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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[ 109312, 2041]*) (*NotebookOutlinePosition[ 110774, 2088]*) (* CellTagsIndexPosition[ 110538, 2078]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Perturbations`DisturbingFunction", StyleBox["`", "MB"] }], "Subsection", CellTags->"Perturbations`DisturbingFunction"], Cell["\<\ S and S' being two planets or satellites, internal and external \ respectively, the disturbing functions of the two bodies are expressed by \ means of the following formulae, where the symbols having a prime refer to \ the external object:\ \>", "Text"], Cell[BoxData[{ \(R = \(\(G\ m'\)\/a'\) \((RD - \(r\/r'\^2\) cos \((\[Psi])\))\)\n\), "\[IndentingNewLine]", \(R' = \(\(G\ m\)\/a'\) \((RD - \(r'\/\(\(r\)\(\ \)\)\^2\) cos \((\[Psi])\))\)\n\), "\[IndentingNewLine]", \(RD = a'\/\@\(r\^2 + r'\^2 - 2\ r\ r' cos \ \((\[Psi])\)\)\[IndentingNewLine]\), "\n", \({m, m'} = mass\n\), "\[IndentingNewLine]", \({r, r'} = radius\ vector\n\), "\[IndentingNewLine]", \({a, a'} = semi - major\ axis\n\), "\[IndentingNewLine]", \(\[Psi] = angle\ \((r\ , r')\)\n\), "\[IndentingNewLine]", \(\[Alpha] = a\/a'\n\), "\[IndentingNewLine]", \(G = gravitational\ constant\)}], "Input", ShowCellBracket->False, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["\<\ RD is the so-called direct part; the other two are denominated \ indirect parts. For further clarity, here is a list of the symbols adopted \ for the notation of elements:\ \>", "Text"], Cell[BoxData[{ \(a = semi - major\ axis\[IndentingNewLine]\), "\n", \(e = eccentricity\[IndentingNewLine]\), "\n", \(\[GothicCapitalI] = inclination\n\), "\[IndentingNewLine]", \(\[CapitalOmega] = longitude\ of\ the\ ascending\ node\n\), "\[IndentingNewLine]", \(\[CurlyPi] = longitude\ of\ the\ pericentre\n\), "\[IndentingNewLine]", \(L = \(M + \[CurlyPi] = mean\ longitude\)\n\), "\[IndentingNewLine]", \(\[Sigma] = sin \((\[GothicCapitalI]\/2)\)\)}], "Input", ShowCellBracket->False, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["In expanding RD, Laplace coefficients are used:", "Text"], Cell[BoxData[{ \(\(b\_s\^\((j)\)\) \((\[Alpha])\)\[IndentingNewLine]\), "\ \[IndentingNewLine]", \(s = 1\/2, 3\/2, 5\/2, \[Ellipsis]\)}], "Input", ShowCellBracket->False, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["and in the package, these are represented in this way:", "Text"], Cell[BoxData[{ \(b[s, j, 0] = \(b\_s\^\((j)\)\) \((\[Alpha])\)\n\), "\n", \(b[s, j, p] = \(d\ \(b\_s\^\((j)\)\) \((\[Alpha])\)\)\/\(d\ \[Alpha]\^p\)\)}], \ "Input", ShowCellBracket->False, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["The expansion of RD has the following form:", "Text"], Cell[BoxData[ \(RD = \[Sum]\+\(j = \(-\[Infinity]\)\)\%\(+\[Infinity]\)RDj\)], "Input", ShowCellBracket->False, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["\<\ j is left unevaluated here, that is, an expansion of RDj is given \ in which the coefficients of the various cos( ) are expressed as a function \ of parameter j, of Laplace coefficients and of the elements e, e', \[Sigma], \ \[Sigma]'. Such coefficients contain a limited number of terms, according to \ the condition:\ \>", "Text"], Cell[TextData[{ " ", Cell[BoxData[ RowBox[{\(\(\[Sigma]\^w\) \(\[Sigma]'\^x\) \(e\^y\) e'\^z\), "=", RowBox[{"0", Cell[""]}]}]]], "if w+x+y+z \[LessEqual] ord" }], "Text"], Cell["\<\ ord is the order (degree) of the expansion. I have found that:\ \>", "Text"], Cell[BoxData[{ \(RDj\ \((ord)\) = \[Sum]\+\(h = H1\)\%H2\(\[Sum]\+\(k = K1\)\%K2\(\[Sum]\ \+\(s = S1\)\%S2\(\[Sum]\+\(t = T1\)\%T2 coef \((h, k, s, t)\) cos \((\((2\ h - j + s + t)\)\ L + j\ L' - s\ \[CurlyPi] - t\ \[CurlyPi]' - \((h + k)\)\ \[CapitalOmega] - \((h - k)\)\ \[CapitalOmega]')\)\)\)\)\n\), "\ \[IndentingNewLine]", \(coef \((h, k, s, t)\) = \[Sum]\+\(z1 = Z1\)\%ZZ1\(\[Sum]\+\(z2 = Z2\)\%ZZ2\(\[Sum]\+\ \(z3 = Z3\)\%ZZ3\(\[Sum]\+\(z4 = Z4\)\%ZZ4 coef1 \((h, k, s, t, z1, z2, z3, z4)\) \[Sigma]\^\(\(\(2\ z1\)\(+\)\) | h + k | \)\ \ \[Sigma]'\^\(\(\(2\ z2\)\(+\)\) | h - k | \)\ e\^\(\(\(2\ z3\)\(+\)\) | s | \ \)\ e'\^\(\(\(2\ z4\)\(+\)\) | t | \)\)\)\)\n\), "\[IndentingNewLine]", \(coef1 \((h, k, s, t, z1, z2, z3, z4)\) = \[Sum]\+\(i = I1\)\%I2\(\[Sum]\+\(p = P1\)\%P2\(\[Sum]\+\(u \ = U1\)\%U2 coef2 \((h, k, s, t, z1, z2, z3, z4, i, p, u)\)\ \[Alpha]\^\(i + u\)\ b[i + 1\/2, \(-h\) + j + p - t, u]\)\)\)}], "Input", ShowCellBracket->False, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell["\<\ coef2 is a polynomial in j, and has rational coefficients. I \ elaborated all the summations in order to supply different terms, preventing \ the emergence of opposite arguments or of equal monomials. coef2 is a \ combination of numerical functions, that I determined by studying the general \ formula for the series expansion of the Kaula functions and of Hansen \ coefficients, making no use of any recursive relations. The package defines the following functions:\ \>", "Text"], Cell[BoxData[ FormBox[ StyleBox[ FrameBox[ StyleBox[GridBox[{ {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"DFExpand[\\\\\\\"\\\", \\\ \"MR\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"ord\\\\\\\"\\\", \ \\\"TI\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"]\\\\\\\"\\\", \ \\\"MR\\\"]\\)\"\>", "\<\"gives the expansion of RDj up to order \ \\!\\(\\*StyleBox[\\\"\\\\\\\"ord\\\\\\\"\\\", \\\"TI\\\"]\\)\"\>"}, {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"DFArguments[\\\\\\\"\\\", \ \\\"MR\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"ord\\\\\\\"\\\", \ \\\"TI\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"]\\\\\\\"\\\", \ \\\"MR\\\"]\\)\"\>", "\<\"gives the list of arguments of RDj up to order \ \\!\\(\\*StyleBox[\\\"\\\\\\\"ord\\\\\\\"\\\", \\\"TI\\\"]\\)\"\>"}, {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"DFCoefficient[\\\\\\\"\\\"\ , \\\"MR\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"arg\\\\\\\"\\\", \\\"TI\\\"]\\)\ \\!\\(\\*StyleBox[\\\"\\\\\\\",\\\\\\\"\\\", \\\"MR\\\"]\\) \ \\!\\(\\*StyleBox[\\\"\\\\\\\"ord\\\\\\\"\\\", \ \\\"TI\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"]\\\\\\\"\\\", \\\"MR\\\"]\\) \ \"\>", "\<\"gives the coefficient of argument \ \\!\\(\\*StyleBox[\\\"\\\\\\\"arg\\\\\\\"\\\", \\\"TI\\\"]\\) of RD or of RDj \ up to order \\!\\(\\*StyleBox[\\\"\\\\\\\"ord\\\\\\\"\\\", \ \\\"TI\\\"]\\)\"\>"}, {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"AddIndirectTerms[\\\\\\\"\ \\\", \\\"MR\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"expr\\\\\\\"\\\", \ \\\"TI\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"]\\\\\\\"\\\", \ \\\"MR\\\"]\\)\"\>", "\<\"introduces the contribution of indirect terms in \ the expression \\!\\(\\*StyleBox[\\\"\\\\\\\"expr\\\\\\\"\\\", \\\"TI\\\"]\\)\ \"\>"} }, ColumnAlignments->{Right, Left}], GridBoxOptions->{RowLines->False}]], "2ColumnBox"], TraditionalForm]], "Text"], Cell["Structure of the disturbing function.", "Caption"], Cell["This loads the package. ", "MathCaption"], Cell["<"In[1]:=", ShowCellBracket->True], Cell["DFExpand", "Subsubsection", CellTags->"DFExpand"], Cell["\<\ The Appendix B of the textbook \"Solar System Dynamics\" di \ C.D.Murray e S.F.Dermott (Cambridge University Press, 1999) [here referred to \ as SSD] contains the expansion of RD up to the fourth order. This is the \ result obtained with DFExpand[4]:\ \>", "MathCaption"], Cell[CellGroupData[{ Cell["DFExpand[4]", "Input", CellLabel->"In[2]:=", ShowCellBracket->True], Cell[BoxData[ \(\((1\/2\ b[1\/2, j, 0] + e\^2\ \((\(-\(1\/2\)\)\ j\^2\ b[1\/2, j, 0] + 1\/4\ \[Alpha]\ b[1\/2, j, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, j, 2])\) + e1\^2\ \((\(-\(1\/2\)\)\ j\^2\ b[1\/2, j, 0] + 1\/4\ \[Alpha]\ b[1\/2, j, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, j, 2])\) + e\^4\ \((\((\(-\(\(9\ j\^2\)\/128\)\) + j\^4\/8)\)\ b[1\/2, j, 0] - 1\/16\ j\^2\ \[Alpha]\ b[1\/2, j, 1] - 1\/16\ j\^2\ \[Alpha]\^2\ b[1\/2, j, 2] + 1\/32\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/128\ \[Alpha]\^4\ b[1\/2, j, 4])\) + e1\^4\ \((\((\(-\(\(17\ j\^2\)\/128\)\) + j\^4\/8)\)\ b[1\/2, j, 0] + \((3\/16 - \(3\ j\^2\)\/16)\)\ \[Alpha]\ b[1\/2, j, 1] + \((9\/32 - j\^2\/16)\)\ \[Alpha]\^2\ b[1\/2, j, 2] + 3\/32\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/128\ \[Alpha]\^4\ b[1\/2, j, 4])\) + e\^2\ e1\^2\ \((1\/2\ j\^4\ b[1\/2, j, 0] + \((1\/8 - j\^2\/2)\)\ \[Alpha]\ b[1\/2, j, 1] + \((7\/16 - j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, j, 2] + 1\/4\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/32\ \[Alpha]\^4\ b[1\/2, j, 4])\) + \[Sigma]\^2\ \((\(-\(1\/4\)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\ b[3\/2, 1 + j, 0])\) + \[Sigma]1\^2\ \((\(-\(1\/4\)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\ b[3\/2, 1 + j, 0])\) + e\^2\ \[Sigma]\^2\ \((\((\(-\(1\/8\)\) + j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2] + \((\(-\(1\/8\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\) + e1\^2\ \[Sigma]\^2\ \((\((\(-\(1\/8\)\) + j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2] + \((\(-\(1\/8\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\) + e\^2\ \[Sigma]1\^2\ \((\((\(-\(1\/8\)\) + j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2] + \((\(-\(1\/8\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\) + e1\^2\ \[Sigma]1\^2\ \((\((\(-\(1\/8\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2] + \((\(-\(1\/8\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\) + \[Sigma]\^4\ \((3\/16\ \[Alpha]\^2\ b[ 5\/2, \(-2\) + j, 0] + 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0] + 3\/16\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\) + \[Sigma]1\^4\ \((3\/16\ \[Alpha]\^2\ b[ 5\/2, \(-2\) + j, 0] + 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0] + 3\/16\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\) + \[Sigma]\^2\ \[Sigma]1\^2\ \((1\/4\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/4\ \[Alpha]\ b[3\/2, 1 + j, 0] + 3\/8\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] + 15\/4\ \[Alpha]\^2\ b[5\/2, j, 0] + 3\/8\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\))\)\ cos[\(-j\)\ L + j\ L1] + e\^4\ \((\((\(-\(\(103\ j\)\/192\)\) + \(283\ j\^2\)\/384 - \(5\ j\^3\)\ \/16 + j\^4\/24)\)\ b[1\/2, j, 0] + \((\(-\(1\/6\)\) + \(59\ j\)\/96 - \(7\ j\^2\)\/16 + j\^3\/12)\)\ \[Alpha]\ b[1\/2, j, 1] + \((1\/8 - \(13\ j\)\/64 + j\^2\/16)\)\ \[Alpha]\^2\ b[ 1\/2, j, 2] + \((\(-\(1\/32\)\) + j\/48)\)\ \[Alpha]\^3\ b[ 1\/2, j, 3] + 1\/384\ \[Alpha]\^4\ b[1\/2, j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - 4\ \[CurlyPi]] + e\^3\ \((\((\(-\(\(13\ j\)\/24\)\) + \(5\ j\^2\)\/8 - j\^3\/6)\)\ b[ 1\/2, j, 0] + \((\(-\(3\/16\)\) + \(9\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[1\/2, j, 1] + \((1\/8 - j\/8)\)\ \[Alpha]\^2\ b[1\/2, j, 2] - 1\/48\ \[Alpha]\^3\ b[1\/2, j, 3])\)\ cos[\((3 - j)\)\ L + j\ L1 - 3\ \[CurlyPi]] + \((e\^2\ \((\((\(-\(\(5\ j\)\/8\)\) + j\^2\/2)\)\ b[1\/2, j, 0] + \((\(-\(1\/4\)\) + j\/2)\)\ \[Alpha]\ b[1\/2, j, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, j, 2])\) + e\^4\ \((\((\(11\ j\)\/48 - \(2\ j\^2\)\/3 + \(5\ j\^3\)\/8 - j\^4\/6)\)\ b[1\/2, j, 0] + \((1\/6 - \(23\ j\)\/48 + j\^2\/2 - j\^3\/6)\)\ \[Alpha]\ b[1\/2, j, 1] + \((\(-\(1\/8\)\) + \(3\ j\)\/32)\)\ \[Alpha]\^2\ b[ 1\/2, j, 2] + 1\/24\ j\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/96\ \[Alpha]\^4\ b[1\/2, j, 4])\) + e\^2\ e1\^2\ \((\((\(5\ j\^3\)\/8 - j\^4\/2)\)\ b[1\/2, j, 0] + \((\(-\(1\/8\)\) - j\/16 + j\^2\/2 - j\^3\/2)\)\ \[Alpha]\ b[1\/2, j, 1] + \((\(-\(1\/16\)\) + \(11\ j\)\/32)\)\ \[Alpha]\^2\ \ b[1\/2, j, 2] + \((1\/8 + j\/8)\)\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/32\ \[Alpha]\^4\ b[1\/2, j, 4])\) + e\^2\ \[Sigma]\^2\ \((\((1\/8 + j\/16 - j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2] + \((1\/8 + j\/16 - j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\) + e\^2\ \[Sigma]1\^2\ \((\((1\/8 + j\/16 - j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2] + \((1\/8 + j\/16 - j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]] + \((e\ \((\(-j\)\ b[1\/2, j, 0] - 1\/2\ \[Alpha]\ b[1\/2, j, 1])\) + e\ e1\^2\ \((j\^3\ b[1\/2, j, 0] + \((\(-\(1\/4\)\) - j\/2 + j\^2\/2)\)\ \[Alpha]\ b[ 1\/2, j, 1] + \((\(-\(1\/2\)\) - j\/4)\)\ \[Alpha]\^2\ b[1\/2, j, 2] - 1\/8\ \[Alpha]\^3\ b[1\/2, j, 3])\) + e\^3\ \((\((j\/8 - \(5\ j\^2\)\/8 + j\^3\/2)\)\ b[1\/2, j, 0] + \((3\/16 - \(7\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 1\/2, j, 1] + \((\(-\(1\/8\)\) - j\/8)\)\ \[Alpha]\^2\ b[1\/2, j, 2] - 1\/16\ \[Alpha]\^3\ b[1\/2, j, 3])\) + e\ \[Sigma]\^2\ \((\((1\/4 + j\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + \((1\/4 + j\/2)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, 1 + j, 1])\) + e\ \[Sigma]1\^2\ \((\((1\/4 + j\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + \((1\/4 + j\/2)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, 1 + j, 1])\))\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi]] + e1\^4\ \((\((1\/16 - \(73\ j\)\/192 + \(211\ j\^2\)\/384 - \(13\ j\^3\)\ \/48 + j\^4\/24)\)\ b[1\/2, \(-4\) + j, 0] + \((\(-\(1\/16\)\) + \(29\ j\)\/96 - \(5\ j\^2\)\/16 + j\^3\/12)\)\ \[Alpha]\ b[1\/2, \(-4\) + j, 1] + \((1\/32 - \(7\ j\)\/64 + j\^2\/16)\)\ \[Alpha]\^2\ b[ 1\/2, \(-4\) + j, 2] + \((\(-\(1\/96\)\) + j\/48)\)\ \[Alpha]\^3\ b[ 1\/2, \(-4\) + j, 3] + 1\/384\ \[Alpha]\^4\ b[1\/2, \(-4\) + j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - 4\ \[CurlyPi]1] + e1\^3\ \((\((\(-\(1\/8\)\) + \(29\ j\)\/48 - \(5\ j\^2\)\/8 + j\^3\/6)\)\ b[1\/2, \(-3\) + j, 0] + \((1\/8 - \(7\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 1\/2, \(-3\) + j, 1] + \((\(-\(1\/16\)\) + j\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-3\) + j, 2] + 1\/48\ \[Alpha]\^3\ b[1\/2, \(-3\) + j, 3])\)\ cos[\((3 - j)\)\ L + j\ L1 - 3\ \[CurlyPi]1] + e\ e1\^3\ \((\((\(-\(3\/8\)\) + \(31\ j\)\/16 - \(119\ j\^2\)\/48 + \(9\ \ j\^3\)\/8 - j\^4\/6)\)\ b[1\/2, \(-3\) + j, 0] + \((3\/8 - \(73\ j\)\/48 + \(11\ j\^2\)\/8 - j\^3\/3)\)\ \[Alpha]\ b[1\/2, \(-3\) + j, 1] + \((\(-\(3\/16\)\) + \(17\ j\)\/32 - j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, \(-3\) + j, 2] + \((1\/16 - j\/12)\)\ \[Alpha]\^3\ b[1\/2, \(-3\) + j, 3] - 1\/96\ \[Alpha]\^4\ b[1\/2, \(-3\) + j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - \[CurlyPi] - 3\ \[CurlyPi]1] + e\ e1\^3\ \((\((3\/8 - \(31\ j\)\/16 + \(119\ j\^2\)\/48 - \(9\ \ j\^3\)\/8 + j\^4\/6)\)\ b[1\/2, \(-3\) + j, 0] + \((\(-\(3\/8\)\) + \(65\ j\)\/48 - j\^2 + j\^3\/6)\)\ \[Alpha]\ b[1\/2, \(-3\) + j, 1] + \((3\/16 - \(11\ j\)\/32)\)\ \[Alpha]\^2\ b[ 1\/2, \(-3\) + j, 2] + \((\(-\(1\/16\)\) - j\/24)\)\ \[Alpha]\^3\ b[ 1\/2, \(-3\) + j, 3] - 1\/96\ \[Alpha]\^4\ b[1\/2, \(-3\) + j, 4])\)\ cos[\((2 - j)\)\ L + j\ L1 + \[CurlyPi] - 3\ \[CurlyPi]1] + \((e1\^2\ \((\((1\/4 - \(7\ j\)\/8 + j\^2\/2)\)\ b[1\/2, \(-2\) + j, 0] + \((\(-\(1\/4\)\) + j\/2)\)\ \[Alpha]\ b[ 1\/2, \(-2\) + j, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, \(-2\) + j, 2])\) + e1\^4\ \((\((1\/8 - \(7\ j\)\/48 - \(5\ j\^2\)\/12 + \(13\ j\^3\)\ \/24 - j\^4\/6)\)\ b[1\/2, \(-2\) + j, 0] + \((\(-\(1\/8\)\) - \(5\ j\)\/48 + j\^2\/2 - j\^3\/6)\)\ \[Alpha]\ b[1\/2, \(-2\) + j, 1] + \((1\/16 + \(9\ j\)\/32)\)\ \[Alpha]\^2\ b[ 1\/2, \(-2\) + j, 2] + \((1\/12 + j\/24)\)\ \[Alpha]\^3\ b[ 1\/2, \(-2\) + j, 3] + 1\/96\ \[Alpha]\^4\ b[1\/2, \(-2\) + j, 4])\) + e\^2\ e1\^2\ \((\((\(-1\) + \(9\ j\)\/2 - \(23\ j\^2\)\/4 + \(23\ \ j\^3\)\/8 - j\^4\/2)\)\ b[1\/2, \(-2\) + j, 0] + \((1 - \(51\ j\)\/16 + \(5\ j\^2\)\/2 - j\^3\/2)\)\ \[Alpha]\ b[1\/2, \(-2\) + j, 1] + \((\(-\(1\/2\)\) + \(25\ j\)\/32)\)\ \[Alpha]\^2\ \ b[1\/2, \(-2\) + j, 2] + \((1\/8 + j\/8)\)\ \[Alpha]\^3\ b[1\/2, \(-2\) + j, 3] + 1\/32\ \[Alpha]\^4\ b[1\/2, \(-2\) + j, 4])\) + e1\^2\ \[Sigma]\^2\ \((\((\(3\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-3\) + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2] + \((\(3\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + e1\^2\ \[Sigma]1\^2\ \((\((\(3\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-3\) + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2] + \((\(3\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1] + e\^2\ e1\^2\ \((\((13\/16 - \(7\ j\)\/2 + \(259\ j\^2\)\/64 - \(7\ j\^3\ \)\/4 + j\^4\/4)\)\ b[1\/2, \(-2\) + j, 0] + \((\(-\(13\/16\)\) + \(11\ j\)\/4 - \(9\ j\^2\)\/4 + j\^3\/2)\)\ \[Alpha]\ b[1\/2, \(-2\) + j, 1] + \((13\/32 - \(15\ j\)\/16 + \(3\ j\^2\)\/8)\)\ \ \[Alpha]\^2\ b[1\/2, \(-2\) + j, 2] + \((\(-\(1\/8\)\) + j\/8)\)\ \[Alpha]\^3\ b[ 1\/2, \(-2\) + j, 3] + 1\/64\ \[Alpha]\^4\ b[1\/2, \(-2\) + j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - 2\ \[CurlyPi]1] + e\ e1\^2\ \((\((1\/2 - 2\ j + \(15\ j\^2\)\/8 - j\^3\/2)\)\ b[ 1\/2, \(-2\) + j, 0] + \((\(-\(1\/2\)\) + \(23\ j\)\/16 - \(3\ j\^2\)\/4)\)\ \ \[Alpha]\ b[1\/2, \(-2\) + j, 1] + \((1\/4 - \(3\ j\)\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-2\) + j, 2] - 1\/16\ \[Alpha]\^3\ b[1\/2, \(-2\) + j, 3])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi] - 2\ \[CurlyPi]1] + e\ e1\^2\ \((\((\(-\(1\/2\)\) + 2\ j - \(15\ j\^2\)\/8 + j\^3\/2)\)\ b[ 1\/2, \(-2\) + j, 0] + \((1\/2 - \(17\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 1\/2, \(-2\) + j, 1] + \((\(-\(1\/4\)\) - j\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-2\) + j, 2] - 1\/16\ \[Alpha]\^3\ b[1\/2, \(-2\) + j, 3])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CurlyPi]1] + \((e1\ \((\((\(-\(1\/2\)\) + j)\)\ b[ 1\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\ b[1\/2, \(-1\) + j, 1])\) + e1\^3\ \((\((\(-\(1\/8\)\) - j\/16 + \(5\ j\^2\)\/8 - j\^3\/2)\)\ b[1\/2, \(-1\) + j, 0] + \((1\/8 + \(9\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[ 1\/2, \(-1\) + j, 1] + \((5\/16 + j\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-1\) + j, 2] + 1\/16\ \[Alpha]\^3\ b[1\/2, \(-1\) + j, 3])\) + e\^2\ e1\ \((\((1\/2 - 2\ j + \(5\ j\^2\)\/2 - j\^3)\)\ b[ 1\/2, \(-1\) + j, 0] + \((\(-\(1\/2\)\) + \(3\ j\)\/2 - j\^2\/2)\)\ \[Alpha]\ b[1\/2, \(-1\) + j, 1] + \((3\/8 + j\/4)\)\ \[Alpha]\^2\ b[ 1\/2, \(-1\) + j, 2] + 1\/8\ \[Alpha]\^3\ b[1\/2, \(-1\) + j, 3])\) + e1\ \[Sigma]\^2\ \((\(-\(1\/2\)\)\ j\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] - 1\/2\ j\ \[Alpha]\ b[3\/2, j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, j, 1])\) + e1\ \[Sigma]1\^2\ \((\(-\(1\/2\)\)\ j\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] - 1\/2\ j\ \[Alpha]\ b[3\/2, j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, j, 1])\))\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi]1] + e\^3\ e1\ \((\((\(-\(2\/3\)\) + \(119\ j\)\/48 - \(137\ j\^2\)\/48 + \ \(29\ j\^3\)\/24 - j\^4\/6)\)\ b[1\/2, \(-1\) + j, 0] + \((2\/3 - \(103\ j\)\/48 + \(13\ j\^2\)\/8 - j\^3\/3)\)\ \[Alpha]\ b[1\/2, \(-1\) + j, 1] + \((\(-\(3\/8\)\) + \(23\ j\)\/32 - j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, \(-1\) + j, 2] + \((5\/48 - j\/12)\)\ \[Alpha]\^3\ b[1\/2, \(-1\) + j, 3] - 1\/96\ \[Alpha]\^4\ b[1\/2, \(-1\) + j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - 3\ \[CurlyPi] - \[CurlyPi]1] + e\^2\ e1\ \((\((\(-\(9\/16\)\) + \(31\ j\)\/16 - \(15\ j\^2\)\/8 + j\^3\/2)\)\ b[1\/2, \(-1\) + j, 0] + \((9\/16 - \(25\ j\)\/16 + \(3\ j\^2\)\/4)\)\ \[Alpha]\ \ b[1\/2, \(-1\) + j, 1] + \((\(-\(5\/16\)\) + \(3\ j\)\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-1\) + j, 2] + 1\/16\ \[Alpha]\^3\ b[1\/2, \(-1\) + j, 3])\)\ cos[\((3 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - \[CurlyPi]1] + \((e\ e1\ \((\((\(-\(1\/2\)\) + \ \(3\ j\)\/2 - j\^2)\)\ b[1\/2, \(-1\) + j, 0] + \((1\/2 - j)\)\ \[Alpha]\ b[ 1\/2, \(-1\) + j, 1] - 1\/4\ \[Alpha]\^2\ b[1\/2, \(-1\) + j, 2])\) + e\ e1\^3\ \((\((\(-\(1\/8\)\) + j\/16 + \(11\ j\^2\)\/16 - \(9\ j\^3\)\/8 + j\^4\/2)\)\ b[1\/2, \(-1\) + j, 0] + \((1\/8 + \(3\ j\)\/16 - j\^2 + j\^3\/2)\)\ \[Alpha]\ b[1\/2, \(-1\) + j, 1] + \((\(-\(1\/16\)\) - \(19\ j\)\/32)\)\ \[Alpha]\^2\ \ b[1\/2, \(-1\) + j, 2] + \((\(-\(3\/16\)\) - j\/8)\)\ \[Alpha]\^3\ b[ 1\/2, \(-1\) + j, 3] - 1\/32\ \[Alpha]\^4\ b[1\/2, \(-1\) + j, 4])\) + e\^3\ e1\ \((\((5\/8 - \(43\ j\)\/16 + \(63\ j\^2\)\/16 - \(19\ j\ \^3\)\/8 + j\^4\/2)\)\ b[1\/2, \(-1\) + j, 0] + \((\(-\(5\/8\)\) + \(37\ j\)\/16 - 2\ j\^2 + j\^3\/2)\)\ \[Alpha]\ b[1\/2, \(-1\) + j, 1] + \((7\/16 - \(17\ j\)\/32)\)\ \[Alpha]\^2\ b[ 1\/2, \(-1\) + j, 2] + \((\(-\(1\/16\)\) - j\/8)\)\ \[Alpha]\^3\ b[ 1\/2, \(-1\) + j, 3] - 1\/32\ \[Alpha]\^4\ b[1\/2, \(-1\) + j, 4])\) + e\ e1\ \[Sigma]\^2\ \((\((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2] + \((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\) + e\ e1\ \[Sigma]1\^2\ \((\((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2] + \((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1] + e\^3\ e1\ \((\((1\/24 - j\/48 - \(13\ j\^2\)\/48 - j\^3\/24 + j\^4\/6)\)\ b[1\/2, 1 + j, 0] + \((\(-\(1\/24\)\) - j\/48 + j\^3\/6)\)\ \[Alpha]\ b[ 1\/2, 1 + j, 1] + \((1\/16 - j\/32)\)\ \[Alpha]\^2\ b[1\/2, 1 + j, 2] + \((\(-\(1\/48\)\) - j\/24)\)\ \[Alpha]\^3\ b[ 1\/2, 1 + j, 3] - 1\/96\ \[Alpha]\^4\ b[1\/2, 1 + j, 4])\)\ cos[\((2 - j)\)\ L + j\ L1 - 3\ \[CurlyPi] + \[CurlyPi]1] + e\^2\ e1\ \((\((1\/16 - j\/16 - \(5\ j\^2\)\/8 - j\^3\/2)\)\ b[1\/2, 1 + j, 0] + \((\(-\(1\/16\)\) - j\/16 - j\^2\/4)\)\ \[Alpha]\ b[1\/2, 1 + j, 1] + \((3\/16 + j\/8)\)\ \[Alpha]\^2\ b[1\/2, 1 + j, 2] + 1\/16\ \[Alpha]\^3\ b[1\/2, 1 + j, 3])\)\ cos[\((1 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] + \[CurlyPi]1] + \((e\ e1\ \((\((1\/2 + \(3\ \ j\)\/2 + j\^2)\)\ b[1\/2, 1 + j, 0] - 1\/2\ \[Alpha]\ b[1\/2, 1 + j, 1] - 1\/4\ \[Alpha]\^2\ b[1\/2, 1 + j, 2])\) + e\ e1\^3\ \((\((1\/8 + j\/16 - \(11\ j\^2\)\/16 - \(9\ j\^3\)\/8 - j\^4\/2)\)\ b[1\/2, 1 + j, 0] + \((\(-\(1\/8\)\) + \(11\ j\)\/16 + \(5\ j\^2\)\/8)\ \)\ \[Alpha]\ b[1\/2, 1 + j, 1] + \((\(-\(11\/16\)\) + \(7\ j\)\/32 + j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, 1 + j, 2] - 5\/16\ \[Alpha]\^3\ b[1\/2, 1 + j, 3] - 1\/32\ \[Alpha]\^4\ b[1\/2, 1 + j, 4])\) + e\^3\ e1\ \((\((\(-\(\(3\ j\)\/16\)\) - \(13\ j\^2\)\/16 - \(9\ j\ \^3\)\/8 - j\^4\/2)\)\ b[1\/2, 1 + j, 0] + \((\(3\ j\)\/16 + \(3\ j\^2\)\/8)\)\ \[Alpha]\ b[ 1\/2, 1 + j, 1] + \((\(-\(1\/8\)\) + \(7\ j\)\/32 + j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, 1 + j, 2] - 3\/16\ \[Alpha]\^3\ b[1\/2, 1 + j, 3] - 1\/32\ \[Alpha]\^4\ b[1\/2, 1 + j, 4])\) + e\ e1\ \[Sigma]\^2\ \((\((\(-\(\(3\ j\)\/4\)\) - j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2] + \((\(-\(\(3\ j\)\/4\)\) - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, 2 + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, 2 + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, 2 + j, 2])\) + e\ e1\ \[Sigma]1\^2\ \((\((\(-\(\(3\ j\)\/4\)\) - j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2] + \((\(-\(\(3\ j\)\/4\)\) - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, 2 + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, 2 + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, 2 + j, 2])\))\)\ cos[\(-j\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1] + e\^2\ e1\^2\ \((\((3\/16 + j + \(109\ j\^2\)\/64 + \(9\ j\^3\)\/8 + j\^4\/4)\)\ b[1\/2, 2 + j, 0] + \((\(-\(3\/16\)\) - \(7\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[1\/2, 2 + j, 1] + \((3\/32 - \(7\ j\)\/32 - j\^2\/8)\)\ \[Alpha]\^2\ b[ 1\/2, 2 + j, 2] + 1\/8\ \[Alpha]\^3\ b[1\/2, 2 + j, 3] + 1\/64\ \[Alpha]\^4\ b[1\/2, 2 + j, 4])\)\ cos[\(-j\)\ L + j\ L1 - 2\ \[CurlyPi] + 2\ \[CurlyPi]1] + 3\/8\ \[Alpha]\^2\ \[Sigma]\^4\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - 4\ \[CapitalOmega]] + \((1\/2\ \[Alpha]\ \[Sigma]\^2\ b[ 3\/2, \(-1\) + j, 0] + e1\^2\ \[Sigma]\^2\ \((\((1\/4 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + e\^2\ \[Sigma]\^2\ \((\((\(-\(7\/4\)\) + 2\ j - j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + \[Sigma]\^2\ \[Sigma]1\^2\ \((\(-\(1\/2\)\)\ \ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 3\/4\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] - 15\/4\ \[Alpha]\^2\ b[5\/2, j, 0])\) + \[Sigma]\^4\ \((\(-\(3\/4\)\)\ \[Alpha]\^2\ b[ 5\/2, \(-2\) + j, 0] - 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CapitalOmega]] + e\^2\ \[Sigma]\^2\ \((\((1 - \(17\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + \((\(-\(1\/2\)\) + j\/4)\)\ \[Alpha]\^2\ b[ 3\/2, \(-1\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - 2\ \[CapitalOmega]] + e\ \[Sigma]\^2\ \((\((3\/4 - j\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi] - 2\ \[CapitalOmega]] + e\ \[Sigma]\^2\ \((\((\(-\(5\/4\)\) + j\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CapitalOmega]] + e\^2\ \[Sigma]\^2\ \((\((3\/4 - \(15\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + \((1\/2 - j\/4)\)\ \[Alpha]\^2\ b[ 3\/2, \(-1\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi] - 2\ \[CapitalOmega]] + e1\^2\ \[Sigma]\^2\ \((\((\(-\(\(3\ j\)\/16\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-3\) + j, 0] + 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]] + e1\ \[Sigma]\^2\ \((1\/2\ j\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi]1 - 2\ \[CapitalOmega]] + e\ e1\ \[Sigma]\^2\ \((\((\(5\ j\)\/4 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] + \((1\/2 - j\/2)\)\ \[Alpha]\^2\ b[ 3\/2, \(-2\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]] + e\ e1\ \[Sigma]\^2\ \((\((\(-\(\(7\ j\)\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] - \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]] + e1\ \[Sigma]\^2\ \((\(-\(1\/2\)\)\ j\ \[Alpha]\ b[3\/2, j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi]1 - 2\ \[CapitalOmega]] + e\ e1\ \[Sigma]\^2\ \((\((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] - 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]] + e\ e1\ \[Sigma]\^2\ \((\((\(3\ j\)\/4 - j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + \((\(-\(1\/2\)\) + j\/2)\)\ \[Alpha]\^2\ b[3\/2, j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\(-j\)\ L + j\ L1 + \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]] + e1\^2\ \[Sigma]\^2\ \((\((\(3\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]] + 3\/8\ \[Alpha]\^2\ \[Sigma]1\^4\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - 4\ \[CapitalOmega]1] - 3\/2\ \[Alpha]\^2\ \[Sigma]\ \[Sigma]1\^3\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - \[CapitalOmega] - 3\ \[CapitalOmega]1] + 3\/2\ \[Alpha]\^2\ \[Sigma]\ \[Sigma]1\^3\ b[5\/2, \(-2\) + j, 0]\ cos[\((2 - j)\)\ L + j\ L1 + \[CapitalOmega] - 3\ \[CapitalOmega]1] + \((1\/2\ \[Alpha]\ \[Sigma]1\^2\ b[ 3\/2, \(-1\) + j, 0] + e1\^2\ \[Sigma]1\^2\ \((\((1\/4 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + e\^2\ \[Sigma]1\^2\ \((\((\(-\(7\/4\)\) + 2\ j - j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + \[Sigma]\^2\ \[Sigma]1\^2\ \((\(-\(1\/2\)\)\ \ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 15\/4\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] - 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0])\) + \[Sigma]1\^4\ \((\(-\(3\/4\)\)\ \[Alpha]\^2\ b[ 5\/2, \(-2\) + j, 0] - 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CapitalOmega]1] + e\^2\ \[Sigma]1\^2\ \((\((1 - \(17\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + \((\(-\(1\/2\)\) + j\/4)\)\ \[Alpha]\^2\ b[ 3\/2, \(-1\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - 2\ \[CapitalOmega]1] + e\ \[Sigma]1\^2\ \((\((3\/4 - j\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi] - 2\ \[CapitalOmega]1] + e\ \[Sigma]1\^2\ \((\((\(-\(5\/4\)\) + j\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CapitalOmega]1] + e\^2\ \[Sigma]1\^2\ \((\((3\/4 - \(15\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + \((1\/2 - j\/4)\)\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi] - 2\ \[CapitalOmega]1] + e1\^2\ \[Sigma]1\^2\ \((\((\(-\(\(3\ j\)\/16\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-3\) + j, 0] + 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e1\ \[Sigma]1\^2\ \((1\/2\ j\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e\ e1\ \[Sigma]1\^2\ \((\((\(5\ j\)\/4 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] + \((1\/2 - j\/2)\)\ \[Alpha]\^2\ b[ 3\/2, \(-2\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e\ e1\ \[Sigma]1\^2\ \((\((\(-\(\(7\ j\)\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] - \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e1\ \[Sigma]1\^2\ \((\(-\(1\/2\)\)\ j\ \[Alpha]\ b[3\/2, j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e\ e1\ \[Sigma]1\^2\ \((\((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, j, 0] - 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e\ e1\ \[Sigma]1\^2\ \((\((\(3\ j\)\/4 - j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + \((\(-\(1\/2\)\) + j\/2)\)\ \[Alpha]\^2\ b[3\/2, j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\(-j\)\ L + j\ L1 + \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e1\^2\ \[Sigma]1\^2\ \((\((\(3\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]1] + 9\/4\ \[Alpha]\^2\ \[Sigma]\^2\ \[Sigma]1\^2\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CapitalOmega] - 2\ \[CapitalOmega]1] - 3\/2\ \[Alpha]\^2\ \[Sigma]\^3\ \[Sigma]1\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - 3\ \[CapitalOmega] - \[CapitalOmega]1] + \((\(-\[Alpha]\)\ \ \[Sigma]\ \[Sigma]1\ b[3\/2, \(-1\) + j, 0] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/2\)\) + j\^2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((7\/2 - 4\ j + j\^2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + \[Sigma]\^3\ \[Sigma]1\ \((1\/2\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 3\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] + 3\/2\ \[Alpha]\^2\ b[5\/2, j, 0])\) + \[Sigma]\ \[Sigma]1\^3\ \((1\/2\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 3\/2\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] + 3\ \[Alpha]\^2\ b[5\/2, j, 0])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CapitalOmega] - \[CapitalOmega]1] + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-2\) + \(17\ j\)\/8 - j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + \((1 - j\/2)\)\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1] + e\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(3\/2\)\) + j)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1] + e\ \[Sigma]\ \[Sigma]1\ \((\((5\/2 - j)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1] + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(3\/2\)\) + \(15\ j\)\/8 - j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + \((\(-1\) + j\/2)\)\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(3\ j\)\/8 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-3\) + j, 0] - 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1] + e1\ \[Sigma]\ \[Sigma]1\ \((\(-j\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] - 1\/2\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(5\ j\)\/2\)\) + j\^2)\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + \((\(-1\) + j)\)\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] + 1\/4\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(7\ j\)\/2 - j\^2)\)\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] + 2\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] + 1\/4\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1] + e1\ \[Sigma]\ \[Sigma]1\ \((j\ \[Alpha]\ b[3\/2, j, 0] - 1\/2\ \[Alpha]\^2\ b[3\/2, j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((j\/2 - j\^2)\)\ \[Alpha]\ b[3\/2, j, 0] + 1\/4\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/2\)\) + j\^2)\)\ \[Alpha]\ b[3\/2, j, 0] + \((1 - j)\)\ \[Alpha]\^2\ b[3\/2, j, 1] + 1\/4\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\(-j\)\ L + j\ L1 + \[CurlyPi] + \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/8\)\) - j\^2\/2)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1] + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/4\)\) - j\/8 + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] + \[CapitalOmega] - \[CapitalOmega]1] + e\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/2\)\) - j)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CapitalOmega] - \[CapitalOmega]1] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/8\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-3\) + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 + \[CapitalOmega] - \[CapitalOmega]1] + e1\ \[Sigma]\ \[Sigma]1\ \((j\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi]1 + \[CapitalOmega] - \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((j\/2 - j\^2)\)\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] - j\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 + \[CapitalOmega] - \ \[CapitalOmega]1] + 3\/2\ \[Alpha]\^2\ \[Sigma]\^3\ \[Sigma]1\ b[5\/2, j, 0]\ cos[\((2 - j)\)\ L + j\ L1 - 3\ \[CapitalOmega] + \[CapitalOmega]1] + \((\[Alpha]\ \[Sigma]\ \ \[Sigma]1\ b[3\/2, 1 + j, 0] + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((1\/2 - j\^2)\)\ \[Alpha]\ b[ 3\/2, 1 + j, 0] + \[Alpha]\^2\ b[3\/2, 1 + j, 1] + 1\/4\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\) + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((1\/2 - j\^2)\)\ \[Alpha]\ b[ 3\/2, 1 + j, 0] + \[Alpha]\^2\ b[3\/2, 1 + j, 1] + 1\/4\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\) + \[Sigma]\^3\ \[Sigma]1\ \((\(-\(1\/2\)\)\ \ \[Alpha]\ b[3\/2, 1 + j, 0] - 3\ \[Alpha]\^2\ b[5\/2, j, 0] - 3\/2\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\) + \[Sigma]\ \[Sigma]1\^3\ \((\(-\(1\/2\)\)\ \ \[Alpha]\ b[3\/2, 1 + j, 0] - 3\ \[Alpha]\^2\ b[5\/2, j, 0] - 3\/2\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\))\)\ cos[\(-j\)\ L + j\ L1 - \[CapitalOmega] + \[CapitalOmega]1] + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/4\)\) - j\/8 + j\^2\/2)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - \[CapitalOmega] + \[CapitalOmega]1] + e\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/2\)\) - j)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/2\ \[Alpha]\^2\ b[3\/2, 1 + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CapitalOmega] + \[CapitalOmega]1] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/8\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - \[CapitalOmega] + \[CapitalOmega]1] + e1\ \[Sigma]\ \[Sigma]1\ \((j\ \[Alpha]\ b[3\/2, j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi]1 - \[CapitalOmega] + \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((j\/2 - j\^2)\)\ \[Alpha]\ b[3\/2, j, 0] - j\ \[Alpha]\^2\ b[3\/2, j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] + \ \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/2\)\) + j\^2)\)\ \[Alpha]\ b[3\/2, j, 0] - \[Alpha]\^2\ b[3\/2, j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\(-j\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] + \ \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(3\ j\)\/2 + j\^2)\)\ \[Alpha]\ b[ 3\/2, 2 + j, 0] - \[Alpha]\^2\ b[3\/2, 2 + j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, 2 + j, 2])\)\ cos[\(-j\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - \[CapitalOmega] + \ \[CapitalOmega]1] + \[Sigma]\^2\ \[Sigma]1\^2\ \((1\/2\ \[Alpha]\ b[3\/2, 1 + j, 0] + 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0] + 3\/2\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\)\ cos[\(-j\)\ L + j\ L1 - 2\ \[CapitalOmega] + 2\ \[CapitalOmega]1]\)], "Output", CellLabel->"Out[2]="] }, Closed]], Cell[TextData[{ "The book shows the grouping of monomials within some coefficients. I have \ not yet been able to identify the general law of this phenomenon or to \ introduce it in the general architecture of the formula. Therefore, I \ attributed to the DFExpand function the option ", StyleBox["Simplification", FontWeight->"Bold"], ", which may assume either ", StyleBox["True", FontWeight->"Bold"], " or ", StyleBox["False", FontWeight->"Bold"], " values (the latter being the default choice). With ", StyleBox["Simplification", FontWeight->"Bold"], "->", StyleBox["True", FontWeight->"Bold"], ", the program applied makes use of ", StyleBox["Collect", FontWeight->"Bold"], " to identify the groupings; I notice that, in the case of more complicated \ expressions, simplification may take a long time. " }], "Text"], Cell[TextData[{ "I could easily have introduced groupings of Laplace coefficients in pairs, \ which we may see in some coefficients in SSD, but I decided against it, as \ this may have interfered with the right functioning of the function ", StyleBox["AddIndirectTerms", FontWeight->"Bold"], "." }], "MathCaption"], Cell[CellGroupData[{ Cell["DFExpand[4,Simplification->True]", "Input", CellLabel->"In[3]:=", ShowCellBracket->True], Cell[BoxData[ \(\((1\/2\ b[1\/2, j, 0] + \((e\^2 + e1\^2)\)\ \((\(-\(1\/2\)\)\ j\^2\ b[1\/2, j, 0] + 1\/4\ \[Alpha]\ b[1\/2, j, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, j, 2])\) + e\^4\ \((\((\(-\(\(9\ j\^2\)\/128\)\) + j\^4\/8)\)\ b[1\/2, j, 0] - 1\/16\ j\^2\ \[Alpha]\ b[1\/2, j, 1] - 1\/16\ j\^2\ \[Alpha]\^2\ b[1\/2, j, 2] + 1\/32\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/128\ \[Alpha]\^4\ b[1\/2, j, 4])\) + e1\^4\ \((\((\(-\(\(17\ j\^2\)\/128\)\) + j\^4\/8)\)\ b[1\/2, j, 0] + \((3\/16 - \(3\ j\^2\)\/16)\)\ \[Alpha]\ b[1\/2, j, 1] + \((9\/32 - j\^2\/16)\)\ \[Alpha]\^2\ b[1\/2, j, 2] + 3\/32\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/128\ \[Alpha]\^4\ b[1\/2, j, 4])\) + e\^2\ e1\^2\ \((1\/2\ j\^4\ b[1\/2, j, 0] + \((1\/8 - j\^2\/2)\)\ \[Alpha]\ b[1\/2, j, 1] + \((7\/16 - j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, j, 2] + 1\/4\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/32\ \[Alpha]\^4\ b[1\/2, j, 4])\) + \((\[Sigma]\^2 + \[Sigma]1\^2)\)\ \ \((\(-\(1\/4\)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\ b[3\/2, 1 + j, 0])\) + \((e\^2 + e1\^2)\)\ \((\[Sigma]\^2 + \[Sigma]1\^2)\)\ \((\((\(-\(1\/8\ \)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2] + \((\(-\(1\/8\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\) + \((\[Sigma]\^4 + \[Sigma]1\^4)\)\ \((3\/16\ \ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] + 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0] + 3\/16\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\) + \[Sigma]\^2\ \[Sigma]1\^2\ \((1\/4\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/4\ \[Alpha]\ b[3\/2, 1 + j, 0] + 3\/8\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] + 15\/4\ \[Alpha]\^2\ b[5\/2, j, 0] + 3\/8\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\))\)\ cos[\(-j\)\ L + j\ L1] + e\^4\ \((\((\(-\(\(103\ j\)\/192\)\) + \(283\ j\^2\)\/384 - \(5\ j\^3\)\ \/16 + j\^4\/24)\)\ b[1\/2, j, 0] + \((\(-\(1\/6\)\) + \(59\ j\)\/96 - \(7\ j\^2\)\/16 + j\^3\/12)\)\ \[Alpha]\ b[1\/2, j, 1] + \((1\/8 - \(13\ j\)\/64 + j\^2\/16)\)\ \[Alpha]\^2\ b[ 1\/2, j, 2] + \((\(-\(1\/32\)\) + j\/48)\)\ \[Alpha]\^3\ b[ 1\/2, j, 3] + 1\/384\ \[Alpha]\^4\ b[1\/2, j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - 4\ \[CurlyPi]] + e\^3\ \((\((\(-\(\(13\ j\)\/24\)\) + \(5\ j\^2\)\/8 - j\^3\/6)\)\ b[ 1\/2, j, 0] + \((\(-\(3\/16\)\) + \(9\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[1\/2, j, 1] + \((1\/8 - j\/8)\)\ \[Alpha]\^2\ b[1\/2, j, 2] - 1\/48\ \[Alpha]\^3\ b[1\/2, j, 3])\)\ cos[\((3 - j)\)\ L + j\ L1 - 3\ \[CurlyPi]] + \((e\^2\ \((\((\(-\(\(5\ j\)\/8\)\) + j\^2\/2)\)\ b[1\/2, j, 0] + \((\(-\(1\/4\)\) + j\/2)\)\ \[Alpha]\ b[1\/2, j, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, j, 2])\) + e\^4\ \((\((\(11\ j\)\/48 - \(2\ j\^2\)\/3 + \(5\ j\^3\)\/8 - j\^4\/6)\)\ b[1\/2, j, 0] + \((1\/6 - \(23\ j\)\/48 + j\^2\/2 - j\^3\/6)\)\ \[Alpha]\ b[1\/2, j, 1] + \((\(-\(1\/8\)\) + \(3\ j\)\/32)\)\ \[Alpha]\^2\ b[ 1\/2, j, 2] + 1\/24\ j\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/96\ \[Alpha]\^4\ b[1\/2, j, 4])\) + e\^2\ e1\^2\ \((\((\(5\ j\^3\)\/8 - j\^4\/2)\)\ b[1\/2, j, 0] + \((\(-\(1\/8\)\) - j\/16 + j\^2\/2 - j\^3\/2)\)\ \[Alpha]\ b[1\/2, j, 1] + \((\(-\(1\/16\)\) + \(11\ j\)\/32)\)\ \[Alpha]\^2\ \ b[1\/2, j, 2] + \((1\/8 + j\/8)\)\ \[Alpha]\^3\ b[1\/2, j, 3] + 1\/32\ \[Alpha]\^4\ b[1\/2, j, 4])\) + e\^2\ \((\[Sigma]\^2 + \[Sigma]1\^2)\)\ \((\((1\/8 + j\/16 - j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2] + \((1\/8 + j\/16 - j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]] + \((e\ \((\(-j\)\ b[1\/2, j, 0] - 1\/2\ \[Alpha]\ b[1\/2, j, 1])\) + e\ e1\^2\ \((j\^3\ b[1\/2, j, 0] + \((\(-\(1\/4\)\) - j\/2 + j\^2\/2)\)\ \[Alpha]\ b[ 1\/2, j, 1] + \((\(-\(1\/2\)\) - j\/4)\)\ \[Alpha]\^2\ b[1\/2, j, 2] - 1\/8\ \[Alpha]\^3\ b[1\/2, j, 3])\) + e\^3\ \((\((j\/8 - \(5\ j\^2\)\/8 + j\^3\/2)\)\ b[1\/2, j, 0] + \((3\/16 - \(7\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 1\/2, j, 1] + \((\(-\(1\/8\)\) - j\/8)\)\ \[Alpha]\^2\ b[1\/2, j, 2] - 1\/16\ \[Alpha]\^3\ b[1\/2, j, 3])\) + e\ \((\[Sigma]\^2 + \[Sigma]1\^2)\)\ \((\((1\/4 + j\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + \((1\/4 + j\/2)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, 1 + j, 1])\))\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi]] + e1\^4\ \((\((1\/16 - \(73\ j\)\/192 + \(211\ j\^2\)\/384 - \(13\ j\^3\)\ \/48 + j\^4\/24)\)\ b[1\/2, \(-4\) + j, 0] + \((\(-\(1\/16\)\) + \(29\ j\)\/96 - \(5\ j\^2\)\/16 + j\^3\/12)\)\ \[Alpha]\ b[1\/2, \(-4\) + j, 1] + \((1\/32 - \(7\ j\)\/64 + j\^2\/16)\)\ \[Alpha]\^2\ b[ 1\/2, \(-4\) + j, 2] + \((\(-\(1\/96\)\) + j\/48)\)\ \[Alpha]\^3\ b[ 1\/2, \(-4\) + j, 3] + 1\/384\ \[Alpha]\^4\ b[1\/2, \(-4\) + j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - 4\ \[CurlyPi]1] + e1\^3\ \((\((\(-\(1\/8\)\) + \(29\ j\)\/48 - \(5\ j\^2\)\/8 + j\^3\/6)\)\ b[1\/2, \(-3\) + j, 0] + \((1\/8 - \(7\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 1\/2, \(-3\) + j, 1] + \((\(-\(1\/16\)\) + j\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-3\) + j, 2] + 1\/48\ \[Alpha]\^3\ b[1\/2, \(-3\) + j, 3])\)\ cos[\((3 - j)\)\ L + j\ L1 - 3\ \[CurlyPi]1] + e\ e1\^3\ \((\((\(-\(3\/8\)\) + \(31\ j\)\/16 - \(119\ j\^2\)\/48 + \(9\ \ j\^3\)\/8 - j\^4\/6)\)\ b[1\/2, \(-3\) + j, 0] + \((3\/8 - \(73\ j\)\/48 + \(11\ j\^2\)\/8 - j\^3\/3)\)\ \[Alpha]\ b[1\/2, \(-3\) + j, 1] + \((\(-\(3\/16\)\) + \(17\ j\)\/32 - j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, \(-3\) + j, 2] + \((1\/16 - j\/12)\)\ \[Alpha]\^3\ b[1\/2, \(-3\) + j, 3] - 1\/96\ \[Alpha]\^4\ b[1\/2, \(-3\) + j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - \[CurlyPi] - 3\ \[CurlyPi]1] + e\ e1\^3\ \((\((3\/8 - \(31\ j\)\/16 + \(119\ j\^2\)\/48 - \(9\ \ j\^3\)\/8 + j\^4\/6)\)\ b[1\/2, \(-3\) + j, 0] + \((\(-\(3\/8\)\) + \(65\ j\)\/48 - j\^2 + j\^3\/6)\)\ \[Alpha]\ b[1\/2, \(-3\) + j, 1] + \((3\/16 - \(11\ j\)\/32)\)\ \[Alpha]\^2\ b[ 1\/2, \(-3\) + j, 2] + \((\(-\(1\/16\)\) - j\/24)\)\ \[Alpha]\^3\ b[ 1\/2, \(-3\) + j, 3] - 1\/96\ \[Alpha]\^4\ b[1\/2, \(-3\) + j, 4])\)\ cos[\((2 - j)\)\ L + j\ L1 + \[CurlyPi] - 3\ \[CurlyPi]1] + \((e1\^2\ \((\((1\/4 - \(7\ j\)\/8 + j\^2\/2)\)\ b[1\/2, \(-2\) + j, 0] + \((\(-\(1\/4\)\) + j\/2)\)\ \[Alpha]\ b[ 1\/2, \(-2\) + j, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, \(-2\) + j, 2])\) + e1\^4\ \((\((1\/8 - \(7\ j\)\/48 - \(5\ j\^2\)\/12 + \(13\ j\^3\)\ \/24 - j\^4\/6)\)\ b[1\/2, \(-2\) + j, 0] + \((\(-\(1\/8\)\) - \(5\ j\)\/48 + j\^2\/2 - j\^3\/6)\)\ \[Alpha]\ b[1\/2, \(-2\) + j, 1] + \((1\/16 + \(9\ j\)\/32)\)\ \[Alpha]\^2\ b[ 1\/2, \(-2\) + j, 2] + \((1\/12 + j\/24)\)\ \[Alpha]\^3\ b[ 1\/2, \(-2\) + j, 3] + 1\/96\ \[Alpha]\^4\ b[1\/2, \(-2\) + j, 4])\) + e\^2\ e1\^2\ \((\((\(-1\) + \(9\ j\)\/2 - \(23\ j\^2\)\/4 + \(23\ \ j\^3\)\/8 - j\^4\/2)\)\ b[1\/2, \(-2\) + j, 0] + \((1 - \(51\ j\)\/16 + \(5\ j\^2\)\/2 - j\^3\/2)\)\ \[Alpha]\ b[1\/2, \(-2\) + j, 1] + \((\(-\(1\/2\)\) + \(25\ j\)\/32)\)\ \[Alpha]\^2\ \ b[1\/2, \(-2\) + j, 2] + \((1\/8 + j\/8)\)\ \[Alpha]\^3\ b[1\/2, \(-2\) + j, 3] + 1\/32\ \[Alpha]\^4\ b[1\/2, \(-2\) + j, 4])\) + e1\^2\ \((\[Sigma]\^2 + \[Sigma]1\^2)\)\ \((\((\(3\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-3\) + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2] + \((\(3\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1] + e\^2\ e1\^2\ \((\((13\/16 - \(7\ j\)\/2 + \(259\ j\^2\)\/64 - \(7\ j\^3\ \)\/4 + j\^4\/4)\)\ b[1\/2, \(-2\) + j, 0] + \((\(-\(13\/16\)\) + \(11\ j\)\/4 - \(9\ j\^2\)\/4 + j\^3\/2)\)\ \[Alpha]\ b[1\/2, \(-2\) + j, 1] + \((13\/32 - \(15\ j\)\/16 + \(3\ j\^2\)\/8)\)\ \ \[Alpha]\^2\ b[1\/2, \(-2\) + j, 2] + \((\(-\(1\/8\)\) + j\/8)\)\ \[Alpha]\^3\ b[ 1\/2, \(-2\) + j, 3] + 1\/64\ \[Alpha]\^4\ b[1\/2, \(-2\) + j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - 2\ \[CurlyPi]1] + e\ e1\^2\ \((\((1\/2 - 2\ j + \(15\ j\^2\)\/8 - j\^3\/2)\)\ b[ 1\/2, \(-2\) + j, 0] + \((\(-\(1\/2\)\) + \(23\ j\)\/16 - \(3\ j\^2\)\/4)\)\ \ \[Alpha]\ b[1\/2, \(-2\) + j, 1] + \((1\/4 - \(3\ j\)\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-2\) + j, 2] - 1\/16\ \[Alpha]\^3\ b[1\/2, \(-2\) + j, 3])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi] - 2\ \[CurlyPi]1] + e\ e1\^2\ \((\((\(-\(1\/2\)\) + 2\ j - \(15\ j\^2\)\/8 + j\^3\/2)\)\ b[ 1\/2, \(-2\) + j, 0] + \((1\/2 - \(17\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 1\/2, \(-2\) + j, 1] + \((\(-\(1\/4\)\) - j\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-2\) + j, 2] - 1\/16\ \[Alpha]\^3\ b[1\/2, \(-2\) + j, 3])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CurlyPi]1] + \((e1\ \((\((\(-\(1\/2\)\) + j)\)\ b[ 1\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\ b[1\/2, \(-1\) + j, 1])\) + e1\^3\ \((\((\(-\(1\/8\)\) - j\/16 + \(5\ j\^2\)\/8 - j\^3\/2)\)\ b[1\/2, \(-1\) + j, 0] + \((1\/8 + \(9\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[ 1\/2, \(-1\) + j, 1] + \((5\/16 + j\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-1\) + j, 2] + 1\/16\ \[Alpha]\^3\ b[1\/2, \(-1\) + j, 3])\) + e\^2\ e1\ \((\((1\/2 - 2\ j + \(5\ j\^2\)\/2 - j\^3)\)\ b[ 1\/2, \(-1\) + j, 0] + \((\(-\(1\/2\)\) + \(3\ j\)\/2 - j\^2\/2)\)\ \[Alpha]\ b[1\/2, \(-1\) + j, 1] + \((3\/8 + j\/4)\)\ \[Alpha]\^2\ b[ 1\/2, \(-1\) + j, 2] + 1\/8\ \[Alpha]\^3\ b[1\/2, \(-1\) + j, 3])\) + e1\ \((\[Sigma]\^2 + \[Sigma]1\^2)\)\ \((\(-\(1\/2\)\)\ j\ \ \[Alpha]\ b[3\/2, \(-2\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] - 