrameMargins->{{12, 4}, {6, 12}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->18, FontWeight->"Plain", FontColor->GrayLevel[1], Background->RGBColor[0.408011, 0.440726, 0.8]], Cell[StyleData["SlideShowSection", "Presentation"], CellFrameMargins->{{18, 10}, {10, 18}}, FontSize->27], Cell[StyleData["SlideShowSection", "SlideShow"], ShowCellBracket->False, PageBreakAbove->True], Cell[StyleData["SlideShowSection", "Printout"], CellMargins->{{18, 30}, {0, 30}}, CellFrameMargins->5, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SlideHyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontSize->26, FontColor->GrayLevel[0.400015], ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonMinHeight->0.85, ButtonMargins->0.5, ButtonNote->None}], Cell[StyleData["SlideHyperlink", "Presentation"], CellMargins->{{10, 10}, {10, 12}}, FontSize->36], Cell[StyleData["SlideHyperlink", "SlideShow"], FontSize->26], Cell[StyleData["SlideHyperlink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SlideTOCLink"], CellMargins->{{24, Inherited}, {Inherited, Inherited}}, StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Helvetica", ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonMargins->1.5, ButtonNote->ButtonData}], Cell[StyleData["SlideTOCLink", "Presentation"], CellMargins->{{35, 10}, {15, 12}}, FontSize->18], Cell[StyleData["SlideTOCLink", "SlideShow"], FontSize->12], Cell[StyleData["SlideTOCLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SlideTOC"], CellDingbat->"\[Bullet]", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, StyleMenuListing->None, FontFamily->"Helvetica"], Cell[StyleData["SlideTOC", "Presentation"], CellMargins->{{25, 10}, {10, 10}}, FontSize->18], Cell[StyleData["SlideTOC", "SlideShow"], FontSize->14], Cell[StyleData["SlideTOC", "Printout"], FontSize->10, FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell["\<\ The following styles are useful for numbered equations, figures, etc. They \ automatically give the cell a FrameLabel containing a reference to a \ particular counter, and also increment that counter.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CellMargins->{{55, 10}, {0, 10}}, CellFrameLabels->{{None, Cell[ TextData[ {"(", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, CounterIncrements->"NumberedEquation", FormatTypeAutoConvert->False], Cell[StyleData["NumberedEquation", "Presentation"], CellMargins->{{80, 10}, {0, 20}}, FontSize->18], Cell[StyleData["NumberedEquation", "Printout"], CellMargins->{{55, 55}, {0, 10}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellMargins->{{55, 145}, {2, 10}}, CellHorizontalScrolling->True, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}], FontWeight -> "Bold"], None}}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False], Cell[StyleData["NumberedFigure", "Presentation"], CellMargins->{{80, 10}, {0, 20}}, FontSize->18], Cell[StyleData["NumberedFigure", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{55, 145}, {2, 10}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}], FontWeight -> "Bold"], None}}, TextAlignment->Center, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False], Cell[StyleData["NumberedTable", "Presentation"], CellMargins->{{80, 10}, {0, 20}}, FontSize->18], Cell[StyleData["NumberedTable", "Printout"], CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{55, 10}, {2, 10}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], CellMargins->{{80, 10}, {5, 25}}, FontSize->18], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ChemicalFormula"], CellMargins->{{55, 