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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[ 82879, 2264]*) (*NotebookOutlinePosition[ 83934, 2297]*) (* CellTagsIndexPosition[ 83890, 2293]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ TagBox[ StyleBox[\(Energy\ Error\ in\ CR\), "Section"], DisplayForm]], "Output"], Cell[BoxData[""], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(z\ = \ m/Lc\ \ x\)], "Input"], Cell[BoxData[ \(\(m\ x\)\/Lc\)], "Output"] }, Open ]], Cell[BoxData[ \(v[k_, q_] := \ \((co\ + \ c2\ x^2 \((k^2\ + \ q^2)\)/2\ + \ c4\ x^4 \((k^2\ + \ q^2)\)^2/4)\) Exp[\(-z^2\) \((k^2\ + \ q^2)\)/2]\)], "Input"], Cell[BoxData[ \(F[s_] := 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c0\^2 + 16\ c0\ k\^2\ x\^2\ \((c2 + c4\ k\^2\ x\^2)\) + k\^4\ x\^4\ \((8\ c2\^2 + 16\ c2\ c4\ k\^2\ x\^2 + 7\ c4\^2\ k\^4\ x\^4)\))\))\) + 256\ b\^9\ k\ \@\[Pi]\ x\ \((c0 + c2\ k\^2\ x\^2 + c4\ k\^4\ \ x\^4)\)\^2\ Erfi[b\ k\ x])\))\)\)\)], "Output"] }, Open ]], Cell[BoxData[ \(t[x_, k_] := \ \(1\/\(512\ b\^9\ x\)\) \((\[ExponentialE]\^\(\(-2\)\ b\^2\ \ k\^2\ x\^2\)\ \@\[Pi]\ \((\(-\[ExponentialE]\^\(b\^2\ k\^2\ x\^2\)\)\ \ \((\(-105\)\ c4\^2 + 30\ b\^2\ c4\ \((4\ c2 + 3\ c4\ k\^2\ x\^2)\) + 8\ b\^6\ \((16\ c0\ c2 + 12\ c2\^2\ k\^2\ x\^2 + 24\ c0\ c4\ k\^2\ x\^2 + 28\ c2\ c4\ k\^4\ x\^4 + 13\ c4\^2\ k\^6\ x\^6)\) + 12\ b\^4\ \((4\ c2\^2 + 16\ c2\ c4\ k\^2\ x\^2 + c4\ \((8\ c0 + 9\ c4\ k\^4\ x\^4)\))\) + 32\ b\^8\ \((8\ c0\^2 + 16\ c0\ k\^2\ x\^2\ \((c2 + c4\ k\^2\ x\^2)\) + k\^4\ x\^4\ \((8\ c2\^2 + 16\ c2\ c4\ k\^2\ x\^2 + 7\ c4\^2\ k\^4\ x\^4)\))\))\) + 256\ b\^9\ k\ \@\[Pi]\ x\ \((c0 + c2\ k\^2\ x\^2 + c4\ k\^4\ \ x\^4)\)\^2\ Erfi[b\ k\ x])\))\)\)], "Input"], Cell[BoxData[ \(\(\(Integrate[ s^5\ t[x, Sqrt[1 - s^2]]/8\ - \ s^3 \((\ 4\ - \ 5 s\ + s^2)\) t[x, 1 - s]/24, {s, 0, 1}]\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[t[x, k], {x, 0, 6}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(\(\((\(-256\)\ b\^8\ c0\^2 - 128\ b\^6\ c0\ c2 - 120\ b\^2\ c2\ c4 + 105\ c4\^2 - 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ \@\[Pi]\)\/\(512\ b\^9\ x\)\), "+", \(\(1\/\(512\ b\^9\)\) \((\((512\ b\^10\ c0\^2\ k\^2 - 512\ b\^8\ c0\ c2\ k\^2 - 192\ b\^4\ c2\ c4\ k\^2 - 90\ b\^2\ c4\^2\ k\^2 - 2\ b\^2\ \((\(-256\)\ b\^8\ c0\^2 - 128\ b\^6\ c0\ c2 - 120\ b\^2\ c2\ c4 + 105\ c4\^2 - 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ k\^2 - b\^2\ \((256\ b\^8\ c0\^2 + 128\ b\^6\ c0\ c2 + 120\ b\^2\ c2\ c4 - 105\ c4\^2 + 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ k\^2 - 8\ b\^6\ \((12\ c2\^2\ k\^2 + 24\ c0\ c4\ k\^2)\))\)\ \@\[Pi]\ x)\)\), "+", \(\(1\/\(512\ b\^9\)\) \((\((\(-224\)\ b\^6\ c2\ c4\ k\^4 - 108\ b\^4\ c4\^2\ k\^4 + 2\ b\^4\ \((\(-256\)\ b\^8\ c0\^2 - 128\ b\^6\ c0\ c2 - 120\ b\^2\ c2\ c4 + 105\ c4\^2 - 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ k\^4 - 1\/2\ b\^4\ \((256\ b\^8\ c0\^2 + 128\ b\^6\ c0\ c2 + 120\ b\^2\ c2\ c4 - 105\ c4\^2 + 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ k\^4 - 32\ b\^8\ \((8\ c2\^2\ k\^4 + 16\ c0\ c4\ k\^4)\) - 2\ b\^2\ k\^2\ \((512\ b\^10\ c0\^2\ k\^2 - 512\ b\^8\ c0\ c2\ k\^2 - 192\ b\^4\ c2\ c4\ k\^2 - 90\ b\^2\ c4\^2\ k\^2 - b\^2\ \((256\ b\^8\ c0\^2 + 128\ b\^6\ c0\ c2 + 120\ b\^2\ c2\ c4 - 105\ c4\^2 + 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ k\^2 - 8\ b\^6\ \((12\ c2\^2\ k\^2 + 24\ c0\ c4\ k\^2)\))\) - b\^2\ k\^2\ \((512\ b\^8\ c0\ c2\ k\^2 + 192\ b\^4\ c2\ c4\ k\^2 + 90\ b\^2\ c4\^2\ k\^2 + 8\ b\^6\ \((12\ c2\^2\ k\^2 + 24\ c0\ c4\ k\^2)\))\) + 256\ b\^9\ k\ \((\(2\ b\^3\ c0\^2\ k\^3\)\/\(3\ \@\[Pi]\) + \ \(4\ b\ c0\ c2\ k\^3\)\/\@\[Pi])\)\ \@\[Pi])\)\ \@\[Pi]\ x\^3)\)\), "+", \(\(1\/\(512\ b\^9\)\) \((\((\(-512\)\ b\^8\ c2\ c4\ k\^6 - 104\ b\^6\ c4\^2\ k\^6 - 4\/3\ b\^6\ \((\(-256\)\ b\^8\ c0\^2 - 128\ b\^6\ c0\ c2 - 120\ b\^2\ c2\ c4 + 105\ c4\^2 - 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ k\^6 - 1\/6\ b\^6\ \((256\ b\^8\ c0\^2 + 128\ b\^6\ c0\ c2 + 120\ b\^2\ c2\ c4 - 105\ c4\^2 + 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ k\^6 + 2\ b\^4\ k\^4\ \((512\ b\^10\ c0\^2\ k\^2 - 512\ b\^8\ c0\ c2\ k\^2 - 192\ b\^4\ c2\ c4\ k\^2 - 90\ b\^2\ c4\^2\ k\^2 - b\^2\ \((256\ b\^8\ c0\^2 + 128\ b\^6\ c0\ c2 + 120\ b\^2\ c2\ c4 - 105\ c4\^2 + 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ k\^2 - 8\ b\^6\ \((12\ c2\^2\ k\^2 + 24\ c0\ c4\ k\^2)\))\) - 1\/2\ b\^4\ k\^4\ \((512\ b\^8\ c0\ c2\ k\^2 + 192\ b\^4\ c2\ c4\ k\^2 + 90\ b\^2\ c4\^2\ k\^2 + 8\ b\^6\ \((12\ c2\^2\ k\^2 + 24\ c0\ c4\ k\^2)\))\) - b\^2\ k\^2\ \((224\ b\^6\ c2\ c4\ k\^4 + 108\ b\^4\ c4\^2\ k\^4 + 32\ b\^8\ \((8\ c2\^2\ k\^4 + 16\ c0\ c4\ k\^4)\))\) - 2\ b\^2\ k\^2\ \((\(-224\)\ b\^6\ c2\ c4\ k\^4 - 108\ b\^4\ c4\^2\ k\^4 - 1\/2\ b\^4\ \((256\ b\^8\ c0\^2 + 128\ b\^6\ c0\ c2 + 120\ b\^2\ c2\ c4 - 105\ c4\^2 + 12\ b\^4\ \((4\ c2\^2 + 8\ c0\ c4)\))\)\ k\^4 - 32\ b\^8\ \((8\ c2\^2\ k\^4 + 16\ c0\ c4\ k\^4)\) - b\^2\ k\^2\ \((512\ b\^8\ c0\ c2\ k\^2 + 192\ b\^4\ c2\ c4\ k\^2 + 90\ b\^2\ c4\^2\ k\^2 + 8\ b\^6\ \((12\ c2\^2\ k\^2 + 24\ c0\ c4\ k\^2)\))\) + 256\ b\^9\ k\ \((\(2\ b\^3\ c0\^2\ k\^3\)\/\(3\ \@\ \[Pi]\) + \(4\ b\ c0\ c2\ k\^3\)\/\@\[Pi])\)\ \@\[Pi])\) + 256\ b\^9\ k\ \((\(b\^5\ c0\^2\ k\^5\)\/\(5\ \@\[Pi]\) + \ \(4\ b\^3\ c0\ c2\ k\^5\)\/\(3\ \@\[Pi]\) + \(2\ b\ k\ \((c2\^2\ k\^4 + 2\ c0\ \ c4\ k\^4)\)\)\/\@\[Pi])\)\ \@\[Pi])\)\ \@\[Pi]\ x\^5)\)\), "+", InterpretationBox[\(O[x]\^7\), SeriesData[ x, 0, {}, -1, 7, 1]]}], SeriesData[ x, 0, { Times[ Rational[ 1, 512], Power[ b, -9], Plus[ Times[ -256, Power[ b, 8], Power[ c0, 2]], Times[ -128, Power[ b, 6], c0, c2], Times[ -120, Power[ b, 2], c2, c4], Times[ 105, Power[ c4, 2]], Times[ -12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ Pi, Rational[ 1, 2]]], 0, Times[ Rational[ 1, 512], Power[ b, -9], Plus[ Times[ 512, Power[ b, 10], Power[ c0, 2], Power[ k, 2]], Times[ -512, Power[ b, 8], c0, c2, Power[ k, 2]], Times[ -192, Power[ b, 4], c2, c4, Power[ k, 2]], Times[ -90, Power[ b, 2], Power[ c4, 2], Power[ k, 2]], Times[ -2, Power[ b, 2], Plus[ Times[ -256, Power[ b, 8], Power[ c0, 2]], Times[ -128, Power[ b, 6], c0, c2], Times[ -120, Power[ b, 2], c2, c4], Times[ 105, Power[ c4, 2]], Times[ -12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ k, 2]], Times[ -1, Power[ b, 2], Plus[ Times[ 256, Power[ b, 8], Power[ c0, 2]], Times[ 128, Power[ b, 6], c0, c2], Times[ 120, Power[ b, 2], c2, c4], Times[ -105, Power[ c4, 2]], Times[ 12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ k, 2]], Times[ -8, Power[ b, 6], Plus[ Times[ 12, Power[ c2, 2], Power[ k, 2]], Times[ 24, c0, c4, Power[ k, 2]]]]], Power[ Pi, Rational[ 1, 2]]], 0, Times[ Rational[ 1, 512], Power[ b, -9], Plus[ Times[ -224, Power[ b, 6], c2, c4, Power[ k, 4]], Times[ -108, Power[ b, 4], Power[ c4, 2], Power[ k, 4]], Times[ 2, Power[ b, 4], Plus[ Times[ -256, Power[ b, 8], Power[ c0, 2]], Times[ -128, Power[ b, 6], c0, c2], Times[ -120, Power[ b, 2], c2, c4], Times[ 105, Power[ c4, 2]], Times[ -12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ k, 4]], Times[ Rational[ -1, 2], Power[ b, 4], Plus[ Times[ 256, Power[ b, 8], Power[ c0, 2]], Times[ 128, Power[ b, 6], c0, c2], Times[ 120, Power[ b, 2], c2, c4], Times[ -105, Power[ c4, 2]], Times[ 12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ k, 4]], Times[ -32, Power[ b, 8], Plus[ Times[ 8, Power[ c2, 2], Power[ k, 4]], Times[ 16, c0, c4, Power[ k, 4]]]], Times[ -2, Power[ b, 2], Power[ k, 2], Plus[ Times[ 512, Power[ b, 10], Power[ c0, 2], Power[ k, 2]], Times[ -512, Power[ b, 8], c0, c2, Power[ k, 2]], Times[ -192, Power[ b, 4], c2, c4, Power[ k, 2]], Times[ -90, Power[ b, 2], Power[ c4, 2], Power[ k, 2]], Times[ -1, Power[ b, 2], Plus[ Times[ 256, Power[ b, 8], Power[ c0, 2]], Times[ 128, Power[ b, 6], c0, c2], Times[ 120, Power[ b, 2], c2, c4], Times[ -105, Power[ c4, 2]], Times[ 12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ k, 2]], Times[ -8, Power[ b, 6], Plus[ Times[ 12, Power[ c2, 2], Power[ k, 2]], Times[ 24, c0, c4, Power[ k, 2]]]]]], Times[ -1, Power[ b, 2], Power[ k, 2], Plus[ Times[ 512, Power[ b, 8], c0, c2, Power[ k, 2]], Times[ 192, Power[ b, 4], c2, c4, Power[ k, 2]], Times[ 90, Power[ b, 2], Power[ c4, 2], Power[ k, 2]], Times[ 8, Power[ b, 6], Plus[ Times[ 12, Power[ c2, 2], Power[ k, 2]], Times[ 24, c0, c4, Power[ k, 2]]]]]], Times[ 256, Power[ b, 9], k, Plus[ Times[ Rational[ 2, 3], Power[ b, 3], Power[ c0, 2], Power[ k, 3], Power[ Pi, Rational[ -1, 2]]], Times[ 4, b, c0, c2, Power[ k, 3], Power[ Pi, Rational[ -1, 2]]]], Power[ Pi, Rational[ 1, 2]]]], Power[ Pi, Rational[ 1, 2]]], 0, Times[ Rational[ 1, 512], Power[ b, -9], Plus[ Times[ -512, Power[ b, 8], c2, c4, Power[ k, 6]], Times[ -104, Power[ b, 6], Power[ c4, 2], Power[ k, 6]], Times[ Rational[ -4, 3], Power[ b, 6], Plus[ Times[ -256, Power[ b, 8], Power[ c0, 2]], Times[ -128, Power[ b, 6], c0, c2], Times[ -120, Power[ b, 2], c2, c4], Times[ 105, Power[ c4, 2]], Times[ -12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ k, 6]], Times[ Rational[ -1, 6], Power[ b, 6], Plus[ Times[ 256, Power[ b, 8], Power[ c0, 2]], Times[ 128, Power[ b, 6], c0, c2], Times[ 120, Power[ b, 2], c2, c4], Times[ -105, Power[ c4, 2]], Times[ 12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ k, 6]], Times[ 2, Power[ b, 4], Power[ k, 4], Plus[ Times[ 512, Power[ b, 10], Power[ c0, 2], Power[ k, 2]], Times[ -512, Power[ b, 8], c0, c2, Power[ k, 2]], Times[ -192, Power[ b, 4], c2, c4, Power[ k, 2]], Times[ -90, Power[ b, 2], Power[ c4, 2], Power[ k, 2]], Times[ -1, Power[ b, 2], Plus[ Times[ 256, Power[ b, 8], Power[ c0, 2]], Times[ 128, Power[ b, 6], c0, c2], Times[ 120, Power[ b, 2], c2, c4], Times[ -105, Power[ c4, 2]], Times[ 12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ k, 2]], Times[ -8, Power[ b, 6], Plus[ Times[ 12, Power[ c2, 2], Power[ k, 2]], Times[ 24, c0, c4, Power[ k, 2]]]]]], Times[ Rational[ -1, 2], Power[ b, 4], Power[ k, 4], Plus[ Times[ 512, Power[ b, 8], c0, c2, Power[ k, 2]], Times[ 192, Power[ b, 4], c2, c4, Power[ k, 2]], Times[ 90, Power[ b, 2], Power[ c4, 2], Power[ k, 2]], Times[ 