1\/2\ j\ \[Alpha]\ b[3\/2, j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, j, 1])\))\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi]1] + e\^3\ e1\ \((\((\(-\(2\/3\)\) + \(119\ j\)\/48 - \(137\ j\^2\)\/48 + \ \(29\ j\^3\)\/24 - j\^4\/6)\)\ b[1\/2, \(-1\) + j, 0] + \((2\/3 - \(103\ j\)\/48 + \(13\ j\^2\)\/8 - j\^3\/3)\)\ \[Alpha]\ b[1\/2, \(-1\) + j, 1] + \((\(-\(3\/8\)\) + \(23\ j\)\/32 - j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, \(-1\) + j, 2] + \((5\/48 - j\/12)\)\ \[Alpha]\^3\ b[1\/2, \(-1\) + j, 3] - 1\/96\ \[Alpha]\^4\ b[1\/2, \(-1\) + j, 4])\)\ cos[\((4 - j)\)\ L + j\ L1 - 3\ \[CurlyPi] - \[CurlyPi]1] + e\^2\ e1\ \((\((\(-\(9\/16\)\) + \(31\ j\)\/16 - \(15\ j\^2\)\/8 + j\^3\/2)\)\ b[1\/2, \(-1\) + j, 0] + \((9\/16 - \(25\ j\)\/16 + \(3\ j\^2\)\/4)\)\ \[Alpha]\ \ b[1\/2, \(-1\) + j, 1] + \((\(-\(5\/16\)\) + \(3\ j\)\/8)\)\ \[Alpha]\^2\ b[ 1\/2, \(-1\) + j, 2] + 1\/16\ \[Alpha]\^3\ b[1\/2, \(-1\) + j, 3])\)\ cos[\((3 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - \[CurlyPi]1] + \((e\ e1\ \((\((\(-\(1\/2\)\) + \ \(3\ j\)\/2 - j\^2)\)\ b[1\/2, \(-1\) + j, 0] + \((1\/2 - j)\)\ \[Alpha]\ b[ 1\/2, \(-1\) + j, 1] - 1\/4\ \[Alpha]\^2\ b[1\/2, \(-1\) + j, 2])\) + e\ e1\^3\ \((\((\(-\(1\/8\)\) + j\/16 + \(11\ j\^2\)\/16 - \(9\ j\^3\)\/8 + j\^4\/2)\)\ b[1\/2, \(-1\) + j, 0] + \((1\/8 + \(3\ j\)\/16 - j\^2 + j\^3\/2)\)\ \[Alpha]\ b[1\/2, \(-1\) + j, 1] + \((\(-\(1\/16\)\) - \(19\ j\)\/32)\)\ \[Alpha]\^2\ \ b[1\/2, \(-1\) + j, 2] + \((\(-\(3\/16\)\) - j\/8)\)\ \[Alpha]\^3\ b[ 1\/2, \(-1\) + j, 3] - 1\/32\ \[Alpha]\^4\ b[1\/2, \(-1\) + j, 4])\) + e\^3\ e1\ \((\((5\/8 - \(43\ j\)\/16 + \(63\ j\^2\)\/16 - \(19\ j\ \^3\)\/8 + j\^4\/2)\)\ b[1\/2, \(-1\) + j, 0] + \((\(-\(5\/8\)\) + \(37\ j\)\/16 - 2\ j\^2 + j\^3\/2)\)\ \[Alpha]\ b[1\/2, \(-1\) + j, 1] + \((7\/16 - \(17\ j\)\/32)\)\ \[Alpha]\^2\ b[ 1\/2, \(-1\) + j, 2] + \((\(-\(1\/16\)\) - j\/8)\)\ \[Alpha]\^3\ b[ 1\/2, \(-1\) + j, 3] - 1\/32\ \[Alpha]\^4\ b[1\/2, \(-1\) + j, 4])\) + e\ e1\ \((\[Sigma]\^2 + \[Sigma]1\^2)\)\ \((\((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2] + \((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1] + e\^3\ e1\ \((\((1\/24 - j\/48 - \(13\ j\^2\)\/48 - j\^3\/24 + j\^4\/6)\)\ b[1\/2, 1 + j, 0] + \((\(-\(1\/24\)\) - j\/48 + j\^3\/6)\)\ \[Alpha]\ b[ 1\/2, 1 + j, 1] + \((1\/16 - j\/32)\)\ \[Alpha]\^2\ b[1\/2, 1 + j, 2] + \((\(-\(1\/48\)\) - j\/24)\)\ \[Alpha]\^3\ b[ 1\/2, 1 + j, 3] - 1\/96\ \[Alpha]\^4\ b[1\/2, 1 + j, 4])\)\ cos[\((2 - j)\)\ L + j\ L1 - 3\ \[CurlyPi] + \[CurlyPi]1] + e\^2\ e1\ \((\((1\/16 - j\/16 - \(5\ j\^2\)\/8 - j\^3\/2)\)\ b[1\/2, 1 + j, 0] + \((\(-\(1\/16\)\) - j\/16 - j\^2\/4)\)\ \[Alpha]\ b[1\/2, 1 + j, 1] + \((3\/16 + j\/8)\)\ \[Alpha]\^2\ b[1\/2, 1 + j, 2] + 1\/16\ \[Alpha]\^3\ b[1\/2, 1 + j, 3])\)\ cos[\((1 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] + \[CurlyPi]1] + \((e\ e1\ \((\((1\/2 + \(3\ \ j\)\/2 + j\^2)\)\ b[1\/2, 1 + j, 0] - 1\/2\ \[Alpha]\ b[1\/2, 1 + j, 1] - 1\/4\ \[Alpha]\^2\ b[1\/2, 1 + j, 2])\) + e\ e1\^3\ \((\((1\/8 + j\/16 - \(11\ j\^2\)\/16 - \(9\ j\^3\)\/8 - j\^4\/2)\)\ b[1\/2, 1 + j, 0] + \((\(-\(1\/8\)\) + \(11\ j\)\/16 + \(5\ j\^2\)\/8)\ \)\ \[Alpha]\ b[1\/2, 1 + j, 1] + \((\(-\(11\/16\)\) + \(7\ j\)\/32 + j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, 1 + j, 2] - 5\/16\ \[Alpha]\^3\ b[1\/2, 1 + j, 3] - 1\/32\ \[Alpha]\^4\ b[1\/2, 1 + j, 4])\) + e\^3\ e1\ \((\((\(-\(\(3\ j\)\/16\)\) - \(13\ j\^2\)\/16 - \(9\ j\ \^3\)\/8 - j\^4\/2)\)\ b[1\/2, 1 + j, 0] + \((\(3\ j\)\/16 + \(3\ j\^2\)\/8)\)\ \[Alpha]\ b[ 1\/2, 1 + j, 1] + \((\(-\(1\/8\)\) + \(7\ j\)\/32 + j\^2\/4)\)\ \[Alpha]\^2\ b[1\/2, 1 + j, 2] - 3\/16\ \[Alpha]\^3\ b[1\/2, 1 + j, 3] - 1\/32\ \[Alpha]\^4\ b[1\/2, 1 + j, 4])\) + e\ e1\ \((\[Sigma]\^2 + \[Sigma]1\^2)\)\ \((\((\(-\(\(3\ j\)\/4\)\ \) - j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2] + \((\(-\(\(3\ j\)\/4\)\) - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, 2 + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, 2 + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, 2 + j, 2])\))\)\ cos[\(-j\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1] + e\^2\ e1\^2\ \((\((3\/16 + j + \(109\ j\^2\)\/64 + \(9\ j\^3\)\/8 + j\^4\/4)\)\ b[1\/2, 2 + j, 0] + \((\(-\(3\/16\)\) - \(7\ j\)\/16 - j\^2\/4)\)\ \[Alpha]\ b[1\/2, 2 + j, 1] + \((3\/32 - \(7\ j\)\/32 - j\^2\/8)\)\ \[Alpha]\^2\ b[ 1\/2, 2 + j, 2] + 1\/8\ \[Alpha]\^3\ b[1\/2, 2 + j, 3] + 1\/64\ \[Alpha]\^4\ b[1\/2, 2 + j, 4])\)\ cos[\(-j\)\ L + j\ L1 - 2\ \[CurlyPi] + 2\ \[CurlyPi]1] + 3\/8\ \[Alpha]\^2\ \[Sigma]\^4\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - 4\ \[CapitalOmega]] + \((1\/2\ \[Alpha]\ \[Sigma]\^2\ b[ 3\/2, \(-1\) + j, 0] + e1\^2\ \[Sigma]\^2\ \((\((1\/4 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + e\^2\ \[Sigma]\^2\ \((\((\(-\(7\/4\)\) + 2\ j - j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + \[Sigma]\^2\ \[Sigma]1\^2\ \((\(-\(1\/2\)\)\ \ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 3\/4\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] - 15\/4\ \[Alpha]\^2\ b[5\/2, j, 0])\) + \[Sigma]\^4\ \((\(-\(3\/4\)\)\ \[Alpha]\^2\ b[ 5\/2, \(-2\) + j, 0] - 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CapitalOmega]] + e\^2\ \[Sigma]\^2\ \((\((1 - \(17\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + \((\(-\(1\/2\)\) + j\/4)\)\ \[Alpha]\^2\ b[ 3\/2, \(-1\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - 2\ \[CapitalOmega]] + e\ \[Sigma]\^2\ \((\((3\/4 - j\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi] - 2\ \[CapitalOmega]] + e\ \[Sigma]\^2\ \((\((\(-\(5\/4\)\) + j\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CapitalOmega]] + e\^2\ \[Sigma]\^2\ \((\((3\/4 - \(15\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + \((1\/2 - j\/4)\)\ \[Alpha]\^2\ b[ 3\/2, \(-1\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi] - 2\ \[CapitalOmega]] + e1\^2\ \[Sigma]\^2\ \((\((\(-\(\(3\ j\)\/16\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-3\) + j, 0] + 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]] + e1\ \[Sigma]\^2\ \((1\/2\ j\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi]1 - 2\ \[CapitalOmega]] + e\ e1\ \[Sigma]\^2\ \((\((\(5\ j\)\/4 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] + \((1\/2 - j\/2)\)\ \[Alpha]\^2\ b[ 3\/2, \(-2\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]] + e\ e1\ \[Sigma]\^2\ \((\((\(-\(\(7\ j\)\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] - \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]] + e1\ \[Sigma]\^2\ \((\(-\(1\/2\)\)\ j\ \[Alpha]\ b[3\/2, j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi]1 - 2\ \[CapitalOmega]] + e\ e1\ \[Sigma]\^2\ \((\((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] - 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]] + e\ e1\ \[Sigma]\^2\ \((\((\(3\ j\)\/4 - j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + \((\(-\(1\/2\)\) + j\/2)\)\ \[Alpha]\^2\ b[3\/2, j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\(-j\)\ L + j\ L1 + \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]] + e1\^2\ \[Sigma]\^2\ \((\((\(3\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]] + 3\/8\ \[Alpha]\^2\ \[Sigma]1\^4\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - 4\ \[CapitalOmega]1] - 3\/2\ \[Alpha]\^2\ \[Sigma]\ \[Sigma]1\^3\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - \[CapitalOmega] - 3\ \[CapitalOmega]1] + 3\/2\ \[Alpha]\^2\ \[Sigma]\ \[Sigma]1\^3\ b[5\/2, \(-2\) + j, 0]\ cos[\((2 - j)\)\ L + j\ L1 + \[CapitalOmega] - 3\ \[CapitalOmega]1] + \((1\/2\ \[Alpha]\ \[Sigma]1\^2\ b[ 3\/2, \(-1\) + j, 0] + e1\^2\ \[Sigma]1\^2\ \((\((1\/4 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + e\^2\ \[Sigma]1\^2\ \((\((\(-\(7\/4\)\) + 2\ j - j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + \[Sigma]\^2\ \[Sigma]1\^2\ \((\(-\(1\/2\)\)\ \ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 15\/4\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] - 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0])\) + \[Sigma]1\^4\ \((\(-\(3\/4\)\)\ \[Alpha]\^2\ b[ 5\/2, \(-2\) + j, 0] - 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CapitalOmega]1] + e\^2\ \[Sigma]1\^2\ \((\((1 - \(17\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + \((\(-\(1\/2\)\) + j\/4)\)\ \[Alpha]\^2\ b[ 3\/2, \(-1\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - 2\ \[CapitalOmega]1] + e\ \[Sigma]1\^2\ \((\((3\/4 - j\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi] - 2\ \[CapitalOmega]1] + e\ \[Sigma]1\^2\ \((\((\(-\(5\/4\)\) + j\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CapitalOmega]1] + e\^2\ \[Sigma]1\^2\ \((\((3\/4 - \(15\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + \((1\/2 - j\/4)\)\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi] - 2\ \[CapitalOmega]1] + e1\^2\ \[Sigma]1\^2\ \((\((\(-\(\(3\ j\)\/16\)\) + j\^2\/4)\)\ \[Alpha]\ b[3\/2, \(-3\) + j, 0] + 1\/4\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e1\ \[Sigma]1\^2\ \((1\/2\ j\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e\ e1\ \[Sigma]1\^2\ \((\((\(5\ j\)\/4 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] + \((1\/2 - j\/2)\)\ \[Alpha]\^2\ b[ 