10}, {2, 10}}, DefaultFormatType->DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", AutoSpacing->False, ScriptLevel->1, ScriptBaselineShifts->{0.6, Automatic}, SingleLetterItalics->False, ZeroWidthTimes->True], Cell[StyleData["ChemicalFormula", "Presentation"], CellMargins->{{80, 10}, {5, 15}}, FontSize->18], Cell[StyleData["ChemicalFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{18, 10}, {Inherited, 6}}, Hyphenation->False, LanguageCategory->"Formula", FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{25, 10}, {8, 20}}, FontSize->16], Cell[StyleData["Program", "Printout"], CellMargins->{{18, 30}, {Inherited, 4}}, FontSize->9.5] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Notation Package Styles", "Section", CellTags->"NotationPackage"], Cell["\<\ The cells below define certain styles needed by the Notation package. These \ styles serve to make visible otherwise invisible tagboxes.\ \>", "Text", CellTags->"NotationPackage"], Cell[StyleData["NotationTemplateStyle"], StyleMenuListing->None, Background->RGBColor[1, 1, 0.850004], TagBoxOptions->{SyntaxForm->"symbol"}, CellTags->"NotationPackage"], Cell[StyleData["NotationPatternWrapperStyle"], StyleMenuListing->None, Background->RGBColor[1, 0.900008, 0.979995], TagBoxOptions->{SyntaxForm->"symbol"}, CellTags->"NotationPackage"], Cell[StyleData["NotationMadeBoxesWrapperStyle"], StyleMenuListing->None, Background->RGBColor[0.900008, 0.889998, 1], TagBoxOptions->{SyntaxForm->"symbol"}, CellTags->"NotationPackage"] }, Closed]] }] ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 220, 7, 169, "Title"], Cell[1999, 62, 81, 2, 49, "Author"], Cell[2083, 66, 167, 7, 81, "Address"], Cell[2253, 75, 782, 19, 110, "Abstract"], Cell[3038, 96, 156, 3, 54, "Text"], Cell[CellGroupData[{ Cell[3219, 103, 36, 0, 77, "SectionFirst"], Cell[3258, 105, 2370, 56, 222, "Text"], Cell[5631, 163, 204, 5, 33, "Text"], Cell[5838, 170, 346, 6, 47, "DisplayFormula"], Cell[6187, 178, 585, 16, 54, "Text"], Cell[6775, 196, 463, 9, 235, "DisplayFormula"], Cell[7241, 207, 1391, 38, 99, "Text"], Cell[8635, 247, 266, 5, 118, "DisplayFormula"], Cell[8904, 254, 528, 16, 54, "Text"], Cell[9435, 272, 657, 12, 96, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[10129, 289, 33, 0, 69, "Section"], Cell[CellGroupData[{ Cell[10187, 293, 43, 0, 52, "Subsection"], Cell[10233, 295, 247, 5, 78, "Input"], Cell[10483, 302, 89, 2, 36, "Input"], Cell[10575, 306, 1011, 30, 99, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[11623, 341, 56, 0, 52, "Subsection"], Cell[11682, 343, 355, 7, 120, "Input"], Cell[12040, 352, 1015, 18, 232, "Input"], Cell[13058, 372, 569, 11, 120, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[13664, 388, 60, 0, 52, "Subsection"], Cell[13727, 390, 5038, 110, 222, "Input"], Cell[18768, 502, 246, 5, 36, "Input"], Cell[19017, 509, 1872, 33, 189, "Input"], Cell[20892, 544, 916, 16, 121, "Input"], Cell[21811, 562, 228, 4, 36, "Input"], Cell[22042, 568, 3070, 52, 459, "Input"], Cell[25115, 622, 156, 4, 36, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[25308, 631, 53, 0, 52, "Subsection"], Cell[25364, 633, 715, 16, 61, "Input"], Cell[26082, 651, 590, 10, 132, "Input"], Cell[26675, 663, 629, 11, 131, "Input"], Cell[27307, 676, 120, 3, 36, "Input"], Cell[27430, 681, 214, 4, 58, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[27693, 691, 49, 0, 69, "Section"], Cell[CellGroupData[{ Cell[27767, 695, 42, 0, 52, "Subsection"], Cell[27812, 697, 114, 3, 33, "Text"], Cell[27929, 702, 1852, 37, 183, "Input"], Cell[29784, 