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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[ 40006, 859]*) (*NotebookOutlinePosition[ 40973, 888]*) (* CellTagsIndexPosition[ 40929, 884]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[{ \(rp = r0 + dr/2\), "\[IndentingNewLine]", \(\(\(rm = r0 - dr/2\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(phipsi[t_, y_] = Series[\(-q[t, y]\)*\((D[q[t, y]*D[chi2[t, y], t], t] + D[q[t, y], y]*D[chi2[t, y], y]/t^2)\), {t, 0, 0}];\)\)}], "Input"], Cell[BoxData[ \(dr\/2 + r0\)], "Output"], Cell[BoxData[ \(\(-\(dr\/2\)\) + r0\)], "Output"] }, Open ]], Cell[BoxData[""], "Input"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(ppp[t_] = Simplify[ Normal[Series[ 2*phi[t4p[t]] + phipsi[t, vp], {dr, 0, 2}]]];\)\)\)], "Input"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(ppm[t_] = Simplify[ Normal[Series[ 2*phi[t4m[t]] + phipsi[t, vm], {dr, 0, 2}]]];\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(P = P /. Simplify[\(Solve[ Normal[Series[ppp[t] + ppm[t], {t, 0, \(-2\)}]] \[Equal] 0, P]\)[\([1]\)]]\[IndentingNewLine]\[IndentingNewLine]\ \[IndentingNewLine]\[IndentingNewLine]\[IndentingNewLine] \(j5 = Normal[Simplify[ Series[Series[Simplify[ppp[t] - ppm[t]], {t, 0, 0}], {dr, 0, 2}]]];\)\)\)\)], "Input"], Cell[BoxData[ \(\(3\ epsilon0\)\/\(4\ k\^2\)\)], "Output"] }, Open ]], Cell[BoxData[ \(\(psi[t_, y_] = Series[q[t, y]^2*\((D[p[t, y], t]*D[chi2[t, y], t] - D[p[t, y], y]*D[chi2[t, y], y]/t^2)\), {t, 0, 0}];\)\)], "Input"], Cell[BoxData[""], "Input"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(pp[t_] = Simplify[ Normal[Series[ phi[t4p[t]] - \((1/Sqrt[6])\)*th[t4p[t]]*delphi[t4p[t]] + psi[t, vp], {dr, 0, 2}]]];\)\)\)], "Input"], Cell[BoxData[ \(\(pm[t_] = Simplify[ Normal[Series[ phi[t4m[t]] - \((1/Sqrt[6])\)*cth[t4m[t]]*delphi[t4m[t]] + psi[t, vm], {dr, 0, 2}]]];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(Simplify[ Normal[Simplify[ Series[Series[Simplify[pp[t] - 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7\ c2)\)\ \[ExponentialE]\^\(5\ v0\)\ k\^2\ L - 96\ \((4\ c1 + 7\ c2)\)\ \[ExponentialE]\^\(6\ v0\)\ k\^2\ L - 384\ \((c1 + c2)\)\ \[ExponentialE]\^\(7\ v0\)\ k\^2\ L - \ \[ExponentialE]\^\(v0/2\)\ epsilon0\ \((234 + 55\ L\^2\ r0)\) + \[ExponentialE]\^\(\(13\ v0\)/2\)\ \ epsilon0\ \((234 + 55\ L\^2\ r0)\) + 72\ \[ExponentialE]\^\(\(7\ v0\)/2\)\ epsilon0\ \((\(-6\) + 17\ L\^2\ r0)\)\ v0 - 2\ \[ExponentialE]\^\(\(11\ v0\)/2\)\ epsilon0\ \((\(-55\)\ L\ \^2\ r0 + 36\ \((\(-7\) + 3\ v0)\))\) - 2\ \[ExponentialE]\^\(\(3\ v0\)/2\)\ epsilon0\ \((55\ L\^2\ \ r0 + 36\ \((7 + 3\ v0)\))\) + \[ExponentialE]\^\(\(5\ v0\)/2\)\ \ epsilon0\ \((\(-18\)\ \((13 + 84\ v0)\) + L\^2\ r0\ \((\(-55\) + 612\ v0)\))\) + \[ExponentialE]\^\(\(9\ v0\)/2\)\ \ epsilon0\ \((234 - 1512\ v0 + L\^2\ r0\ \((55 + 612\ v0)\))\))\))\)/\((1152\ \((\(-1\) + \ \[ExponentialE]\^v0)\)\ \((1 + \[ExponentialE]\^v0)\)\^3\ \((1 + \ \[ExponentialE]\^\(2\ v0\))\)\ k\^2)\) + \((\[ExponentialE]\^\(\(-v0\)/2\)\ \ \((768\ \((\(-c1\) + c2)\)\ k\^2\ L\^3\ r0 - 768\ \((c1 + c2)\)\ \[ExponentialE]\^\(9\ v0\)\ k\^2\ L\^3\ r0 + 192\ \[ExponentialE]\^\(5\ v0\)\ k\^2\ L\ \((\(-18\)\ c2 + 5\ c1\ L\^2\ r0)\) + 192\ \[ExponentialE]\^\(4\ v0\)\ k\^2\ L\ \((18\ c2 + 5\ c1\ L\^2\ r0)\) + 192\ \[ExponentialE]\^v0\ k\^2\ L\ \((\(-c1\)\ L\^2\ r0 + 2\ c2\ \((9 + 2\ L\^2\ r0)\))\) - 192\ \[ExponentialE]\^\(8\ v0\)\ k\^2\ L\ \((c1\ L\^2\ r0 + 2\ c2\ \((9 + 2\ L\^2\ r0)\))\) + 192\ \[ExponentialE]\^\(2\ v0\)\ k\^2\ L\ \((3\ c1\ L\^2\ r0 \ - 2\ c2\ \((9 + 4\ L\^2\ r0)\))\) + 192\ \[ExponentialE]\^\(6\ v0\)\ k\^2\ L\ \((\(-3\)\ c1\ L\^2\ \ r0 + 2\ c2\ \((9 + 4\ L\^2\ r0)\))\) - 192\ \[ExponentialE]\^\(3\ v0\)\ k\^2\ L\ \((3\ c1\ L\^2\ r0 \ + 2\ c2\ \((9 + 4\ L\^2\ r0)\))\) + 192\ \[ExponentialE]\^\(7\ v0\)\ k\^2\ L\ \((3\ c1\ L\^2\ r0 \ + 2\ c2\ \((9 + 4\ L\^2\ r0)\))\) + \[ExponentialE]\^\(v0/2\)\ \ \((288\ L\^2\ \((3\ epsilon2 - 2\ b1\ k\^2 + 2\ k\^4\ Q)\) - epsilon0\ \((\(-2052\) + 216\ k\^2\ L\^2 + 468\ L\^2\ r0 + 175\ L\^4\ r0\^2)\))\) + \[ExponentialE]\^\(\(17\ \ v0\)/2\)\ \((\(-288\)\ L\^2\ \((3\ epsilon2 + 2\ k\^2\ \((b1 + k\^2\ Q)\))\) + epsilon0\ \((\(-2052\) + 216\ k\^2\ L\^2 + 468\ L\^2\ r0 + 175\ L\^4\ r0\^2)\))\) - 72\ \[ExponentialE]\^\(\(9\ v0\)/2\)\ \((16\ b1\ k\^2\ L\^2 + 32\ b2\ k\^2\ L\^2 + 3\ epsilon0\ \((156 + 36\ L\^2\ r0 - L\^4\ r0\^2)\)\ v0)\) + 6\ \[ExponentialE]\^\(\(3\ v0\)/2\)\ epsilon0\ \((\(-96\)\ \ L\^2\ r0\ \((\(-2\) + 3\ v0)\) + 5\ L\^4\ r0\^2\ \((\(-11\) + 6\ v0)\) + 36\ \((\(-41\) + 18\ v0)\))\) + 6\ \[ExponentialE]\^\(\(15\ v0\)/2\)\ epsilon0\ \((\(-96\)\ L\ \^2\ r0\ \((2 + 3\ v0)\) + 5\ L\^4\ r0\^2\ \((11 + 6\ v0)\) + 36\ \((41 + 18\ v0)\))\) + 6\ \[ExponentialE]\^\(\(7\ v0\)/2\)\ epsilon0\ \((288\ L\^2\ \ r0\ \((\(-2\) + v0)\) + L\^4\ r0\^2\ \((\(-55\) + 6\ v0)\) + 36\ \((\(-41\) + 138\ v0)\))\) + 6\ \[ExponentialE]\^\(\(11\ v0\)/2\)\ epsilon0\ \((288\ L\^2\ \ r0\ \((2 + v0)\) + L\^4\ r0\^2\ \((55 + 6\ v0)\) + 36\ \((41 + 138\ v0)\))\) + 2\ \[ExponentialE]\^\(\(5\ v0\)/2\)\ \((288\ L\^2\ \((\(-3\)\ \ epsilon2 + 2\ k\^2\ \((b1 + b2 - k\^2\ Q)\))\) + epsilon0\ \((6804 + 216\ k\^2\ L\^2 - 8424\ v0 + L\^4\ r0\^2\ \((\(-155\) + 54\ v0)\) + 36\ L\^2\ r0\ \((13 + 54\ v0)\))\))\) + 2\ \[ExponentialE]\^\(\(13\ v0\)/2\)\ \((288\ L\^2\ \((3\ \ epsilon2 + 2\ k\^2\ \((b1 + b2 + k\^2\ Q)\))\) + epsilon0\ \((\(-216\)\ k\^2\ L\^2 - 324\ \((21 + 26\ v0)\) + 36\ L\^2\ r0\ \((\(-13\) + 54\ v0)\) + L\^4\ r0\^2\ \((155 + 54\ v0)\))\))\))\))\)/\((1152\ \((\(-1\) + \ \[ExponentialE]\^\(2\ v0\))\)\^3\ \((1 + \[ExponentialE]\^\(2\ v0\))\)\ k\^2\ \ L\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(j2 = Normal[Simplify[ Series[Series[Simplify[pp[t] - k^2*Q/3], {t, 0, 0}], {dr, 0, 2}]]] /. {t -> 1}\)\)\)], "Input"], Cell[BoxData[ \(\(-\(\(k\^2\ Q\)\/3\)\) - \(dr\^2\ epsilon0\ L\^2\ \((3 + 3\ \ \[ExponentialE]\^\(3\ v0\) + \[ExponentialE]\^\(2\ v0\)\ \((13 - 6\ v0)\) + \ \[ExponentialE]\^v0\ \((13 + 6\ v0)\))\)\)\/\(384\ \((1 + \ \[ExponentialE]\^v0)\)\^3\ k\^2\) - \(\(1\/\(48\ \((1 + \ \[ExponentialE]\^v0)\)\^3\ k\^2\)\)\((dr\ \((8\ c2\ \[ExponentialE]\^\(v0/2\)\ \ \((1 + \[ExponentialE]\^v0)\)\^2\ k\^2\ L + epsilon0\ \((3 + L\^2\ r0 + \[ExponentialE]\^\(3\ v0\)\ \((3 + L\^2\ r0)\) - 3\ \[ExponentialE]\^\(2\ v0\)\ \((1 - L\^2\ r0 + 6\ v0)\) + 3\ \[ExponentialE]\^v0\ \((\(-1\) + L\^2\ r0 + 6\ v0)\))\))\))\)\) + \((\(-48\)\ \((\(-1\) + \ \[ExponentialE]\^\(2\ v0\))\)\^2\ L\ \((12\ c2\ \[ExponentialE]\^\(v0/2\)\ \ \((\(-1\) + \[ExponentialE]\^v0)\)\ k\^2 + \((1 + \[ExponentialE]\^v0)\)\ L\ \ \((3\ \((\(-1\) + \[ExponentialE]\^v0)\)\ epsilon2 - 2\ c1\ \[ExponentialE]\^\(v0/2\)\ k\^2\ L\ r0)\))\) + epsilon0\ \((324 - 36\ k\^2\ L\^2 - 13\ L\^4\ r0\^2 + \[ExponentialE]\^\(6\ v0\)\ \((\(-324\) + 36\ k\^2\ L\^2 + 13\ L\^4\ r0\^2)\) - 108\ \[ExponentialE]\^\(3\ v0\)\ \((\(-36\) + L\^4\ r0\^2)\)\ v0 + 2\ \[ExponentialE]\^\(5\ v0\)\ \((L\^4\ r0\^2\ \((50 - 9\ v0)\) + 324\ \((2 + v0)\))\) - 2\ \[ExponentialE]\^v0\ \((\(-324\)\ \((\(-2\) + v0)\) + L\^4\ r0\^2\ \((50 + 9\ v0)\))\) - \[ExponentialE]\^\(4\ v0\)\ \ \((108\ k\^2\ L\^2 + 324\ \((5 + 8\ v0)\) + L\^4\ r0\^2\ \((\(-161\) + 72\ v0)\))\) + \[ExponentialE]\^\(2\ v0\)\ \ \((108\ k\^2\ L\^2 - 324\ \((\(-5\) + 8\ v0)\) - L\^4\ r0\^2\ \((161 + 72\ v0)\))\))\))\)/\((288\ \((\(-1\) + \ \[ExponentialE]\^\(2\ v0\))\)\^3\ k\^2\ L\^2)\)\)], "Output"] }, Open ]], Cell[BoxData[""], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\(QQ = Q /. \(Solve[j2 \[Equal] 0, Q]\)[\([1]\)]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\) \)\)\)], "Input"], Cell[BoxData[ \(\(-\(\(1\/k\^2\)\((3\ \((\(dr\^2\ epsilon0\ L\^2\ \((3 + 3\ \ \[ExponentialE]\^\(3\ v0\) + \[ExponentialE]\^\(2\ v0\)\ \((13 - 6\ v0)\) + \ \[ExponentialE]\^v0\ \((13 + 6\ v0)\))\)\)\/\(384\ \((1 + \ \[ExponentialE]\^v0)\)\^3\ k\^2\) + \(\(1\/\(48\ \((1 + \ \[ExponentialE]\^v0)\)\^3\ k\^2\)\)\((dr\ \((8\ c2\ \[ExponentialE]\^\(v0/2\)\ \ \((1 + \[ExponentialE]\^v0)\)\^2\ k\^2\ L + epsilon0\ \((3 + L\^2\ r0 + \[ExponentialE]\^\(3\ v0\)\ \((3 + L\^2\ r0)\) - 3\ \[ExponentialE]\^\(2\ v0\)\ \((1 - L\^2\ r0 + 6\ v0)\) + 3\ \[ExponentialE]\^v0\ \((\(-1\) + L\^2\ r0 + 6\ v0)\))\))\))\)\) - \((\(-48\)\ \((\(-1\) \ + \[ExponentialE]\^\(2\ v0\))\)\^2\ L\ \((12\ c2\ \[ExponentialE]\^\(v0/2\)\ \ \((\(-1\) + \[ExponentialE]\^v0)\)\ k\^2 + \((1 + \[ExponentialE]\^v0)\)\ L\ \ \((3\ \((\(-1\) + \[ExponentialE]\^v0)\)\ epsilon2 - 2\ c1\ \[ExponentialE]\^\(v0/2\)\ k\^2\ L\ \ r0)\))\) + epsilon0\ \((324 - 36\ k\^2\ L\^2 - 13\ L\^4\ r0\^2 + \[ExponentialE]\^\(6\ v0\)\ \ \((\(-324\) + 36\ k\^2\ L\^2 + 13\ L\^4\ r0\^2)\) - 108\ \[ExponentialE]\^\(3\ v0\)\ \((\(-36\) + L\^4\ r0\^2)\)\ v0 + 2\ \[ExponentialE]\^\(5\ v0\)\ \((L\^4\ r0\^2\ \ \((50 - 9\ v0)\) + 324\ \((2 + v0)\))\) - 2\ \[ExponentialE]\^v0\ \((\(-324\)\ \((\(-2\) + v0)\) + L\^4\ r0\^2\ \((50 + 9\ v0)\))\) - \[ExponentialE]\^\(4\ \ v0\)\ \((108\ k\^2\ L\^2 + 324\ \((5 + 8\ v0)\) + L\^4\ r0\^2\ \((\(-161\) + 72\ v0)\))\) + \[ExponentialE]\^\(2\ v0\ \)\ \((108\ k\^2\ L\^2 - 324\ \((\(-5\) + 8\ v0)\) - L\^4\ r0\^2\ \((161 + 72\ v0)\))\))\))\)/\((288\ \((\(-1\) + \ \[ExponentialE]\^\(2\ v0\))\)\^3\ k\^2\ L\^2)\))\))\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(j4 = Simplify[j1 /. {Q \[Rule] QQ}]\)\)\)], "Input"], Cell[BoxData[ \(\((\[ExponentialE]\^\(\(-v0\)/2\)\ \((1536\ \((c1 - c2)\)\ \[ExponentialE]\^v0\ k\^2\ L\^3\ \((dr - 2\ r0)\) + 1536\ \((c1 + c2)\)\ \[ExponentialE]\^\(8\ v0\)\ k\^2\ L\^3\ \((dr - 2\ r0)\) - 1536\ \((c1 - c2)\)\ k\^2\ L\^3\ \((dr + 2\ r0)\) - 1536\ \((c1 + c2)\)\ \[ExponentialE]\^\(9\ v0\)\ k\^2\ L\^3\ \((dr + 2\ r0)\) - 3072\ \[ExponentialE]\^\(4\ v0\)\ k\^2\ L\^3\ \((c2\ dr - 2\ c1\ r0)\) + 3072\ \[ExponentialE]\^\(5\ v0\)\ k\^2\ L\^3\ \((c2\ dr + 2\ c1\ r0)\) + 3072\ \[ExponentialE]\^\(2\ v0\)\ k\^2\ L\^3\ \((c1\ dr - 2\ c2\ r0)\) - 3072\ \[ExponentialE]\^\(6\ v0\)\ k\^2\ L\^3\ \((c1\ dr - 2\ c2\ r0)\) - 3072\ \[ExponentialE]\^\(3\ v0\)\ k\^2\ L\^3\ \((c1\ dr + 2\ c2\ r0)\) + 3072\ \[ExponentialE]\^\(7\ v0\)\ k\^2\ L\^3\ \((c1\ dr + 2\ c2\ r0)\) + \[ExponentialE]\^\(\(17\ v0\)/2\)\ \ \((\(-2304\)\ b1\ k\^2\ L\^2 + epsilon0\ \((109\ dr\^2\ L\^4 + 4\ dr\ L\^2\ \((342 + 91\ L\^2\ r0)\) + 4\ \((\(-108\) + 468\ L\^2\ r0 + 97\ L\^4\ r0\^2)\))\))\) - \ \[ExponentialE]\^\(v0/2\)\ \((2304\ b1\ k\^2\ L\^2 + epsilon0\ \((109\ dr\^2\ L\^4 + 4\ dr\ L\^2\ \((342 + 91\ L\^2\ r0)\) + 4\ \((\(-108\) + 468\ L\^2\ r0 + 97\ L\^4\ r0\^2)\))\))\) - 72\ \[ExponentialE]\^\(\(9\ v0\)/2\)\ \((64\ b1\ k\^2\ L\^2 + 128\ b2\ k\^2\ L\^2 + epsilon0\ \((144 - 168\ dr\ L\^2 - 15\ dr\^2\ L\^4 + 432\ L\^2\ r0 + 68\ dr\ L\^4\ r0 - 60\ L\^4\ r0\^2)\)\ v0)\) - 6\ \[ExponentialE]\^\(\(15\ v0\)/2\)\ epsilon0\ \((24\ dr\ L\^2\ \ \((11 + 24\ v0)\) + dr\^2\ L\^4\ \((\(-37\) + 48\ v0)\) - 12\ \((60 - 32\ L\^2\ r0\ \((2 + 3\ v0)\) + L\^4\ r0\^2\ \((\(-15\) + 16\ v0)\))\))\) - 6\ \[ExponentialE]\^\(\(3\ v0\)/2\)\ epsilon0\ \((24\ dr\ L\^2\ \ \((\(-11\) + 24\ v0)\) + dr\^2\ L\^4\ \((37 + 48\ v0)\) - 12\ \((\(-60\) + 32\ L\^2\ r0\ \((2 - 3\ v0)\) + L\^4\ r0\^2\ \((15 + 16\ v0)\))\))\) - 6\ \[ExponentialE]\^\(\(11\ v0\)/2\)\ epsilon0\ \((24\ dr\ L\^2\ \ \((11 + 60\ v0)\) + dr\^2\ L\^4\ \((\(-37\) + 132\ v0)\) - 12\ \((60 + 144\ v0 + 96\ L\^2\ r0\ \((2 + v0)\) + L\^4\ r0\^2\ \((\(-15\) + 44\ v0)\))\))\) - 6\ \[ExponentialE]\^\(\(7\ v0\)/2\)\ epsilon0\ \((24\ dr\ L\^2\ \ \((\(-11\) + 60\ v0)\) + dr\^2\ L\^4\ \((37 + 132\ v0)\) - 12\ \((\(-60\) + 96\ L\^2\ r0\ \((\(-2\) + v0)\) + 144\ v0 + L\^4\ r0\^2\ \((15 + 44\ v0)\))\))\) + 2\ \[ExponentialE]\^\(\(13\ v0\)/2\)\ \((2304\ \((b1 + b2)\)\ k\^2\ L\^2 + epsilon0\ \((dr\^2\ L\^4\ \((\(-331\) + 270\ v0)\) + 4\ \((\(-324\)\ \((3 + 2\ v0)\) + 36\ L\^2\ r0\ \((\(-13\) + 54\ v0)\) + L\^4\ r0\^2\ \((\(-367\) + 270\ v0)\))\) + 4\ dr\ L\^2\ \((54 + 756\ v0 + L\^2\ r0\ \((\(-91\) + 306\ v0)\))\))\))\) + 2\ \[ExponentialE]\^\(\(5\ v0\)/2\)\ \((2304\ \((b1 + b2)\)\ k\^2\ L\^2 + epsilon0\ \((dr\^2\ L\^4\ \((331 + 270\ v0)\) + 4\ \((972 - 648\ v0 + 36\ L\^2\ r0\ \((13 + 54\ v0)\) + L\^4\ r0\^2\ \((367 + 270\ v0)\))\) + 4\ dr\ L\^2\ \((\(-54\) + 756\ v0 + L\^2\ r0\ \((91 + 306\ v0)\))\))\))\))\))\)/\((4608\ \ \((\(-1\) + \[ExponentialE]\^\(2\ v0\))\)\^3\ \((1 + \[ExponentialE]\^\(2\ v0\ \))\)\ k\^2\ L\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(bb1 = Normal[Simplify[ Series[b1 /. \(Solve[{j4 \[Equal] 0, j5 \[Equal] 0}, {b1, b2}]\)[\([1]\)], {dr, 0, 2}]]]\)], "Input"], Cell[BoxData[ \(\((dr\^2\ epsilon0\ L\^2\ \((\(-109\) + 109\ \[ExponentialE]\^\(4\ v0\) - 36\ \[ExponentialE]\^\(2\ v0\)\ v0 - 8\ \[ExponentialE]\^\(3\ v0\)\ \((\(-55\) + 36\ v0)\) - 8\ \[ExponentialE]\^v0\ \((55 + 36\ v0)\))\))\)/\((2304\ \((1 + \ \[ExponentialE]\^v0)\)\^2\ \((1 + \[ExponentialE]\^\(2\ v0\))\)\ k\^2)\) + \ \((\[ExponentialE]\^\(\(-v0\)/2\)\ \((768\ \((\(-c1\) + c2)\)\ k\^2\ L\^3\ r0 - 768\ \((c1 - c2)\)\ \[ExponentialE]\^v0\ k\^2\ L\^3\ r0 + 768\ \((c1 + c2)\)\ \[ExponentialE]\^\(2\ v0\)\ k\^2\ L\^3\ r0 + 768\ \((c1 + c2)\)\ \[ExponentialE]\^\(3\ v0\)\ k\^2\ L\^3\ r0 + 768\ \((c1 - c2)\)\ \[ExponentialE]\^\(4\ v0\)\ k\^2\ L\^3\ r0 + 768\ \((c1 - c2)\)\ \[ExponentialE]\^\(5\ v0\)\ k\^2\ L\^3\ r0 - 768\ \((c1 + c2)\)\ \[ExponentialE]\^\(6\ v0\)\ k\^2\ L\^3\ r0 - 768\ \((c1 + c2)\)\ \[ExponentialE]\^\(7\ v0\)\ k\^2\ L\^3\ r0 - \ \[ExponentialE]\^\(v0/2\)\ epsilon0\ \((\(-108\) + 468\ L\^2\ r0 + 97\ L\^4\ r0\^2)\) + \[ExponentialE]\^\(\(13\ v0\)/2\)\ \ epsilon0\ \((\(-108\) + 468\ L\^2\ r0 + 97\ L\^4\ r0\^2)\) + 72\ \[ExponentialE]\^\(\(7\ v0\)/2\)\ epsilon0\ \((36 - 48\ L\^2\ r0 + 7\ L\^4\ r0\^2)\)\ v0 + 18\ \[ExponentialE]\^\(\(11\ v0\)/2\)\ epsilon0\ \((60 - 32\ L\^2\ r0\ \((2 + 3\ v0)\) + L\^4\ r0\^2\ \((\(-15\) + 16\ v0)\))\) + 18\ \[ExponentialE]\^\(\(3\ v0\)/2\)\ epsilon0\ \((\(-60\) + 32\ L\^2\ r0\ \((2 - 3\ v0)\) + L\^4\ r0\^2\ \((15 + 16\ v0)\))\) + 3\ \[ExponentialE]\^\(\(9\ v0\)/2\)\ epsilon0\ \((\(-36\)\ \ \((17 + 12\ v0)\) + 12\ L\^2\ r0\ \((13 + 108\ v0)\) + L\^4\ r0\^2\ \((\(-277\) + 180\ v0)\))\) + 3\ \[ExponentialE]\^\(\(5\ v0\)/2\)\ epsilon0\ \((612 - 432\ v0 + 12\ L\^2\ r0\ \((\(-13\) + 108\ v0)\) + L\^4\ r0\^2\ \((277 + 180\ v0)\))\))\))\)/\((576\ \((\(-1\) + \ \[ExponentialE]\^\(2\ v0\))\)\^2\ \((1 + \[ExponentialE]\^\(2\ v0\))\)\ k\^2\ \ L\^2)\) + \((dr\ \[ExponentialE]\^\(\(-v0\)/2\)\ \((384\ \((\(-c1\) + c2)\)\ k\^2\ L + 384\ \((c1 - c2)\)\ \[ExponentialE]\^v0\ k\^2\ L - 384\ \((c1 + c2)\)\ \[ExponentialE]\^\(2\ v0\)\ k\^2\ L + 384\ \((c1 + c2)\)\ \[ExponentialE]\^\(3\ v0\)\ k\^2\ L + 384\ \((c1 - c2)\)\ \[ExponentialE]\^\(4\ v0\)\ k\^2\ L - 384\ \((c1 - c2)\)\ \[ExponentialE]\^\(5\ v0\)\ k\^2\ L + 384\ \((c1 + c2)\)\ \[ExponentialE]\^\(6\ v0\)\ k\^2\ L - 384\ \((c1 + c2)\)\ \[ExponentialE]\^\(7\ v0\)\ k\^2\ L - \ \[ExponentialE]\^\(v0/2\)\ epsilon0\ \((342 + 91\ L\^2\ r0)\) + \[ExponentialE]\^\(\(13\ v0\)/2\)\ \ epsilon0\ \((342 + 91\ L\^2\ r0)\) - 1296\ \[ExponentialE]\^\(\(7\ v0\)/2\)\ epsilon0\ v0 - 36\ \[ExponentialE]\^\(\(3\ v0\)/2\)\ epsilon0\ \((\(-11\) + 24\ v0)\) - 36\ \[ExponentialE]\^\(\(11\ v0\)/2\)\ epsilon0\ \((11 + 24\ v0)\) + \[ExponentialE]\^\(\(9\ v0\)/2\)\ epsilon0\ \ \((18\ \((\(-13\) + 84\ v0)\) + L\^2\ r0\ \((\(-53\) + 612\ v0)\))\) + \[ExponentialE]\^\(\(5\ v0\)/2\)\ \ epsilon0\ \((18\ \((13 + 84\ v0)\) + L\^2\ r0\ \((53 + 612\ v0)\))\))\))\)/\((576\ \((\(-1\) + \ \[ExponentialE]\^\(2\ v0\))\)\^2\ \((1 + \[ExponentialE]\^\(2\ v0\))\)\ k\^2)\ \)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(bb2 = Normal[Simplify[ Series[b2 /. \(Solve[{j4 \[Equal] 0, j5 \[Equal] 0}, {b1, b2}]\)[\([1]\)], {dr, 0, 2}]]]\)\)\)], "Input"], Cell[BoxData[ \(\(-\(\((dr\^2\ epsilon0\ L\^2\ \((\(-109\) + 109\ \[ExponentialE]\^\(4\ v0\) - 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18\ \[ExponentialE]\^\(\(11\ v0\)/2\)\ epsilon0\ \((60 + 32\ L\^2\ r0\ \((2 + 3\ v0)\) + L\^4\ r0\^2\ \((\(-15\) + 16\ v0)\))\) - 18\ \[ExponentialE]\^\(\(3\ v0\)/2\)\ epsilon0\ \((\(-60\) + 32\ L\^2\ r0\ \((\(-2\) + 3\ v0)\) + L\^4\ r0\^2\ \((15 + 16\ v0)\))\) - 3\ \[ExponentialE]\^\(\(9\ v0\)/2\)\ epsilon0\ \((\(-36\)\ \ \((17 + 12\ v0)\) - 12\ L\^2\ r0\ \((13 + 108\ v0)\) + L\^4\ r0\^2\ \((\(-277\) + 180\ v0)\))\) - 3\ \[ExponentialE]\^\(\(5\ v0\)/2\)\ epsilon0\ \((612 + 12\ L\^2\ r0\ \((13 - 108\ v0)\) - 432\ v0 + L\^4\ r0\^2\ \((277 + 180\ v0)\))\))\))\)/\((576\ \((\(-1\) + \ \[ExponentialE]\^\(2\ v0\))\)\^2\ \((1 + \[ExponentialE]\^\(2\ v0\))\)\ k\^2\ \ L\^2)\) + \((dr\ \[ExponentialE]\^\(\(-v0\)/2\)\ \((\(-384\)\ \((c1 + c2)\)\ k\^2\ L + 384\ \((c1 + c2)\)\ \[ExponentialE]\^v0\ k\^2\ L - 384\ \((c1 - c2)\)\ \[ExponentialE]\^\(2\ v0\)\ k\^2\ L + 384\ \((c1 - c2)\)\ \[ExponentialE]\^\(3\ v0\)\ k\^2\ L + 384\ \((c1 + c2)\)\ \[ExponentialE]\^\(4\ v0\)\ k\^2\ L - 384\ \((c1 + c2)\)\ \[ExponentialE]\^\(5\ v0\)\ k\^2\ L + 384\ \((c1 - c2)\)\ \[ExponentialE]\^\(6\ v0\)\ k\^2\ L - 384\ \((c1 - c2)\)\ \[ExponentialE]\^\(7\ v0\)\ k\^2\ L - 19\ \[ExponentialE]\^\(v0/2\)\ epsilon0\ \((\(-18\) + L\^2\ r0)\) + 19\ \[ExponentialE]\^\(\(13\ v0\)/2\)\ epsilon0\ \((\(-18\) + L\^2\ r0)\) + 1296\ \[ExponentialE]\^\(\(7\ v0\)/2\)\ epsilon0\ v0 + 36\ \[ExponentialE]\^\(\(3\ v0\)/2\)\ epsilon0\ \((\(-11\) + 24\ v0)\) + 36\ \[ExponentialE]\^\(\(11\ v0\)/2\)\ epsilon0\ \((11 + 24\ v0)\) + \[ExponentialE]\^\(\(5\ v0\)/2\)\ epsilon0\ \ \((\(-18\)\ \((13 + 84\ v0)\) + L\^2\ r0\ \((\(-163\) + 612\ v0)\))\) + \[ExponentialE]\^\(\(9\ v0\)/2\)\ \ epsilon0\ \((234 - 1512\ v0 + L\^2\ r0\ \((163 + 612\ v0)\))\))\))\)/\((576\ \((\(-1\) + \ \[ExponentialE]\^\(2\ v0\))\)\^2\ \((1 + \[ExponentialE]\^\(2\ v0\))\)\ k\^2)\ \)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(j10 = Normal[Series[ Simplify[ Series[\((ppp[t] - k^2*chi2[t, vp])\) /. {b1 \[Rule] bb1, b2 \[Rule] bb2, Q \[Rule] QQ}, {dr, 0, 2}]], {t, 0, 0}]] /. {t \[Rule] 1}\)\)\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[BoxData[""], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\ \(Normal[Simplify[ Series[Series[\((k^2*chi2[t, vp]/3 - zetam[1])\) /. {Q \[Rule] QQ}, {t, 0, 0}], {dr, 0, 2}]]] /. {t \[Rule] 1}\)\)\)], "Input"], Cell[BoxData[ \(\(-\(\(dr\^2\ epsilon0\ L\^2\ \((3 + 3\ \[ExponentialE]\^\(3\ v0\) + \[ExponentialE]\^\(2\ v0\)\ \ \((13 - 6\ v0)\) + \[ExponentialE]\^v0\ \((13 + 6\ v0)\))\)\)\/\(384\ \((1 + \ \[ExponentialE]\^v0)\)\^3\ k\^2\)\)\) - \(\(1\/\(48\ \((1 + \ \[ExponentialE]\^v0)\)\^3\ k\^2\)\)\((dr\ \((8\ c2\ \[ExponentialE]\^\(v0/2\)\ \ \((1 + \[ExponentialE]\^v0)\)\^2\ k\^2\ L + epsilon0\ \((3 + L\^2\ r0 + \[ExponentialE]\^\(3\ v0\)\ \((3 + L\^2\ r0)\) - 3\ \[ExponentialE]\^\(2\ v0\)\ \((1 - L\^2\ r0 + 6\ v0)\) + 3\ \[ExponentialE]\^v0\ \((\(-1\) + L\^2\ r0 + 6\ v0)\))\))\))\)\) + \((32\ \ \[ExponentialE]\^\(v0/2\)\ \((\(-1\) + \[ExponentialE]\^\(2\ v0\))\)\^2\ k\^2\ \ L\ \((\(-6\)\ c2\ \((\(-1\) + \[ExponentialE]\^v0)\) + c1\ \((1 + \[ExponentialE]\^v0)\)\ L\^2\ r0)\) - epsilon0\ \((\(-108\) - L\^4\ r0\^2 + \[ExponentialE]\^\(6\ v0\)\ \((108 + L\^4\ r0\^2)\) + 36\ \[ExponentialE]\^\(3\ v0\)\ \((\(-36\) + L\^4\ r0\^2)\)\ v0 + 6\ \[ExponentialE]\^\(5\ v0\)\ \((L\^4\ r0\^2\ \((\(-2\) + v0)\) - 36\ \((2 + v0)\))\) + 6\ \[ExponentialE]\^v0\ \((\(-36\)\ \((\(-2\) + v0)\) + L\^4\ r0\^2\ \((2 + v0)\))\) + 3\ \[ExponentialE]\^\(4\ v0\)\ \((L\^4\ r0\^2\ \((\(-9\) + 8\ v0)\) + 36\ \((5 + 8\ v0)\))\) + 3\ \[ExponentialE]\^\(2\ v0\)\ \((36\ \((\(-5\) + 8\ v0)\) + L\^4\ r0\^2\ \((9 + 8\ v0)\))\))\))\)/\((96\ \((\(-1\) + \ \[ExponentialE]\^\(2\ v0\))\)\^3\ k\^2\ L\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(Normal[ Simplify[Series[ Series[ppp[t] /. {b1 \[Rule] bb1, b2 \[Rule] bb2, Q \[Rule] QQ}, {t, 0, 0}], {dr, 0, 2}]]]\)\)\)], "Input"], Cell[BoxData[ \(\(-\(\(dr\^2\ epsilon0\ L\^2\ \((3 + 3\ \[ExponentialE]\^\(3\ v0\) + \[ExponentialE]\^\(2\ v0\)\ \ \((13 - 6\ v0)\) + \[ExponentialE]\^v0\ \((13 + 6\ v0)\))\)\)\/\(128\ \((1 + \ \[ExponentialE]\^v0)\)\^3\ k\^2\)\)\) - \(\(1\/\(16\ \((1 + \ \[ExponentialE]\^v0)\)\^3\ k\^2\)\)\((dr\ \((8\ c2\ \[ExponentialE]\^\(v0/2\)\ \ \((1 + \[ExponentialE]\^v0)\)\^2\ k\^2\ L + epsilon0\ \((3 + L\^2\ r0 + \[ExponentialE]\^\(3\ v0\)\ \((3 + L\^2\ r0)\) - 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