3\/2, \(-2\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e\ e1\ \[Sigma]1\^2\ \((\((\(-\(\(7\ j\)\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] - \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e1\ \[Sigma]1\^2\ \((\(-\(1\/2\)\)\ j\ \[Alpha]\ b[3\/2, j, 0] + 1\/4\ \[Alpha]\^2\ b[3\/2, j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e\ e1\ \[Sigma]1\^2\ \((\((\(-\(j\/4\)\) + j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, j, 0] - 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e\ e1\ \[Sigma]1\^2\ \((\((\(3\ j\)\/4 - j\^2\/2)\)\ \[Alpha]\ b[3\/2, j, 0] + \((\(-\(1\/2\)\) + j\/2)\)\ \[Alpha]\^2\ b[3\/2, j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\(-j\)\ L + j\ L1 + \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]1] + e1\^2\ \[Sigma]1\^2\ \((\((\(3\ j\)\/16 + j\^2\/4)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] + 1\/16\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]1] + 9\/4\ \[Alpha]\^2\ \[Sigma]\^2\ \[Sigma]1\^2\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CapitalOmega] - 2\ \[CapitalOmega]1] - 3\/2\ \[Alpha]\^2\ \[Sigma]\^3\ \[Sigma]1\ b[5\/2, \(-2\) + j, 0]\ cos[\((4 - j)\)\ L + j\ L1 - 3\ \[CapitalOmega] - \[CapitalOmega]1] + \((\(-\[Alpha]\)\ \ \[Sigma]\ \[Sigma]1\ b[3\/2, \(-1\) + j, 0] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/2\)\) + j\^2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((7\/2 - 4\ j + j\^2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] - \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\) + \[Sigma]\^3\ \[Sigma]1\ \((1\/2\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 3\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] + 3\/2\ \[Alpha]\^2\ b[5\/2, j, 0])\) + \[Sigma]\ \[Sigma]1\^3\ \((1\/2\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 3\/2\ \[Alpha]\^2\ b[5\/2, \(-2\) + j, 0] + 3\ \[Alpha]\^2\ b[5\/2, j, 0])\))\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CapitalOmega] - \[CapitalOmega]1] + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-2\) + \(17\ j\)\/8 - j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + \((1 - j\/2)\)\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1] + e\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(3\/2\)\) + j)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1] + e\ \[Sigma]\ \[Sigma]1\ \((\((5\/2 - j)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1] + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(3\/2\)\) + \(15\ j\)\/8 - j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + \((\(-1\) + j\/2)\)\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(3\ j\)\/8 - j\^2\/2)\)\ \[Alpha]\ b[ 3\/2, \(-3\) + j, 0] - 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1] + e1\ \[Sigma]\ \[Sigma]1\ \((\(-j\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] - 1\/2\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1])\)\ cos[\((3 - j)\)\ L + j\ L1 - \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(5\ j\)\/2\)\) + j\^2)\)\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + \((\(-1\) + j)\)\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] + 1\/4\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((4 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(7\ j\)\/2 - j\^2)\)\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] + 2\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] + 1\/4\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1] + e1\ \[Sigma]\ \[Sigma]1\ \((j\ \[Alpha]\ b[3\/2, j, 0] - 1\/2\ \[Alpha]\^2\ b[3\/2, j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 + \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((j\/2 - j\^2)\)\ \[Alpha]\ b[3\/2, j, 0] + 1\/4\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/2\)\) + j\^2)\)\ \[Alpha]\ b[3\/2, j, 0] + \((1 - j)\)\ \[Alpha]\^2\ b[3\/2, j, 1] + 1\/4\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\(-j\)\ L + j\ L1 + \[CurlyPi] + \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/8\)\) - j\^2\/2)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] - 1\/8\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\)\ cos[\(-j\)\ L + j\ L1 + 2\ \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1] + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/4\)\) - j\/8 + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] + \[CapitalOmega] - \[CapitalOmega]1] + e\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/2\)\) - j)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/2\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CapitalOmega] - \[CapitalOmega]1] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/8\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-3\) + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-3\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-3\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 + \[CapitalOmega] - \[CapitalOmega]1] + e1\ \[Sigma]\ \[Sigma]1\ \((j\ \[Alpha]\ b[3\/2, \(-2\) + j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi]1 + \[CapitalOmega] - \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((j\/2 - j\^2)\)\ \[Alpha]\ b[ 3\/2, \(-2\) + j, 0] - j\ \[Alpha]\^2\ b[3\/2, \(-2\) + j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, \(-2\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 + \[CapitalOmega] - \ \[CapitalOmega]1] + 3\/2\ \[Alpha]\^2\ \[Sigma]\^3\ \[Sigma]1\ b[5\/2, j, 0]\ cos[\((2 - j)\)\ L + j\ L1 - 3\ \[CapitalOmega] + \[CapitalOmega]1] + \((\[Alpha]\ \[Sigma]\ \ \[Sigma]1\ b[3\/2, 1 + j, 0] + \((e\^2 + e1\^2)\)\ \[Sigma]\ \[Sigma]1\ \((\((1\/2 - j\^2)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] + \[Alpha]\^2\ b[ 3\/2, 1 + j, 1] + 1\/4\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\) + \[Sigma]\ \[Sigma]1\ \((\[Sigma]\^2 + \[Sigma]1\ \^2)\)\ \((\(-\(1\/2\)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 3\ \[Alpha]\^2\ b[5\/2, j, 0] - 3\/2\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\))\)\ cos[\(-j\)\ L + j\ L1 - \[CapitalOmega] + \[CapitalOmega]1] + e\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/4\)\) - j\/8 + j\^2\/2)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, 1 + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, 1 + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - \[CapitalOmega] + \[CapitalOmega]1] + e\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(1\/2\)\) - j)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/2\ \[Alpha]\^2\ b[3\/2, 1 + j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CapitalOmega] + \[CapitalOmega]1] + e1\^2\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/8\)\) + j\^2\/2)\)\ \[Alpha]\ b[3\/2, \(-1\) + j, 0] + 1\/2\ j\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1] + 1\/8\ \[Alpha]\^3\ b[3\/2, \(-1\) + j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - \[CapitalOmega] + \[CapitalOmega]1] + e1\ \[Sigma]\ \[Sigma]1\ \((j\ \[Alpha]\ b[3\/2, j, 0] + 1\/2\ \[Alpha]\^2\ b[3\/2, j, 1])\)\ cos[\((1 - j)\)\ L + j\ L1 - \[CurlyPi]1 - \[CapitalOmega] + \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((j\/2 - j\^2)\)\ \[Alpha]\ b[3\/2, j, 0] - j\ \[Alpha]\^2\ b[3\/2, j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\((2 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] + \ \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(-\(\(3\ j\)\/2\)\) + j\^2)\)\ \[Alpha]\ b[3\/2, j, 0] - \[Alpha]\^2\ b[3\/2, j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, j, 2])\)\ cos[\(-j\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] + \ \[CapitalOmega]1] + e\ e1\ \[Sigma]\ \[Sigma]1\ \((\((\(3\ j\)\/2 + j\^2)\)\ \[Alpha]\ b[ 3\/2, 2 + j, 0] - \[Alpha]\^2\ b[3\/2, 2 + j, 1] - 1\/4\ \[Alpha]\^3\ b[3\/2, 2 + j, 2])\)\ cos[\(-j\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - \[CapitalOmega] + \ \[CapitalOmega]1] + \[Sigma]\^2\ \[Sigma]1\^2\ \((1\/2\ \[Alpha]\ b[3\/2, 1 + j, 0] + 3\/4\ \[Alpha]\^2\ b[5\/2, j, 0] + 3\/2\ \[Alpha]\^2\ b[5\/2, 2 + j, 0])\)\ cos[\(-j\)\ L + j\ L1 - 2\ \[CapitalOmega] + 2\ \[CapitalOmega]1]\)], "Output", CellLabel->"Out[3]="] }, Closed]], Cell["\<\ The term 4D0.9 found in SSD does not appear in the previous result, \ but it is replaced by a slightly different one. Bearing in mind that in each \ term of RD, parameter j must assume the values ranging from -infinity to \ +infinity, it is deduced that it is possible to use the replacement j->-j, \ without altering the value of the result. By doing this, it can easily be \ seen that the term in SSD coincides with the one generated by the function.\ \ \>", "Text"], Cell["DFArguments", "Subsubsection", CellTags->"DFArguments"], Cell["\<\ DFArguments[ord] supplies an ordered list of all the arguments \ present in the DFExpand[ord]. Therefore, it proves useful when only some \ terms of the expansion are to be studied with DFCoefficient. \ \>", "Text"], Cell["\<\ By looking through the arguments in SSD, we will find j L1-j L+\ \[CurlyPi]-\[CurlyPi]1-\[CapitalOmega]+\[CapitalOmega]1 in place of j L1-j L-\ \[CurlyPi]+\[CurlyPi]1+\[CapitalOmega]-\[CapitalOmega]1, but we accounted for \ this apparent divergence in the previous section.