741, 211, 5, 33, "Text"], Cell[CellGroupData[{ Cell[30020, 750, 85, 2, 36, "Input"], Cell[30108, 754, 14438, 233, 2119, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[44583, 992, 85, 2, 36, "Input"], Cell[44671, 996, 456, 7, 51, "Message"], Cell[45130, 1005, 48258, 801, 6222, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[93425, 1811, 87, 2, 36, "Input"], Cell[93515, 1815, 456, 7, 51, "Message"], Cell[93974, 1824, 56822, 1006, 6600, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[150833, 2835, 87, 2, 36, "Input"], Cell[150923, 2839, 456, 7, 51, "Message"], Cell[151382, 2848, 43725, 783, 4637, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[195156, 3637, 43, 0, 52, "Subsection"], Cell[195202, 3639, 124, 3, 33, "Text"], Cell[195329, 3644, 520, 9, 144, "Input"], Cell[195852, 3655, 91, 2, 33, "Text"], Cell[195946, 3659, 431, 9, 102, "Input"], Cell[196380, 3670, 158, 4, 33, "Text"], Cell[CellGroupData[{ Cell[196563, 3678, 435, 8, 58, "Input"], Cell[197001, 3688, 973, 21, 144, "Output"] }, Open ]], Cell[197989, 3712, 158, 5, 33, "Text"], Cell[CellGroupData[{ Cell[198172, 3721, 442, 8, 58, "Input"], Cell[198617, 3731, 945, 21, 144, "Output"] }, Open ]], Cell[199577, 3755, 179, 5, 33, "Text"], Cell[CellGroupData[{ Cell[199781, 3764, 138, 3, 36, "Input"], Cell[199922, 3769, 6687, 172, 144, "Output"] }, Open ]], Cell[206624, 3944, 255, 8, 33, "Text"], Cell[CellGroupData[{ Cell[206904, 3956, 208, 4, 53, "Input"], Cell[207115, 3962, 115, 2, 53, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[207267, 3969, 173, 4, 52, "Input"], Cell[207443, 3975, 115, 2, 53, "Output"] }, Open ]], Cell[207573, 3980, 687, 17, 75, "Text"], Cell[CellGroupData[{ Cell[208285, 4001, 191, 3, 53, "Input"], Cell[208479, 4006, 571, 11, 97, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[209087, 4022, 154, 3, 52, "Input"], Cell[209244, 4027, 304, 5, 53, "Output"] }, Open ]], Cell[209563, 4035, 220, 8, 33, "Text"], Cell[CellGroupData[{ Cell[209808, 4047, 192, 4, 53, "Input"], Cell[210003, 4053, 441, 7, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[210481, 4065, 153, 3, 52, "Input"], Cell[210637, 4070, 547, 10, 53, "Output"] }, Open ]], Cell[211199, 4083, 353, 11, 33, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[211589, 4099, 53, 0, 52, "Subsection"], Cell[211645, 4101, 1399, 36, 138, "Text"], Cell[CellGroupData[{ Cell[213069, 4141, 37, 0, 39, "Subsubsection"], Cell[213109, 4143, 416, 9, 93, "Input"], Cell[213528, 4154, 463, 9, 57, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[214028, 4168, 56, 0, 39, "Subsubsection"], Cell[214087, 4170, 358, 7, 93, "Input"], Cell[214448, 4179, 447, 9, 57, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[214944, 4194, 46, 0, 52, "Subsection"], Cell[214993, 4196, 248, 4, 54, "Text"], Cell[215244, 4202, 376, 7, 78, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[215669, 4215, 37, 0, 69, "Section"], Cell[215709, 4217, 713, 14, 96, "Text"], Cell[216425, 4233, 1122, 24, 117, "Text"], Cell[CellGroupData[{ Cell[217572, 4261, 103, 4, 52, "Subsection"], Cell[217678, 4267, 3409, 61, 387, "Input"], Cell[221090, 4330, 98, 2, 36, "Input"], Cell[221191, 4334, 49, 0, 33, "Text"], Cell[CellGroupData[{ Cell[221265, 4338, 165, 4, 53, "Input"], Cell[221433, 4344, 113, 2, 36, "Output"] }, Open ]], Cell[221561, 4349, 67, 0, 33, "Text"], Cell[CellGroupData[{ Cell[221653, 4353, 135, 3, 37, "Input"], Cell[221791, 4358, 114, 2, 36, "Output"] }, Open ]], Cell[221920, 4363, 278, 9, 33, "Text"], Cell[CellGroupData[{ Cell[222223, 4376, 178, 4, 36, "Input"], Cell[222404, 