8, Power[ b, 6], Plus[ Times[ 12, Power[ c2, 2], Power[ k, 2]], Times[ 24, c0, c4, Power[ k, 2]]]]]], Times[ -1, Power[ b, 2], Power[ k, 2], Plus[ Times[ 224, Power[ b, 6], c2, c4, Power[ k, 4]], Times[ 108, Power[ b, 4], Power[ c4, 2], Power[ k, 4]], Times[ 32, Power[ b, 8], Plus[ Times[ 8, Power[ c2, 2], Power[ k, 4]], Times[ 16, c0, c4, Power[ k, 4]]]]]], Times[ -2, Power[ b, 2], Power[ k, 2], Plus[ Times[ -224, Power[ b, 6], c2, c4, Power[ k, 4]], Times[ -108, Power[ b, 4], Power[ c4, 2], Power[ k, 4]], Times[ Rational[ -1, 2], Power[ b, 4], Plus[ Times[ 256, Power[ b, 8], Power[ c0, 2]], Times[ 128, Power[ b, 6], c0, c2], Times[ 120, Power[ b, 2], c2, c4], Times[ -105, Power[ c4, 2]], Times[ 12, Power[ b, 4], Plus[ Times[ 4, Power[ c2, 2]], Times[ 8, c0, c4]]]], Power[ k, 4]], Times[ -32, Power[ b, 8], Plus[ Times[ 8, Power[ c2, 2], Power[ k, 4]], Times[ 16, c0, c4, Power[ k, 4]]]], Times[ -1, Power[ b, 2], Power[ k, 2], Plus[ Times[ 512, Power[ b, 8], c0, c2, Power[ k, 2]], Times[ 192, Power[ b, 4], c2, c4, Power[ k, 2]], Times[ 90, Power[ b, 2], Power[ c4, 2], Power[ k, 2]], Times[ 8, Power[ b, 6], Plus[ Times[ 12, Power[ c2, 2], Power[ k, 2]], Times[ 24, c0, c4, Power[ k, 2]]]]]], Times[ 256, Power[ b, 9], k, Plus[ Times[ Rational[ 2, 3], Power[ b, 3], Power[ c0, 2], Power[ k, 3], Power[ Pi, Rational[ -1, 2]]], Times[ 4, b, c0, c2, Power[ k, 3], Power[ Pi, Rational[ -1, 2]]]], Power[ Pi, Rational[ 1, 2]]]]], Times[ 256, Power[ b, 9], k, Plus[ Times[ Rational[ 1, 5], Power[ b, 5], Power[ c0, 2], Power[ k, 5], Power[ Pi, Rational[ -1, 2]]], Times[ Rational[ 4, 3], Power[ b, 3], c0, c2, Power[ k, 5], Power[ Pi, Rational[ -1, 2]]], Times[ 2, b, k, Plus[ Times[ Power[ c2, 2], Power[ k, 4]], Times[ 2, c0, c4, Power[ k, 4]]], Power[ Pi, Rational[ -1, 2]]]], Power[ Pi, Rational[ 1, 2]]]], Power[ Pi, Rational[ 1, 2]]]}, -1, 7, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[%]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(-\(\(\((256\ b\^8\ c0\^2 + 128\ b\^6\ c0\ c2 + 120\ b\^2\ c2\ c4 - 105\ c4\^2 + 48\ b\^4\ \((c2\^2 + 2\ c0\ c4)\))\)\ \@\[Pi]\)\/\(512\ b\^9\ x\)\)\), "+", \(\(3\ \((256\ b\^8\ c0\^2 - 128\ b\^6\ c0\ c2 - 24\ b\^2\ c2\ c4 - 65\ c4\^2 - 16\ b\^4\ \((c2\^2 + 2\ c0\ c4)\))\)\ k\^2\ \@\[Pi]\ x\)\/\(512\ b\^7\)\), "-", \(\(23\ \((\((256\ b\^8\ c0\^2 - 384\ b\^6\ c0\ c2 + 24\ b\^2\ c2\ c4 - 9\ c4\^2 + 48\ b\^4\ \((c2\^2 + 2\ c0\ c4)\))\)\ k\^4\ \@\[Pi])\)\ x\^3\)\/\(3072\ \ b\^5\)\), "+", \(\(13\ \((1792\ b\^8\ c0\^2 - 4480\ b\^6\ c0\ c2 - 840\ b\^2\ c2\ c4 - 135\ c4\^2 + 1680\ b\^4\ \((c2\^2 + 2\ c0\ c4)\))\)\ k\^6\ \@\[Pi]\ x\^5\)\/\(15360\ b\^3\ \)\), "+", InterpretationBox[\(O[x]\^7\), SeriesData[ x, 0, {}, -1, 7, 1]]}], SeriesData[ x, 0, { Times[ Rational[ -1, 512], Power[ b, -9], Plus[ Times[ 256, Power[ b, 8], Power[ c0, 2]], Times[ 128, Power[ b, 6], c0, c2], Times[ 120, Power[ b, 2], c2, c4], Times[ -105, Power[ c4, 2]], Times[ 48, Power[ b, 4], Plus[ Power[ c2, 2], Times[ 2, c0, c4]]]], Power[ Pi, Rational[ 1, 2]]], 0, Times[ Rational[ 3, 512], Power[ b, -7], Plus[ Times[ 256, Power[ b, 8], Power[ c0, 2]], Times[ -128, Power[ b, 6], c0, c2], Times[ -24, Power[ b, 2], c2, c4], Times[ -65, Power[ c4, 2]], Times[ -16, Power[ b, 4], Plus[ Power[ c2, 2], Times[ 2, c0, c4]]]], Power[ k, 2], Power[ Pi, Rational[ 1, 2]]], 0, Times[ Rational[ -23, 3072], Power[ b, -5], Plus[ Times[ 256, Power[ b, 8], Power[ c0, 2]], Times[ -384, Power[ b, 6], c0, c2], Times[ 24, Power[ b, 2], c2, c4], Times[ -9, Power[ c4, 2]], Times[ 48, Power[ b, 4], Plus[ Power[ c2, 2], Times[ 2, c0, c4]]]], Power[ k, 4], Power[ Pi, Rational[ 1, 2]]], 0, Times[ Rational[ 13, 15360], Power[ b, -3], Plus[ Times[ 1792, Power[ b, 8], Power[ c0, 2]], Times[ -4480, Power[ b, 6], c0, c2], Times[ -840, Power[ b, 2], c2, c4], Times[ -135, Power[ c4, 2]], Times[ 1680, Power[ b, 4], Plus[ Power[ c2, 2], Times[ 2, c0, c4]]]], Power[ k, 6], Power[ Pi, Rational[ 1, 2]]]}, -1, 7, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Integrate[ s^5\ E2cr[Sqrt[1 - s^2]]/8\ - \ s^3 \((\ 4\ - \ 5 s\ + s^2)\) E2cr[1 - s]/24, {s, 0, 1}]\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \(\(35\ c4\^2\ \@\[Pi]\)\/\(12288\ w\^9\ x\) - \(5\ c2\ c4\ \ \@\[Pi]\)\/\(1536\ w\^7\ x\) - \(c2\^2\ \@\[Pi]\)\/\(768\ w\^5\ x\) - \(c0\ \ c4\ \@\[Pi]\)\/\(384\ w\^5\ x\) - \(c0\ c2\ \@\[Pi]\)\/\(288\ w\^3\ x\) - \ \(c0\^2\ \@\[Pi]\)\/\(144\ w\ x\) - \(13\ c4\^2\ \@\[Pi]\ x\)\/\(8192\ w\^7\) \ - \(3\ c2\ c4\ \@\[Pi]\ x\)\/\(5120\ w\^5\) - \(c2\^2\ \@\[Pi]\ x\)\/\(2560\ \ w\^3\) - \(c0\ c4\ \@\[Pi]\ x\)\/\(1280\ w\^3\) - \(c0\ c2\ \@\[Pi]\ \ x\)\/\(320\ w\) + 1\/160\ c0\^2\ \@\[Pi]\ w\ x + \(69\ c4\^2\ \@\[Pi]\ x\^3\)\/\(573440\ \ w\^5\) - \(23\ c2\ c4\ \@\[Pi]\ x\^3\)\/\(71680\ w\^3\) - \(23\ c2\^2\ \ \@\[Pi]\ x\^3\)\/\(35840\ w\) - \(23\ c0\ c4\ \@\[Pi]\ x\^3\)\/\(17920\ w\) + \ \(23\ c0\ c2\ \@\[Pi]\ w\ x\^3\)\/4480 - \(23\ c0\^2\ \@\[Pi]\ w\^3\ \ x\^3\)\/6720\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[Simplify[%]]\)], "Input"], Cell[BoxData[ \(\(-\(\(c0\^2\ m\ \@\[Pi]\)\/\(144\ L\ x\)\)\) - \(c0\ c2\ m\^3\ \@\[Pi]\ \)\/\(288\ L\^3\ x\) - \(c2\^2\ m\^5\ \@\[Pi]\)\/\(768\ L\^5\ x\) - \(c0\ c4\ \ m\^5\ \@\[Pi]\)\/\(384\ L\^5\ x\) - \(5\ c2\ c4\ m\^7\ \@\[Pi]\)\/\(1536\ \ L\^7\ x\) + \(35\ c4\^2\ m\^9\ \@\[Pi]\)\/\(12288\ L\^9\ x\) + \(c0\^2\ L\ \@\ \[Pi]\ x\)\/\(160\ m\) - \(c0\ c2\ m\ \@\[Pi]\ x\)\/\(320\ L\) - \(c2\^2\ \ m\^3\ \@\[Pi]\ x\)\/\(2560\ L\^3\) - \(c0\ c4\ m\^3\ \@\[Pi]\ x\)\/\(1280\ \ L\^3\) - \(3\ c2\ c4\ m\^5\ \@\[Pi]\ x\)\/\(5120\ L\^5\) - \(13\ c4\^2\ m\^7\ \ \@\[Pi]\ x\)\/\(8192\ L\^7\) - \(23\ c0\^2\ L\^3\ \@\[Pi]\ x\^3\)\/\(6720\ \ m\^3\) + \(23\ c0\ c2\ L\ \@\[Pi]\ x\^3\)\/\(4480\ m\) - \(23\ c2\^2\ m\ \@\ \[Pi]\ x\^3\)\/\(35840\ L\) - \(23\ c0\ c4\ m\ \@\[Pi]\ x\^3\)\/\(17920\ L\) \ - \(23\ c2\ c4\ m\^3\ \@\[Pi]\ x\^3\)\/\(71680\ L\^3\) + \(69\ c4\^2\ m\^5\ \ \@\[Pi]\ x\^3\)\/\(573440\ L\^5\)\)], "Output"] }, Open ]], Cell[BoxData[ \(Vt[y_, q_] := \ \((\ c0\ + \ c2\ \((\ y^2\ + \ q^2)\)/2\ + \ c4\ \((\ y^2\ + \ q^2)\)^2/4)\)\ Exp[\(-\((y^2\ + \ q^2)\)\)/ 2]\)], "Input"], Cell[BoxData[ \(Flp[y_, q_] := \ q^2\ Vt[y, q]^2/\((y^2\ - q^2)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[Flp[y, q], {q, 0, \[Infinity]}, PrincipalValue \[Rule] True, Assumptions \[Rule] y^2 > 0]\)], "Input"], Cell[BoxData[ \(\(\(1\/\(32\ y\^2\)\)\((\[ExponentialE]\^\(-y\^2\)\ \[Pi]\ \((\(8\ \ \[ImaginaryI]\ c0\^2\ \[ExponentialE]\^\(-y\^2\)\ \((\(-2\)\ \@\[Pi] - \(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) + 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\)\)\/\(\@\[Pi]\ \((1\/y\^2)\)\^\(3/2\)\) + \(8\ \ \[ImaginaryI]\ c0\ c2\ \[ExponentialE]\^\(-y\^2\)\ y\^2\ \((\(-2\)\ \@\[Pi] - \ \(2\ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) + 2\ \@\[Pi]\ \((1 - \ \(\@\(-y\^2\)\ Erfi[\@y\^2]\)\/\@y\^2)\))\)\)\/\(\@\[Pi]\ \ \((1\/y\^2)\)\^\(3/2\)\) + \(2\ \[ImaginaryI]\ c2\^2\ \ \[ExponentialE]\^\(-y\^2\)\ y\^4\ \((\(-2\)\ \@\[Pi] - \(2\ \[ExponentialE]\^\ \(y\^2\)\)\/\@\(-y\^2\) + 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\)\)\/\(\@\[Pi]\ \((1\/y\^2)\)\^\(3/2\)\) + \(4\ \ \[ImaginaryI]\ c0\ c4\ \[ExponentialE]\^\(-y\^2\)\ y\^4\ \((\(-2\)\ \@\[Pi] - \ \(2\ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) + 2\ \@\[Pi]\ \((1 - \ \(\@\(-y\^2\)\ Erfi[\@y\^2]\)\/\@y\^2)\))\)\)\/\(\@\[Pi]\ \ \((1\/y\^2)\)\^\(3/2\)\) + \(2\ \[ImaginaryI]\ c2\ c4\ \ \[ExponentialE]\^\(-y\^2\)\ y\^6\ \((\(-2\)\ \@\[Pi] - \(2\ \[ExponentialE]\^\ \(y\^2\)\)\/\@\(-y\^2\) + 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\)\)\/\(\@\[Pi]\ \((1\/y\^2)\)\^\(3/2\)\) + \(\ \[ImaginaryI]\ c4\^2\ \[ExponentialE]\^\(-y\^2\)\ y\^8\ \((\(-2\)\ \@\[Pi] - \ \(2\ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) + 2\ \@\[Pi]\ \((1 - \ \(\@\(-y\^2\)\ Erfi[\@y\^2]\)\/\@y\^2)\))\)\)\/\(2\ \@\[Pi]\ \ \((1\/y\^2)\)\^\(3/2\)\) + \(15\ \[ImaginaryI]\ c2\^2\ \ \[ExponentialE]\^\(-y\^2\)\ \((\(-\(\(8\ \@\[Pi]\)\/15\)\) - \(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\(5\ \((\(-y\^2\))\)\^\(5/2\)\) + 2\/5\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \((\(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\)\)\/\(2\ \@\[Pi]\ \((1\/y\^2)\)\^\(7/2\)\) \ + \(15\ \[ImaginaryI]\ c0\ c4\ \[ExponentialE]\^\(-y\^2\)\ \((\(-\(\(8\ \@\ \[Pi]\)\/15\)\) - \(2\ \[ExponentialE]\^\(y\^2\)\)\/\(5\ \ \((\(-y\^2\))\)\^\(5/2\)\) + 2\/5\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \ \((\(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\)\)\/\(\@\[Pi]\ \((1\/y\^2)\)\^\(7/2\)\) + \ \(45\ \[ImaginaryI]\ c2\ c4\ \[ExponentialE]\^\(-y\^2\)\ y\^2\ \((\(-\(\(8\ \ \@\[Pi]\)\/15\)\) - \(2\ \[ExponentialE]\^\(y\^2\)\)\/\(5\ \ \((\(-y\^2\))\)\^\(5/2\)\) + 2\/5\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \ \((\(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\)\)\/\(2\ \@\[Pi]\ \((1\/y\^2)\)\^\(7/2\)\) \ + \(45\ \[ImaginaryI]\ c4\^2\ \[ExponentialE]\^\(-y\^2\)\ y\^4\ \((\(-\(\(8\ \ \@\[Pi]\)\/15\)\) - \(2\ \[ExponentialE]\^\(y\^2\)\)\/\(5\ \ \((\(-y\^2\))\)\^\(5/2\)\) + 2\/5\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \ \((\(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\)\)\/\(4\ \@\[Pi]\ \((1\/y\^2)\)\^\(7/2\)\) \ + \(8\ \[ImaginaryI]\ c0\ c2\ \[ExponentialE]\^\(-y\^2\)\ \((\(-2\)\ \@\[Pi] \ + 3\/2\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \((\(-y\^2\))\)\^\(3/2\)\) \ - 2\/3\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \ \(\@\(-y\^2\)\ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\)\)\/\(\@\[Pi]\ \((1\/y\^2)\)\ \^\(5/2\)\) + \(4\ \[ImaginaryI]\ c2\^2\ \[ExponentialE]\^\(-y\^2\)\ y\^2\ \ \((\(-2\)\ \@\[Pi] + 3\/2\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \ \((\(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\)\)\/\(\@\[Pi]\ \((1\/y\^2)\)\^\(5/2\)\) + \ \(8\ \[ImaginaryI]\ c0\ c4\ \[ExponentialE]\^\(-y\^2\)\ y\^2\ \((\(-2\)\ \@\ \[Pi] + 3\/2\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \ \((\(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\)\)\/\(\@\[Pi]\ \((1\/y\^2)\)\^\(5/2\)\) + \ \(6\ \[ImaginaryI]\ c2\ c4\ \[ExponentialE]\^\(-y\^2\)\ y\^4\ \((\(-2\)\ \@\ \[Pi] + 3\/2\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \ \((\(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\)\)\/\(\@\[Pi]\ \((1\/y\^2)\)\^\(5/2\)\) + \ \(2\ \[ImaginaryI]\ c4\^2\ \[ExponentialE]\^\(-y\^2\)\ y\^6\ \((\(-2\)\ \@\ \[Pi] + 3\/2\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \ \((\(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\)\)\/\(\@\[Pi]\ \((1\/y\^2)\)\^\(5/2\)\) - \ \(\(1\/\(4\ \@\[Pi]\ \((1\/y\^2)\)\^\(9/2\)\)\)\((105\ \[ImaginaryI]\ c2\ c4\ \ \[ExponentialE]\^\(-y\^2\)\ \((\(16\ \@\[Pi]\)\/105 - \(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\(7\ \((\(-y\^2\))\)\^\(7/2\)\) + 2\/7\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(5\ \ \((\(-y\^2\))\)\^\(5/2\)\) - 2\/5\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \((\ \(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\@\ \(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ Erfi[\ \@y\^2]\)\/\@y\^2)\))\))\))\))\))\)\) - \(\(1\/\(4\ \@\[Pi]\ \ \((1\/y\^2)\)\^\(9/2\)\)\)\((105\ \[ImaginaryI]\ c4\^2\ \[ExponentialE]\^\(-y\ \^2\)\ y\^2\ \((\(16\ \@\[Pi]\)\/105 - \(2\ \[ExponentialE]\^\(y\^2\)\)\/\(7\ \ \((\(-y\^2\))\)\^\(7/2\)\) + 2\/7\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(5\ \ \((\(-y\^2\))\)\^\(5/2\)\) - 2\/5\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \((\ \(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\@\ \(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ Erfi[\ \@y\^2]\)\/\@y\^2)\))\))\))\))\))\)\) + \(\(1\/\(32\ \@\[Pi]\ \((1\/y\^2)\)\^\ \(11/2\)\)\)\((945\ \[ImaginaryI]\ c4\^2\ \[ExponentialE]\^\(-y\^2\)\ \ \((\(-\(\(32\ \@\[Pi]\)\/945\)\) - \(2\ \[ExponentialE]\^\(y\^2\)\)\/\(9\ \((\ \(-y\^2\))\)\^\(9/2\)\) + 2\/9\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(7\ \ \((\(-y\^2\))\)\^\(7/2\)\) - 2\/7\ \((\(2\ \[ExponentialE]\^\(y\^2\)\)\/\(5\ \((\ \(-y\^2\))\)\^\(5/2\)\) - 2\/5\ \((\(2\ \ \[ExponentialE]\^\(y\^2\)\)\/\(3\ \((\(-y\^2\))\)\^\(3/2\)\) - 2\/3\ \((\(2\ \[ExponentialE]\^\(y\^2\)\ \)\/\@\(-y\^2\) - 2\ \@\[Pi]\ \((1 - \(\@\(-y\^2\)\ \ Erfi[\@y\^2]\)\/\@y\^2)\))\))\))\))\))\))\)\))\))\)\)\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Frlpart[y_] = Simplify[%, {y > 0, y^2 > 0}]\)], "Input"], Cell[BoxData[ \(1\/512\ \[ExponentialE]\^\(\(-2\)\ y\^2\)\ \@\[Pi]\ \((\(-\ \[ExponentialE]\^\(y\^2\)\)\ \((256\ c0\^2 + 16\ c2\^2\ \((3 + 6\ y\^2 + 16\ y\^4)\) + 8\ c2\ c4\ \((15 + 24\ y\^2 + 28\ y\^4 + 64\ y\^6)\) + c4\^2\ \((105 + 150\ y\^2 + 132\ y\^4 + 120\ y\^6 + 256\ y\^8)\) + 32\ c0\ \((4\ c2\ \((1 + 4\ y\^2)\) + c4\ \((3 + 6\ y\^2 + 16\ y\^4)\))\))\) + 256\ \@\[Pi]\ y\ \((c0 + c2\ y\^2 + c4\ y\^4)\)\^2\ Erfi[ y])\)\)], "Output"] }, Open ]], Cell[BoxData[ \( (*\ Frlpart[ y_] := \(1\/\(512\ y\)\) \((\ \[ExponentialE]\^\(\(-2\)\ y\^2\)\ \@\ \[Pi]\ \((\(-\[ExponentialE]\^\(y\^2\)\)\ y\ \((256\ c0\^2 + 16\ c2\^2\ \((3 + 6\ y\^2 + 16\ y\^4)\) + 8\ c2\ c4\ \((15 + 12\ y\^2 + 20\ y\^4 + 32\ y\^6 + 36\ y\^8 + 16\ y\^10)\) + c4\^2\ \((105 + 30\ y\^2 + 132\ y\^4 + 56\ y\^6 + 120\ y\^8 + 112\ y\^10 + 128\ y\^12 + 64\ y\^14 + 16\ y\^16)\) + 32\ c0\ \((4\ c2\ \((1 + 4\ y\^2)\) + c4\ \((3 + 2\ y\^2 + 8\ y\^4 + 8\ y\^6 + 4\ y\^8)\))\))\) + 16\ \@\[Pi]\ \((4\ c0\ y + y\^3\ \((4\ c2 + c4\ \((y + \ y\^3)\)\^2)\))\)\^2\ Erfi[y])\))\)\ *) \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Vcr[p_]\ = \ Vt[p/Lc, p/Lc]\ /. \ {c0 \[Rule] C0, c2 \[Rule] C2\ Lc^2, c4 \[Rule] C4\ Lc^4}\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-\(p\^2\/Lc\^2\)\)\ \((C0 + C2\ p\^2 + C4\ p\^4)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(VGVcr[p_]\ = \ Simplify[M\ Lc/\((2\ Pi^2)\) Frlpart[p/Lc] /. \ \ {c0 \[Rule] C0, c2 \[Rule] C2\ Lc^2, c4 \[Rule] C4\ Lc^4}]\)], "Input"], Cell[BoxData[ \(\(\(1\/\(1024\ \[Pi]\^\(3/2\)\)\)\((\[ExponentialE]\^\(-\(\(2\ \ p\^2\)\/Lc\^2\)\)\ Lc\ M\ \((\(-\[ExponentialE]\^\(p\^2\/Lc\^2\)\)\ \((256\ \ C0\^2 + 16\ C2\^2\ \((3\ Lc\^4 + 6\ Lc\^2\ p\^2 + 16\ p\^4)\) + 8\ C2\ C4\ \((15\ Lc\^6 + 24\ Lc\^4\ p\^2 + 28\ Lc\^2\ p\^4 + 64\ p\^6)\) + C4\^2\ \((105\ Lc\^8 + 150\ Lc\^6\ p\^2 + 132\ Lc\^4\ p\^4 + 120\ Lc\^2\ p\^6 + 256\ p\^8)\) + 32\ C0\ \((4\ C2\ \((Lc\^2 + 4\ p\^2)\) + C4\ \((3\ Lc\^4 + 6\ Lc\^2\ p\^2 + 16\ p\^4)\))\))\) + \(256\ p\ \((C0 + C2\ p\^2 \ + C4\ p\^4)\)\^2\ \@\[Pi]\ Erfi[p\/Lc]\)\/Lc)\))\)\)\)], "Output"] }, Open ]], Cell[BoxData[ \(\(Tcr1[p_]\ = \ Series[\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Series[Vcr[p]\ + \ VGVcr[p], {p, 0, 0}]\ /. \ {C0 \[Rule] C01\ alpha\ + \ C02\ alpha^2, C2 \[Rule] 0, C4 \[Rule] 0}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {alpha, 0, 2}];\)\)], "Input"], Cell[BoxData[ \(\(Tcr2[p_]\ = \ Series[\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Series[ Vcr[p]\ + \ VGVcr[p], {p, 0, 2}]\ /. \ {C0 \[Rule] C01\ alpha\ + \ C02\ alpha^2, C2 \[Rule] C21\ alpha\ + \ C22\ alpha^2, C4 \[Rule] 0}, {alpha, 0, 2}];\)\)], "Input"], Cell[BoxData[ \(\(Tcr3[p_]\ = \ Series[\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Series[Vcr[p]\ + \ VGVcr[p], {p, 0, 4}] /. \ {C0 \[Rule] C01\ alpha\ + \ C02\ alpha^2, C2 \[Rule] C21\ alpha\ + \ C22\ alpha^2, C4 \[Rule] C41\ alpha\ + \ C42\ alpha^2}, {alpha, 0, 2}];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Vtrue[p_]\ = \ \((4\ Pi/M)\)\ alpha\ m^3/\((p^2 + m^2)\)^2\)], "Input"], Cell[BoxData[ \(\(4\ alpha\ m\^3\ \[Pi]\)\/\(M\ \((m\^2 + p\^2)\)\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(VGVtrue[ p_]\ = \ \((4\ Pi/M)\)\ alpha^2\ m^6/\((p^2 + m^2)\)^4\ \(\((p^2 - m^2)\)\(/\)\((2 m)\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(\(2\ alpha\^2\ m\^5\ \((\(-m\^2\) + p\^2)\)\ \[Pi]\)\/\(M\ \((m\^2 + \ p\^2)\)\^4\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Ttrue1[p_]\ = \ Series[Vtrue[p]\ + \ VGVtrue[p], {p, 0, 0}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\((\(4\ alpha\ \[Pi]\)\/\(m\ M\) - \(2\ alpha\^2\ \[Pi]\)\/\(m\ \ M\))\), "+", InterpretationBox[\(O[p]\^1\), SeriesData[ p, 0, {}, 0, 1, 1]]}], SeriesData[ p, 0, { Plus[ Times[ 4, alpha, Power[ m, -1], Power[ M, -1], Pi], Times[ -2, Power[ alpha, 2], Power[ m, -1], Power[ M, -1], Pi]]}, 0, 1, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Ttrue2[p_]\ = \ Series[Vtrue[p]\ + \ VGVtrue[p], {p, 0, 2}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\((\(4\ alpha\ \[Pi]\)\/\(m\ M\) - \(2\ alpha\^2\ \[Pi]\)\/\(m\ \ M\))\), "+", \(\((\(-\(\(8\ alpha\ \[Pi]\)\/\(m\^3\ M\)\)\) + \(10\ alpha\^2\ \ \[Pi]\)\/\(m\^3\ M\))\)\ p\^2\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Plus[ Times[ 4, alpha, Power[ m, -1], Power[ M, -1], Pi], Times[ -2, Power[ alpha, 2], Power[ m, -1], Power[ M, -1], Pi]], 0, Plus[ Times[ -8, alpha, Power[ m, -3], Power[ M, -1], Pi], Times[ 10, Power[ alpha, 2], Power[ m, -3], Power[ M, -1], Pi]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Ttrue3[p_]\ = \ Series[Vtrue[p]\ + \ VGVtrue[p], {p, 0, 4}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\((\(4\ alpha\ \[Pi]\)\/\(m\ M\) - \(2\ alpha\^2\ \[Pi]\)\/\(m\ \ M\))\), "+", \(\((\(-\(\(8\ alpha\ \[Pi]\)\/\(m\^3\ M\)\)\) + \(10\ alpha\^2\ \ \[Pi]\)\/\(m\^3\ M\))\)\ p\^2\), "+", \(\((\(12\ alpha\ \[Pi]\)\/\(m\^5\ M\) - \(28\ alpha\^2\ \[Pi]\)\ \/\(m\^5\ M\))\)\ p\^4\), "+", InterpretationBox[\(O[p]\^5\), SeriesData[ p, 0, {}, 0, 5, 1]]}], SeriesData[ p, 0, { Plus[ Times[ 4, alpha, Power[ m, -1], Power[ M, -1], Pi], Times[ -2, Power[ alpha, 2], Power[ m, -1], Power[ M, -1], Pi]], 0, Plus[ Times[ -8, alpha, Power[ m, -3], Power[ M, -1], Pi], Times[ 10, Power[ alpha, 2], Power[ m, -3], Power[ M, -1], Pi]], 0, Plus[ Times[ 12, alpha, Power[ m, -5], Power[ M, -1], Pi], Times[ -28, Power[ alpha, 2], Power[ m, -5], Power[ M, -1], Pi]]}, 0, 5, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(solve1\ = \ \(Solve[ Tcr1[p] \[Equal] Ttrue1[p], {C01, C02}]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \({C02 \[Rule] \(-\(\(2\ \((\(-2\)\ Lc\ \@\[Pi] + m\ \[Pi])\)\)\/\(m\^2\ M\)\)\), C01 \[Rule] \(4\ \[Pi]\)\/\(m\ M\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(solve2\ = \ \(Solve[ Tcr2[p] \[Equal] Ttrue2[p], {C01, C02, C21, C22}]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \({C22 \[Rule] \(12\ Lc\^5\ \@\[Pi] - 44\ Lc\^3\ m\^2\ \@\[Pi] + 3\ Lc\ m\ \^4\ \@\[Pi] + 20\ Lc\^2\ m\^3\ \[Pi] - 4\ m\^5\ \[Pi]\)\/\(2\ Lc\^2\ m\^6\ M\ \), C02 \[Rule] \(12\ Lc\^5\ \@\[Pi] - 28\ Lc\^3\ m\^2\ \@\[Pi] + 27\ Lc\ \ m\^4\ \@\[Pi] - 8\ m\^5\ \[Pi]\)\/\(4\ m\^6\ M\), C01 \[Rule] \(4\ \[Pi]\)\/\(m\ M\), C21 \[Rule] \(4\ \((\[Pi]\/Lc\^2 - \(2\ \[Pi]\)\/m\^2)\)\)\/\(m\ \ M\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(solve3\ = \ \(Solve[ Tcr3[p] \[Equal] Ttrue3[p], {C01, C02, C21, C22, C41, C42}]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \({C42 \[Rule] \(\(1\/\(192\ Lc\^4\ m\^10\ M\)\)\((3564\ Lc\^9\ \@\[Pi] - 8784\ Lc\^7\ m\^2\ \@\[Pi] + 15156\ Lc\^5\ m\^4\ \@\[Pi] - 2808\ Lc\^3\ m\^6\ \@\[Pi] - 77\ Lc\ m\^8\ \@\[Pi] - 5376\ Lc\^4\ m\^5\ \[Pi] + 1920\ Lc\^2\ m\^7\ \[Pi] - 192\ m\^9\ \[Pi])\)\), C22 \[Rule] \(2700\ Lc\^9\ \@\[Pi] - 5904\ Lc\^7\ m\^2\ \@\[Pi] + 6708\ \ Lc\^5\ m\^4\ \@\[Pi] - 5336\ Lc\^3\ m\^6\ \@\[Pi] + 651\ Lc\ m\^8\ \@\[Pi] + \ 1280\ Lc\^2\ m\^7\ \[Pi] - 256\ m\^9\ \[Pi]\)\/\(128\ Lc\^2\ m\^10\ M\), C02 \[Rule] \(3780\ Lc\^9\ \@\[Pi] - 7920\ Lc\^7\ m\^2\ \@\[Pi] + 8220\ \ Lc\^5\ m\^4\ \@\[Pi] - 4840\ Lc\^3\ m\^6\ \@\[Pi] + 2265\ Lc\ m\^8\ \@\[Pi] - \ 512\ m\^9\ \[Pi]\)\/\(256\ m\^10\ M\), C01 \[Rule] \(4\ \[Pi]\)\/\(m\ M\), C21 \[Rule] \(4\ \((\[Pi]\/Lc\^2 - \(2\ \[Pi]\)\/m\^2)\)\)\/\(m\ M\), C41 \[Rule] \(2\ \((\[Pi]\/Lc\^4 + \(6\ \[Pi]\)\/m\^4 - \(4\ \ \[Pi]\)\/\(Lc\^2\ m\^2\))\)\)\/\(m\ M\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[C02/\((\(-4\)\ Pi/\((M\ m)\))\)\ /. \ solve1]\)], "Input"], Cell[BoxData[ \(1\/2 - Lc\/\(m\ \@\[Pi]\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\(C02/C01\ /. \ solve2\)\ \ /. \ Lc\ \[Rule] \ m\ eta]\)], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\) + \(27\ eta\)\/\(16\ \@\[Pi]\) - \(7\ eta\^3\)\/\(4\ \@\ \[Pi]\) + \(3\ eta\^5\)\/\(4\ \@\[Pi]\)\)], "Output"] }, Open ]], Cell[BoxData[ \(\(D0\ = \ \(-\ \((27\ - \ 28\ eta^2\ + \ 12\ eta^4)\)\);\)\)], "Input"], Cell[BoxData[ \(\(Expand[\(-1\)/2\ - \ eta\ D0/\((16\ Sqrt[Pi])\)];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandAll[solve2]\)], "Input"], Cell[BoxData[ \({C22 \[Rule] \(6\ Lc\^3\ \@\[Pi]\)\/\(m\^6\ M\) - \(22\ Lc\ \@\[Pi]\)\/\ \(m\^4\ M\) + \(3\ \@\[Pi]\)\/\(2\ Lc\ m\^2\ M\) + \(10\ \[Pi]\)\/\(m\^3\ M\) \ - \(2\ \[Pi]\)\/\(Lc\^2\ m\ M\), C02 \[Rule] \(3\ Lc\^5\ \@\[Pi]\)\/\(m\^6\ M\) - \(7\ Lc\^3\ \ \@\[Pi]\)\/\(m\^4\ M\) + \(27\ Lc\ \@\[Pi]\)\/\(4\ m\^2\ M\) - \(2\ \[Pi]\)\/\ \(m\ M\), C01 \[Rule] \(4\ \[Pi]\)\/\(m\ M\), C21 \[Rule] \(-\(\(8\ \[Pi]\)\/\(m\^3\ M\)\)\) + \(4\ \[Pi]\)\/\(Lc\^2\ \ m\ M\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandAll[solve3]\)], "Input"], Cell[BoxData[ \({C42 \[Rule] \(297\ Lc\^5\ \@\[Pi]\)\/\(16\ m\^10\ M\) - \(183\ Lc\^3\ \ \@\[Pi]\)\/\(4\ m\^8\ M\) + \(1263\ Lc\ \@\[Pi]\)\/\(16\ m\^6\ M\) - \(117\ \ \@\[Pi]\)\/\(8\ Lc\ m\^4\ M\) - \(77\ \@\[Pi]\)\/\(192\ Lc\^3\ m\^2\ M\) - \ \(28\ \[Pi]\)\/\(m\^5\ M\) + \(10\ \[Pi]\)\/\(Lc\^2\ m\^3\ M\) - \ \[Pi]\/\(Lc\^4\ m\ M\), C22 \[Rule] \(675\ Lc\^7\ \@\[Pi]\)\/\(32\ m\^10\ M\) - \(369\ Lc\^5\ \ \@\[Pi]\)\/\(8\ m\^8\ M\) + \(1677\ Lc\^3\ \@\[Pi]\)\/\(32\ m\^6\ M\) - \(667\ \ Lc\ \@\[Pi]\)\/\(16\ m\^4\ M\) + \(651\ \@\[Pi]\)\/\(128\ Lc\ m\^2\ M\) + \ \(10\ \[Pi]\)\/\(m\^3\ M\) - \(2\ \[Pi]\)\/\(Lc\^2\ m\ M\), C02 \[Rule] \(945\ Lc\^9\ \@\[Pi]\)\/\(64\ m\^10\ M\) - \(495\ Lc\^7\ \ \@\[Pi]\)\/\(16\ m\^8\ M\) + \(2055\ Lc\^5\ \@\[Pi]\)\/\(64\ m\^6\ M\) - \ \(605\ Lc\^3\ \@\[Pi]\)\/\(32\ m\^4\ M\) + \(2265\ Lc\ \@\[Pi]\)\/\(256\ m\^2\ \ M\) - \(2\ \[Pi]\)\/\(m\ M\), C01 \[Rule] \(4\ \[Pi]\)\/\(m\ M\), C21 \[Rule] \(-\(\(8\ \[Pi]\)\/\(m\^3\ M\)\)\) + \(4\ \[Pi]\)\/\(Lc\^2\ \ m\ M\), C41 \[Rule] \(12\ \[Pi]\)\/\(m\^5\ M\) - \(8\ \[Pi]\)\/\(Lc\^2\ m\^3\ \ M\) + \(2\ \[Pi]\)\/\(Lc\^4\ m\ M\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(C41\ - \ \((C01/Lc^4)\) \((3\ eta^4\ - \ 2\ eta^2\ + \ 1/2)\)\ \ /. \ eta \[Rule] Lc/m\)\ /. \ solve3]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(C2t\ = \ 1 - \ 2 eta^2;\)\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(C2t\)\" is similar to \ existing symbol \"\!\(c2t\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(C4t\ = \ 3 eta^4\ - \ 2 eta^2 + 1/2;\)\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(C4t\)\" is similar to \ existing symbol \"\!\(c4t\)\"."\)], "Message"] }, Open ]], Cell[BoxData[ \(\(h0\ = \ 256\ + \ 32\ \((4\ C2t\ + \ 3\ C4t)\)\ + \ 3 \((16 C2t^2\ + \ 40\ C2t\ C4t\ + \ 35\ C4t^2)\);\)\)], "Input"], Cell[BoxData[ \(\(B\ = \ eta/\((256\ Sqrt[Pi])\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\ \(C02\ - \ C01 \((\(-1\)/2 + \ h0\ B)\)\ /. \ eta \[Rule] Lc/m\)\ /. \ solve3]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[BoxData[ \(\(h2\ \ = \ 3\ \((\(-256\) + 32 \((4\ C2t + C4t)\)\ + \ \((16\ C2t^2\ + \ 24\ C2t\ C4t\ + \ 15\ C4t^2)\))\);\)\)], "Input"], Cell[BoxData[ \(\(h4\ = \ 23/6\ \((\ 256\ + \ 96 \((\(-4\)\ C2t\ + \ C4t)\)\ + \ 3\ \((16\ C2t^2\ + \ 8\ C2t\ C4t\ + \ 3\ C4t^2)\))\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(C22\ - \ C01/Lc^2\ \((\(-1\)/2\ + \ 5/2\ eta^2\ + \ B \((h0 + h2)\))\)\ /. \ eta \[Rule] Lc/m\)\ /. \ solve3]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[\(C42\ - \ C01/Lc^4 \((\(-1\)/4\ + \ 5/2 eta^2\ - \ 7\ eta^4\ + \ B \((h0/2\ + \ h2\ + \ h4)\))\)\ /. \ eta \[Rule] Lc/m\)\ /. \ solve3]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(Vtest[ p_] := \(\((c0\ + \ c2\ y^2 p^2\ + \ c4\ y^4\ p^4)\) \(Exp[\(-y^2\) p^2/a^2]\)\(\ \ \)\)\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(\[IndentingNewLine]\)\(Series[ Integrate[ s^5\ Vtest[Sqrt[1 - s^2]]/8\ - \ s^3 \((\ 4\ - \ 5 s\ + s^2)\) Vtest[1 - s]/24, {s, 0, 1}], {y, 0, 4}]\)\)\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\((1\/48\ a\^2\ \((\(-\(\(40\ c0\)\/\(3\ a\^2\)\)\) - 16\ c2 - 16\ a\^2\ c4 - \(2\ \((\(-8\)\ c0 + 48\ a\^4\ c4)\)\)\/a\^2 \ + \(2\ \((4\ a\^2\ c0 + 24\ a\^4\ c2 + 120\ a\^6\ c4)\)\)\/a\^4 - \(4\ \((8\ \ a\^4\ c0 + 24\ a\^6\ c2 + 96\ a\^8\ c4)\)\)\/\(3\ a\^6\))\) + 1\/48\ a\^2\ \((\(16\ c0\)\/\(3\ a\^2\) + 8\ c2 + 20\ a\^2\ c4 + \(\(-8\)\ c0 - 12\ a\^2\ c2 - 24\ a\^4\ c4\)\ \/a\^2 - \(\(-4\)\ a\^2\ c0 + 24\ a\^6\ c4\)\/\(2\ a\^4\) + \(8\ a\^4\ c0 + \ 24\ a\^6\ c2 + 96\ a\^8\ c4\)\/\(6\ a\^6\))\))\), "+", \(\((1\/48\ a\^2\ \((\(32\ c0\)\/\(15\ a\^4\) + \(16\ c2\)\/\(5\ \ a\^2\) + 8\ c4 + \(\(-\(\(16\ c0\)\/\(3\ a\^2\)\)\) - 8\ c2 - 20\ a\^2\ \ c4\)\/a\^2 - \(\(-8\)\ c0 - 12\ a\^2\ c2 - 24\ a\^4\ c4\)\/\(2\ a\^4\) + \ \(\(-4\)\ a\^2\ c0 + 24\ a\^6\ c4\)\/\(6\ a\^6\) - \(8\ a\^4\ c0 + 24\ a\^6\ \ c2 + 96\ a\^8\ c4\)\/\(24\ a\^8\))\) + 1\/48\ a\^2\ \((\(-\(\(32\ c0\)\/\(3\ a\^4\)\)\) - \(16\ \ c2\)\/a\^2 - 32\ c4 - \(2\ \((\(-\(\(40\ c0\)\/\(3\ a\^2\)\)\) - 16\ \ c2 - 16\ a\^2\ c4)\)\)\/a\^2 + \(2\ \((\(-8\)\ c0 + 48\ a\^4\ c4)\)\)\/a\^4 - \ \(4\ \((4\ a\^2\ c0 + 24\ a\^4\ c2 + 120\ a\^6\ c4)\)\)\/\(3\ a\^6\) + \(2\ \ \((8\ a\^4\ c0 + 24\ a\^6\ c2 + 96\ a\^8\ c4)\)\)\/\(3\ a\^8\))\))\)\ y\^2\), "+", \(\((1\/48\ a\^2\ \((\(-\(\(88\ c0\)\/\(15\ a\^6\)\)\) - \(48\ \ c2\)\/\(5\ a\^4\) - \(2\ \((\(-\(\(32\ c0\)\/\(3\ a\^4\)\)\) - \(16\ \ c2\)\/a\^2 - 32\ c4)\)\)\/a\^2 - \(112\ c4\)\/\(5\ a\^2\) + \(2\ \((\(-\(\(40\ \ c0\)\/\(3\ a\^2\)\)\) - 16\ c2 - 16\ a\^2\ c4)\)\)\/a\^4 - \(4\ \((\(-8\)\ \ c0 + 48\ a\^4\ c4)\)\)\/\(3\ a\^6\) + \(2\ \((4\ a\^2\ c0 + 24\ a\^4\ c2 + \ 120\ a\^6\ c4)\)\)\/\(3\ a\^8\) - \(4\ \((8\ a\^4\ c0 + 24\ a\^6\ c2 + 96\ \ a\^8\ c4)\)\)\/\(15\ a\^10\))\) + 1\/48\ a\^2\ \((\(64\ c0\)\/\(105\ a\^6\) + \(32\ c2\)\/\(35\ a\ \^4\) + \(\(-\(\(32\ c0\)\/\(15\ a\^4\)\)\) - \(16\ c2\)\/\(5\ a\^2\) - 8\ c4\ \)\/a\^2 + \(16\ c4\)\/\(7\ a\^2\) - \(\(-\(\(16\ c0\)\/\(3\ a\^2\)\)\) - 8\ \ c2 - 20\ a\^2\ c4\)\/\(2\ a\^4\) + \(\(-8\)\ c0 - 12\ a\^2\ c2 - 24\ a\^4\ c4\ \)\/\(6\ a\^6\) - \(\(-4\)\ a\^2\ c0 + 24\ a\^6\ c4\)\/\(24\ a\^8\) + \(8\ \ a\^4\ c0 + 24\ a\^6\ c2 + 96\ a\^8\ c4\)\/\(120\ a\^10\))\))\)\ y\^4\), "+", InterpretationBox[\(O[y]\^5\), SeriesData[ y, 0, {}, 0, 5, 1]]}], SeriesData[ y, 0, { Plus[ Times[ Rational[ 1, 48], Power[ a, 2], Plus[ Times[ Rational[ -40, 3], Power[ a, -2], c0], Times[ -16, c2], Times[ -16, Power[ a, 2], c4], Times[ -2, Power[ a, -2], Plus[ Times[ -8, c0], Times[ 48, Power[ a, 4], c4]]], Times[ 2, Power[ a, -4], Plus[ Times[ 4, Power[ a, 2], c0], Times[ 24, Power[ a, 4], c2], Times[ 120, Power[ a, 6], c4]]], Times[ Rational[ -4, 3], Power[ a, -6], Plus[ Times[ 8, Power[ a, 4], c0], Times[ 24, Power[ a, 6], c2], Times[ 96, Power[ a, 8], c4]]]]], Times[ Rational[ 1, 48], Power[ a, 2], Plus[ Times[ Rational[ 16, 3], Power[ a, -2], c0], Times[ 8, c2], Times[ 20, Power[ a, 2], c4], Times[ Power[ a, -2], Plus[ Times[ -8, c0], Times[ -12, Power[ a, 2], c2], Times[ -24, Power[ a, 4], c4]]], Times[ Rational[ -1, 2], Power[ a, -4], Plus[ Times[ -4, Power[ a, 2], c0], Times[ 24, Power[ a, 6], c4]]], Times[ Rational[ 1, 6], Power[ a, -6], Plus[ Times[ 8, Power[ a, 4], c0], Times[ 24, Power[ a, 6], c2], Times[ 96, Power[ a, 8], c4]]]]]], 0, Plus[ Times[ Rational[ 1, 48], Power[ a, 2], Plus[ Times[ Rational[ 32, 15], Power[ a, -4], c0], Times[ Rational[ 16, 5], Power[ a, -2], c2], Times[ 8, c4], Times[ Power[ a, -2], Plus[ Times[ Rational[ -16, 3], Power[ a, -2], c0], Times[ -8, c2], Times[ -20, Power[ a, 2], c4]]], Times[ Rational[ -1, 2], Power[ a, -4], Plus[ Times[ -8, c0], Times[ -12, Power[ a, 2], c2], Times[ -24, Power[ a, 4], c4]]], Times[ Rational[ 1, 6], Power[ a, -6], Plus[ Times[ -4, Power[ a, 2], c0], Times[ 24, Power[ a, 6], c4]]], Times[ Rational[ -1, 24], Power[ a, -8], Plus[ Times[ 8, Power[ a, 4], c0], Times[ 24, Power[ a, 6], c2], Times[ 96, Power[ a, 8], c4]]]]], Times[ Rational[ 1, 48], Power[ a, 2], Plus[ Times[ Rational[ -32, 3], Power[ a, -4], c0], Times[ -16, Power[ a, -2], c2], Times[ -32, c4], Times[ -2, Power[ a, -2], Plus[ Times[ Rational[ -40, 3], Power[ a, -2], c0], Times[ -16, c2], Times[ -16, Power[ a, 2], c4]]], Times[ 2, Power[ a, -4], Plus[ Times[ -8, c0], Times[ 48, Power[ a, 4], c4]]], Times[ Rational[ -4, 3], Power[ a, -6], Plus[ Times[ 4, Power[ a, 2], c0], Times[ 24, Power[ a, 4], c2], Times[ 120, Power[ a, 6], c4]]], Times[ Rational[ 2, 3], Power[ a, -8], Plus[ Times[ 8, Power[ a, 4], c0], Times[ 24, Power[ a, 6], c2], Times[ 96, Power[ a, 8], c4]]]]]], 0, Plus[ Times[ Rational[ 1, 48], Power[ a, 2], Plus[ Times[ Rational[ -88, 15], Power[ a, -6], c0], Times[ Rational[ -48, 5], Power[ a, -4], c2], Times[ -2, Power[ a, -2], Plus[ Times[ Rational[ -32, 3], Power[ a, -4], c0], Times[ -16, Power[ a, -2], c2], Times[ -32, c4]]], Times[ Rational[ -112, 5], Power[ a, -2], c4], Times[ 2, Power[ a, -4], Plus[ Times[ Rational[ -40, 3], Power[ a, -2], c0], Times[ -16, c2], Times[ -16, Power[ a, 2], c4]]], Times[ Rational[ -4, 3], Power[ a, -6], Plus[ Times[ -8, c0], Times[ 48, Power[ a, 4], c4]]], Times[ Rational[ 2, 3], Power[ a, -8], Plus[ Times[ 4, Power[ a, 2], c0], Times[ 24, Power[ a, 4], c2], Times[ 120, Power[ a, 6], c4]]], Times[ Rational[ -4, 15], Power[ a, -10], Plus[ Times[ 8, Power[ a, 4], c0], Times[ 24, Power[ a, 6], c2], Times[ 96, Power[ a, 8], c4]]]]], Times[ Rational[ 1, 48], Power[ a, 2], Plus[ Times[ Rational[ 64, 105], Power[ a, -6], c0], Times[ Rational[ 32, 35], Power[ a, -4], c2], Times[ Power[ a, -2], Plus[ Times[ Rational[ -32, 15], Power[ a, -4], c0], Times[ Rational[ -16, 5], Power[ a, -2], c2], Times[ -8, c4]]], Times[ Rational[ 16, 7], Power[ a, -2], c4], Times[ Rational[ -1, 2], Power[ a, -4], Plus[ Times[ Rational[ -16, 3], Power[ a, -2], c0], Times[ -8, c2], Times[ -20, Power[ a, 2], c4]]], Times[ Rational[ 1, 6], Power[ a, -6], Plus[ Times[ -8, c0], Times[ -12, Power[ a, 2], c2], Times[ -24, Power[ a, 4], c4]]], Times[ Rational[ -1, 24], Power[ a, -8], Plus[ Times[ -4, Power[ a, 2], c0], Times[ 24, Power[ a, 6], c4]]], Times[ Rational[ 1, 120], Power[ a, -10], Plus[ Times[ 8, Power[ a, 4], c0], Times[ 24, Power[ a, 6], c2], Times[ 96, Power[ a, 8], c4]]]]]]}, 0, 5, 1]]], "Output"] }, Closed]], 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