\ \>", "MathCaption"], Cell[CellGroupData[{ Cell["DFArguments[4]", "Input", CellLabel->"In[4]:=", ShowCellBracket->True], Cell[BoxData[ \({\(-j\)\ L + j\ L1 - 2\ \[CapitalOmega] + 2\ \[CapitalOmega]1, \(-j\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - \[CapitalOmega] + \ \[CapitalOmega]1, \(-j\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] + \ \[CapitalOmega]1, \(-j\)\ L + j\ L1 - \[CapitalOmega] + \[CapitalOmega]1, \(-j\)\ L + j\ L1 + 2\ \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1, \(-j\)\ L + j\ L1 + \[CurlyPi] + \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1, \(-j\)\ L + j\ L1 + 2\ \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1, \(-j\)\ L + j\ L1 + 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]1, \(-j\)\ L + j\ L1 + \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]1, \(-j\)\ L + j\ L1 + 2\ \[CurlyPi] - 2\ \[CapitalOmega]1, \(-j\)\ L + j\ L1 + 2\ \[CurlyPi]1 - 2\ \[CapitalOmega], \(-j\)\ L + j\ L1 + \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega], \(-j\)\ L + j\ L1 + 2\ \[CurlyPi] - 2\ \[CapitalOmega], \(-j\)\ L + j\ L1 - 2\ \[CurlyPi] + 2\ \[CurlyPi]1, \(-j\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1, \(-j\)\ L + j\ L1, \((1 - j)\)\ L + j\ L1 - \[CurlyPi]1 - \[CapitalOmega] + \[CapitalOmega]1, \((1 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CapitalOmega] + \[CapitalOmega]1, \((1 - j)\)\ L + j\ L1 - \[CurlyPi]1 + \[CapitalOmega] - \[CapitalOmega]1, \((1 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CapitalOmega] - \[CapitalOmega]1, \((1 - j)\)\ L + j\ L1 + \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1, \((1 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1, \((1 - j)\)\ L + j\ L1 + \[CurlyPi]1 - 2\ \[CapitalOmega]1, \((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CapitalOmega]1, \((1 - j)\)\ L + j\ L1 + \[CurlyPi]1 - 2\ \[CapitalOmega], \((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CapitalOmega], \((1 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] + \[CurlyPi]1, \((1 - j)\)\ L + j\ L1 - \[CurlyPi]1, \((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CurlyPi]1, \((1 - j)\)\ L + j\ L1 - \[CurlyPi], \((2 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] + \ \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - \[CapitalOmega] + \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - \[CapitalOmega] + \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - 3\ \[CapitalOmega] + \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 + \[CapitalOmega] - \ \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 + \[CapitalOmega] - \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] + \[CapitalOmega] - \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - \[CapitalOmega] - \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - 2\ \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 + \[CapitalOmega] - 3\ \[CapitalOmega]1, \((2 - j)\)\ L + j\ L1 - \[CurlyPi] + \[CurlyPi]1 - 2\ \[CapitalOmega], \((2 - j)\)\ L + j\ L1 + \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega], \((2 - j)\)\ L + j\ L1 - 2\ \[CapitalOmega], \((2 - j)\)\ L + j\ L1 - 3\ \[CurlyPi] + \[CurlyPi]1, \((2 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1, \((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1, \((2 - j)\)\ L + j\ L1 + \[CurlyPi] - 3\ \[CurlyPi]1, \((2 - j)\)\ L + j\ L1 - 2\ \[CurlyPi], \((3 - j)\)\ L + j\ L1 - \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1, \((3 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1, \((3 - j)\)\ L + j\ L1 - \[CurlyPi]1 - 2\ \[CapitalOmega]1, \((3 - j)\)\ L + j\ L1 - \[CurlyPi] - 2\ \[CapitalOmega]1, \((3 - j)\)\ L + j\ L1 - \[CurlyPi]1 - 2\ \[CapitalOmega], \((3 - j)\)\ L + j\ L1 - \[CurlyPi] - 2\ \[CapitalOmega], \((3 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - \[CurlyPi]1, \((3 - j)\)\ L + j\ L1 - \[CurlyPi] - 2\ \[CurlyPi]1, \((3 - j)\)\ L + j\ L1 - 3\ \[CurlyPi]1, \((3 - j)\)\ L + j\ L1 - 3\ \[CurlyPi], \((4 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - \[CapitalOmega] - \ \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - \[CapitalOmega] - \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - \[CapitalOmega] - \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - 3\ \[CapitalOmega] - \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - 2\ \[CapitalOmega] - 2\ \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - 2\ \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - 2\ \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - \[CapitalOmega] - 3\ \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - 4\ \[CapitalOmega]1, \((4 - j)\)\ L + j\ L1 - \[CurlyPi] - \[CurlyPi]1 - 2\ \[CapitalOmega], \((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi]1 - 2\ \[CapitalOmega], \((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - 2\ \[CapitalOmega], \((4 - j)\)\ L + j\ L1 - 4\ \[CapitalOmega], \((4 - j)\)\ L + j\ L1 - 3\ \[CurlyPi] - \[CurlyPi]1, \((4 - j)\)\ L + j\ L1 - 2\ \[CurlyPi] - 2\ \[CurlyPi]1, \((4 - j)\)\ L + j\ L1 - \[CurlyPi] - 3\ \[CurlyPi]1, \((4 - j)\)\ L + j\ L1 - 4\ \[CurlyPi]1, \((4 - j)\)\ L + j\ L1 - 4\ \[CurlyPi]}\)], "Output", CellLabel->"Out[4]="] }, Closed]], Cell["DFCoefficient", "Subsubsection", CellTags->"DFCoefficient"], Cell["\<\ DFCoefficient[arg, ord] is the coefficient of cos(arg) in RDj (or \ RD).\ \>", "Text"], Cell["Let us check the term 4D1.5 on page 543 in SSD:", "MathCaption"], Cell[CellGroupData[{ Cell[BoxData[ \(DFCoefficient[\((1 - j)\)\ L + j\ L1 + \[CurlyPi] - 2\ \[CapitalOmega], 4]\)], "Input", CellLabel->"In[5]:=", ShowCellBracket->True], Cell[BoxData[ \(e\ \[Sigma]\^2\ \((\((\(-\(5\/4\)\) + j\/2)\)\ \[Alpha]\ b[ 3\/2, \(-1\) + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, \(-1\) + j, 1])\)\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell["\<\ If we effect the replacement j->-j in the argument, we can see it \ can be found correctly in the coefficient as well (functions b are normalised \ so that j is always preceded by the sign +, changing the sign in the second \ argument when necessary).\ \>", "MathCaption"], Cell[CellGroupData[{ Cell[BoxData[ \(DFCoefficient[\((1 + j)\)\ L - j\ L1 + \[CurlyPi] - 2\ \[CapitalOmega], 4]\)], "Input", CellLabel->"In[6]:=", ShowCellBracket->True], Cell[BoxData[ \(e\ \[Sigma]\^2\ \((\((\(-\(5\/4\)\) - j\/2)\)\ \[Alpha]\ b[3\/2, 1 + j, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, 1 + j, 1])\)\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell["\<\ The argument may also have all its coefficients in numerical form \ (without j); for instance, let us consider j=-5 in the previous result:\ \>", \ "MathCaption"], Cell[CellGroupData[{ Cell[BoxData[ \(DFCoefficient[\((1 + j)\)\ L - j\ L1 + \[CurlyPi] - 2\ \[CapitalOmega] /. j \[Rule] \(-5\), 4]\)], "Input", CellLabel->"In[7]:=", ShowCellBracket->True], Cell[BoxData[ \(e\ \[Sigma]\^2\ \((5\/4\ \[Alpha]\ b[3\/2, 4, 0] - 1\/4\ \[Alpha]\^2\ b[3\/2, 4, 1])\)\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell["\<\ DFExpand[ord] generates certain terms which, as the values for j \ change, produce infinite other arguments, though not their opposite values \ (this is because, as I stated at the beginning, the algorithm prevents \ repetitions). However, cos(arg)=cos(-arg), so using an argument like arg or \ like its opposite is perfectly equivalent. Since the function DFCoefficient \ recognizes this, it may be used for any valid forms of the arguments:\ \>", \ "MathCaption"], Cell[CellGroupData[{ Cell[BoxData[ \(DFCoefficient[\(-\((5\ L - 2\ L1 - \[CurlyPi] - 2\ \[CapitalOmega]1)\)\), 6] === DFCoefficient[5\ L - 2\ L1 - \[CurlyPi] - 2\ \[CapitalOmega]1, 6]\)], "Input", CellLabel->"In[8]:=", ShowCellBracket->True], Cell[BoxData[ \(True\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell["AddIndirectTerms", "Subsubsection", CellTags->"AddIndirectTerms"], Cell[TextData[{ "AddIndirectTerms[expr] introduces the contribution of the indirect terms \ in one given expression, on condition that this is in an entirely numerical \ form (without parameter j). The option ", StyleBox["Perturber", FontWeight->"Bold"], " may assume the values: ", StyleBox["External", FontWeight->"Bold"], " (default choice) and ", StyleBox["Internal", FontWeight->"Bold"], StyleBox[":", FontVariations->{"CompatibilityType"->0}] }], "MathCaption"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(df = DFExpand[2, Simplification \[Rule] True] /. j \[Rule] 2;\)\), "\[IndentingNewLine]", \(AddIndirectTerms[df]\)}], "Input", CellLabel->"In[9]:=", ShowCellBracket->True], Cell[BoxData[ \(\((1\/2\ b[1\/2, 2, 0] + \((e\^2 + e1\^2)\)\ \((\(-2\)\ b[1\/2, 2, 0] + 1\/4\ \[Alpha]\ b[1\/2, 2, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, 2, 2])\) + \((\[Sigma]\^2 + \[Sigma]1\^2)\)\ \ \((\(-\(1\/4\)\)\ \[Alpha]\ b[3\/2, 1, 0] - 1\/4\ \[Alpha]\ b[3\/2, 3, 0])\))\)\ cos[\(-2\)\ L + 2\ L1] + e\^2\ \((3\/4\ b[1\/2, 2, 0] + 3\/4\ \[Alpha]\ b[1\/2, 2, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, 2, 2])\)\ cos[2\ L1 - 2\ \[CurlyPi]] + e\ \((\(-2\)\ b[1\/2, 2, 0] - 1\/2\ \[Alpha]\ b[1\/2, 2, 1])\)\ cos[\(-L\) + 2\ L1 - \[CurlyPi]] + e1\^2\ \((1\/2\ b[1\/2, 0, 0] + 3\/4\ \[Alpha]\ b[1\/2, 0, 1] + 1\/8\ \[Alpha]\^2\ b[1\/2, 0, 2])\)\ cos[ 2\ L1 - 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