4382, 610, 14, 104, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[223051, 4401, 199, 4, 36, "Input"], Cell[223253, 4407, 563, 14, 104, "Output"] }, Open ]], Cell[223831, 4424, 973, 27, 96, "Text"], Cell[CellGroupData[{ Cell[224829, 4455, 63, 2, 36, "Input"], Cell[224895, 4459, 431, 8, 78, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[225363, 4472, 63, 2, 36, "Input"], Cell[225429, 4476, 747, 13, 99, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[226213, 4494, 78, 2, 36, "Input"], Cell[226294, 4498, 7423, 151, 750, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[233754, 4654, 78, 2, 36, "Input"], Cell[233835, 4658, 7160, 143, 795, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[241032, 4806, 77, 2, 36, "Input"], Cell[241112, 4810, 1493, 27, 316, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[242642, 4842, 77, 2, 36, "Input"], Cell[242722, 4846, 4804, 88, 886, "Output"] }, Open ]], Cell[247541, 4937, 2154, 70, 138, "Text"], Cell[CellGroupData[{ Cell[249720, 5011, 490, 9, 58, "Input"], Cell[250213, 5022, 1543, 30, 126, "Output"] }, Open ]], Cell[251771, 5055, 129, 3, 33, "Text"], Cell[251903, 5060, 296, 5, 114, "DisplayFormula"], Cell[252202, 5067, 253, 6, 33, "Text"], Cell[CellGroupData[{ Cell[252480, 5077, 197, 4, 36, "Input"], Cell[252680, 5083, 564, 14, 104, "Output"] }, Open ]], Cell[253259, 5100, 304, 8, 33, "Text"], Cell[CellGroupData[{ Cell[253588, 5112, 205, 4, 36, "Input"], Cell[253796, 5118, 533, 14, 104, "Output"] }, Open ]], Cell[254344, 5135, 273, 8, 33, "Text"], Cell[254620, 5145, 1200, 27, 117, "Text"], Cell[CellGroupData[{ Cell[255845, 5176, 216, 5, 57, "Input"], Cell[256064, 5183, 591, 11, 78, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[256692, 5199, 168, 3, 36, "Input"], Cell[256863, 5204, 294, 5, 57, "Output"] }, Open ]], Cell[257172, 5212, 731, 21, 75, "Text"], Cell[CellGroupData[{ Cell[257928, 5237, 168, 4, 57, "Input"], Cell[258099, 5243, 587, 11, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[258723, 5259, 248, 5, 57, "Input"], Cell[258974, 5266, 306, 6, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[259317, 5277, 95, 2, 36, "Input"], Cell[259415, 5281, 4016, 52, 372, "Output"] }, Open ]], Cell[263446, 5336, 308, 8, 54, "Text"], Cell[CellGroupData[{ Cell[263779, 5348, 200, 5, 57, "Input"], Cell[263982, 5355, 363, 6, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[264382, 5366, 251, 5, 57, "Input"], Cell[264636, 5373, 299, 5, 57, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[264984, 5384, 103, 4, 52, "Subsection"], Cell[265090, 5390, 3402, 61, 387, "Input"], Cell[268495, 5453, 98, 2, 36, "Input"], Cell[268596, 5457, 49, 0, 33, "Text"], Cell[CellGroupData[{ Cell[268670, 5461, 165, 4, 53, "Input"], Cell[268838, 5467, 114, 2, 36, "Output"] }, Open ]], Cell[268967, 5472, 250, 6, 33, "Text"], Cell[CellGroupData[{ Cell[269242, 5482, 178, 4, 36, "Input"], Cell[269423, 5488, 614, 14, 104, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[270074, 5507, 199, 4, 36, "Input"], Cell[270276, 5513, 565, 14, 104, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[270878, 5532, 216, 5, 57, "Input"], Cell[271097, 5539, 591, 11, 78, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[271725, 5555, 169, 4, 57, "Input"], Cell[271897, 5561, 588, 11, 57, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[272534, 5578, 43, 0, 52, "Subsection"], Cell[272580, 5580, 116, 5, 33, "Text"], Cell[272699, 5587, 3421, 61, 387, "Input"], Cell[276123, 5650, 90, 2, 36, "Input"], Cell[CellGroupData[{ Cell[276238, 5656, 166, 4, 53, "Input"], Cell[276407, 5662, 115, 2, 36, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[276559, 5669, 179, 4, 36, "Input"], Cell[276741, 5675, 608, 14, 104, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[277386, 5694, 200, 4, 36, "Input"], Cell[277589, 5700, 564, 14, 104, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[278190, 5719, 208, 4, 57, "Input"], Cell[278401, 5725, 532, 14, 104, "Output"] }, Open ]], Cell[278948, 5742, 747, 19, 75, "Text"], Cell[CellGroupData[{ Cell[279720, 5765, 217, 5, 57, "Input"], Cell[279940, 5772, 534, 10, 75, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[280511, 5787, 169, 3, 36, "Input"], Cell[280683, 5792, 298, 6, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[281018, 5803, 169, 4, 57, "Input"], Cell[281190, 5809, 577, 11, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[281804, 5825, 249, 5, 57, "Input"], Cell[282056, 5832, 304, 5, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[282397, 5842, 198, 4, 36, "Input"], Cell[282598, 5848, 564, 14, 104, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[283211, 5868, 43, 0, 52, "Subsection"], Cell[283257, 5870, 116, 5, 33, "Text"], Cell[CellGroupData[{ Cell[283398, 5879, 3278, 57, 387, "Input"], Cell[286679, 5938, 2516, 41, 225, "Output"] }, Open ]], Cell[289210, 5982, 90, 2, 36, "Input"], Cell[CellGroupData[{ Cell[289325, 5988, 166, 4, 53, "Input"], Cell[289494, 5994, 114, 2, 36, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[289645, 6001, 179, 4, 36, "Input"], Cell[289827, 6007, 611, 14, 104, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[290475, 6026, 200, 4, 36, "Input"], Cell[290678, 6032, 567, 14, 104, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[291282, 6051, 208, 4, 57, "Input"], Cell[291493, 6057, 566, 14, 104, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[292096, 6076, 198, 4, 36, "Input"], Cell[292297, 6082, 565, 14, 104, "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[292923, 6103, 45, 0, 69, "Section"], Cell[292971, 6105, 237, 4, 54, "Text"], Cell[293211, 6111, 196, 4, 58, "Input"], Cell[CellGroupData[{ Cell[293432, 6119, 105, 2, 36, "Input"], Cell[293540, 6123, 98, 2, 36, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[293675, 6130, 106, 2, 36, "Input"], Cell[293784, 6134, 95, 2, 36, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[293916, 6141, 160, 4, 50, "Input"], Cell[294079, 6147, 95, 2, 36, "Output"] }, Open ]], Cell[294189, 6152, 87, 2, 33, "Text"], Cell[294279, 6156, 659, 13, 136, "Input"], Cell[CellGroupData[{ Cell[294963, 6173, 446, 8, 78, "Input"], Cell[295412, 6183, 1096, 25, 122, "Output"] }, Open ]], Cell[296523, 6211, 413, 10, 54, "Text"], Cell[CellGroupData[{ Cell[296961, 6225, 116, 2, 36, "Input"], Cell[297080, 6229, 586, 11, 113, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[297703, 6245, 137, 3, 36, "Input"], Cell[297843, 6250, 312, 6, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[298192, 6261, 117, 2, 36, "Input"], Cell[298312, 6265, 592, 12, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[298941, 6282, 137, 3, 36, "Input"], Cell[299081, 6287, 315, 5, 57, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[299433, 6297, 171, 4, 50, "Input"], Cell[299607, 6303, 548, 11, 88, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[300192, 6319, 137, 3, 36, "Input"], Cell[300332, 6324, 320, 6, 57, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[300701, 6336, 30, 0, 69, "Section"], Cell[300734, 6338, 2898, 76, 230, "Text"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)