(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 12924, 459]*) (*NotebookOutlinePosition[ 13901, 489]*) (* CellTagsIndexPosition[ 13857, 485]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Nuclear Matter Paper: CR Equation Checks", FontSize->18]], "Title", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Overview:\nIn this notebook, we look at log vs. power divergences \ using perturbative matching.\n\nPlan:\nTry to make the equations and \ solutions as general as possible so that we can expand\n\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" strategy:\n", Evaluatable->False, AspectRatioFixed->True], "Write equations as they appear in the notes, using := to define lhs's. \ Then the evaluation of the rhs is not performed until the lhs is actually \ used somewhere else. This avoids the mess of having explicit arguments for \ everything.\n", StyleBox["\n", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Clear symbols", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ In order to avoid interference from symbols defined in other \ notebooks, we first Clear all symbols. We assume that the relevant symbols \ are in the Global` context.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear["\"]\)], "Input", AspectRatioFixed->True], Cell["\<\ Suppress spelling warnings (only do this after the notebook is \ debugged!).\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(\(\ \)\(Off[General::"\", General::"\"]\)\(\ \ \)\)\)], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Date", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(\((date = Date[\(-4\)]; \[IndentingNewLine]Print["\", date[\([2]\)], "\", date[\([3]\)], "\", date[\([1]\)], "\< at \>", date[\([4]\)], "\<:\>", date[\([5]\)]])\)\)], "Input"], Cell[BoxData[ InterpretationBox[\("Notebook run on: \ "\[InvisibleSpace]4\[InvisibleSpace]"/"\[InvisibleSpace]12\[InvisibleSpace]"/\ "\[InvisibleSpace]2000\[InvisibleSpace]" at "\[InvisibleSpace]8\ \[InvisibleSpace]":"\[InvisibleSpace]58\), SequenceForm[ "Notebook run on: ", 4, "/", 12, "/", 2000, " at ", 8, ":", 58], Editable->False]], "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Define potentials and integrals", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["CR Potential", "Subsection"], Cell["Define the forward CR potential:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(C0cr\ = \ 4\ alpha\ Pi/\((m\ M)\);\)\)], "Input"], Cell[BoxData[ \(V0cr[k_, q_]\ := \ \((C0cr\ )\)\ \ Exp[\(-k^2\)/\((2\ \ Lc^2)\)]\ \ Exp[\(-q^2\)/\((2\ \ Lc^2)\)]\)], "Input"], Cell[BoxData[ \(\(\(V0cr2[k_, q_]\)\(\ \)\(:=\)\(\ \)\(1\)\(\ \)\( (*\((C0cr\ )\)*) \)\(\ \)\)\)], \ "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[V0cr2[k, k], {k, 0, 2}]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell["Define integrals needed to solve for the K matrix:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(j\ /: \ IntegerQ[j]\ = \ True\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell["\<\ CR2integral[p_, n_,dim_] := FullSimplify[1/(2 Pi^(dim-1))* Integrate[ k^((dim-1)+2*n) V0cr2[p,k] 1/(p^2-k^2) V0cr2[k,p], {k, 0, Lc}, PrincipalValue -> True, Assumptions ->{Im[p] == 0, p > 0, Im[Lc]==0,Re[Lc^2] > 0, Lc>0, n >= \ 0}], {Lc>0,p>0}]\ \>", "Input"], Cell[BoxData[ \(CR3integral[p_, n_, dim_] := FullSimplify[ 1/\((2\ Pi^\((dim - 1)\))\)* Integrate[ k^\((\((dim - 1)\) + 2*n)\)\ V0cr2[p, k]\ 1/\((p^2 + k^2)\)\ V0cr2[k, p], {k, 0, Lc}, PrincipalValue \[Rule] True, Assumptions \[Rule] {Im[p] \[Equal] 0, p > 0, Im[Lc] \[Equal] 0, Re[Lc^2] > 0, Lc > 0, n \[GreaterEqual] 0}], {Lc > 0, p > 0}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[CR2integral[p, 0, 3], {p, 0, 2}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(-\(Lc\/\(2\ \[Pi]\^2\)\)\), "+", \(p\^2\/\(2\ Lc\ \[Pi]\^2\)\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Times[ Rational[ -1, 2], Lc, Power[ Pi, -2]], 0, Times[ Rational[ 1, 2], Power[ Lc, -1], Power[ Pi, -2]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Series[CR2integral[p, 1, 3], {p, 0, 2}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(-\(Lc\^3\/\(6\ \[Pi]\^2\)\)\), "-", \(\(Lc\ p\^2\)\/\(2\ \[Pi]\^2\)\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Times[ Rational[ -1, 6], Power[ Lc, 3], Power[ Pi, -2]], 0, Times[ Rational[ -1, 2], Lc, Power[ Pi, -2]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Series[CR2integral[p, 2, 3], {p, 0, 2}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(-\(Lc\^5\/\(10\ \[Pi]\^2\)\)\), "-", \(\(Lc\^3\ p\^2\)\/\(6\ \[Pi]\^2\)\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Times[ Rational[ -1, 10], Power[ Lc, 5], Power[ Pi, -2]], 0, Times[ Rational[ -1, 6], Power[ Lc, 3], Power[ Pi, -2]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Series[CR2integral[p, 3, 3], {p, 0, 2}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(-\(Lc\^7\/\(14\ \[Pi]\^2\)\)\), "-", \(\(Lc\^5\ p\^2\)\/\(10\ \[Pi]\^2\)\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Times[ Rational[ -1, 14], Power[ Lc, 7], Power[ Pi, -2]], 0, Times[ Rational[ -1, 10], Power[ Lc, 5], Power[ Pi, -2]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ Series[CR3integral[p, 0, 3], {p, 0, 2}], {Im[p] \[Equal] 0, p > 0, Im[Lc] \[Equal] 0, Re[Lc^2] > 0, Lc > 0}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(Lc\/\(2\ \[Pi]\^2\)\), "-", \(p\/\(4\ \[Pi]\)\), "+", \(p\^2\/\(2\ Lc\ \[Pi]\^2\)\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Times[ Rational[ 1, 2], Lc, Power[ Pi, -2]], Times[ Rational[ -1, 4], Power[ Pi, -1]], Times[ Rational[ 1, 2], Power[ Lc, -1], Power[ Pi, -2]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ Series[CR3integral[p, 1, 3], {p, 0, 4}], {Im[p] \[Equal] 0, p > 0, Im[Lc] \[Equal] 0, Re[Lc^2] > 0, Lc > 0}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(Lc\^3\/\(6\ \[Pi]\^2\)\), "-", \(\(Lc\ p\^2\)\/\(2\ \[Pi]\^2\)\), "+", \(p\^3\/\(4\ \[Pi]\)\), "-", \(p\^4\/\(2\ \((Lc\ \[Pi]\^2)\)\)\), "+", InterpretationBox[\(O[p]\^5\), SeriesData[ p, 0, {}, 0, 5, 1]]}], SeriesData[ p, 0, { Times[ Rational[ 1, 6], Power[ Lc, 3], Power[ Pi, -2]], 0, Times[ Rational[ -1, 2], Lc, Power[ Pi, -2]], Times[ Rational[ 1, 4], Power[ Pi, -1]], Times[ Rational[ -1, 2], Power[ Lc, -1], Power[ Pi, -2]]}, 0, 5, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ Series[CR2integral[p, 0, 2], {p, 0, 2}], {Im[p] \[Equal] 0, p > 0, Im[Lc] \[Equal] 0, Re[Lc^2] > 0, Lc > 0}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(Log[p\/Lc]\/\(2\ \[Pi]\)\), "+", \(p\^2\/\(4\ Lc\^2\ \[Pi]\)\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Times[ Rational[ 1, 2], Power[ Pi, -1], Log[ Times[ Power[ Lc, -1], p]]], 0, Times[ Rational[ 1, 4], Power[ Lc, -2], Power[ Pi, -1]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ Series[CR2integral[p, 1, 2], {p, 0, 2}], {Im[p] \[Equal] 0, p > 0, Im[Lc] \[Equal] 0, Re[Lc^2] > 0, Lc > 0}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(-\(Lc\^2\/\(4\ \[Pi]\)\)\), "+", \(\(Log[p\/Lc]\ p\^2\)\/\(2\ \[Pi]\)\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Times[ Rational[ -1, 4], Power[ Lc, 2], Power[ Pi, -1]], 0, Times[ Rational[ 1, 2], Power[ Pi, -1], Log[ Times[ Power[ Lc, -1], p]]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ Series[CR3integral[p, 0, 2], {p, 0, 2}], {Im[p] \[Equal] 0, p > 0, Im[Lc] \[Equal] 0, Re[Lc^2] > 0, Lc > 0}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(Log[Lc\/p]\/\(2\ \[Pi]\)\), "+", \(p\^2\/\(4\ Lc\^2\ \[Pi]\)\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Times[ Rational[ 1, 2], Power[ Pi, -1], Log[ Times[ Lc, Power[ p, -1]]]], 0, Times[ Rational[ 1, 4], Power[ Lc, -2], Power[ Pi, -1]]}, 0, 3, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ Series[CR3integral[p, 1, 2], {p, 0, 2}], {Im[p] \[Equal] 0, p > 0, Im[Lc] \[Equal] 0, Re[Lc^2] > 0, Lc > 0}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(Lc\^2\/\(4\ \[Pi]\)\), "+", \(\(Log[p\/Lc]\ p\^2\)\/\(2\ \[Pi]\)\), "+", InterpretationBox[\(O[p]\^3\), SeriesData[ p, 0, {}, 0, 3, 1]]}], SeriesData[ p, 0, { Times[ Rational[ 1, 4], Power[ Lc, 2], Power[ Pi, -1]], 0, Times[ Rational[ 1, 2], Power[ Pi, -1], Log[ Times[ Power[ Lc, -1], p]]]}, 0, 3, 1]]], "Output"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{663, 853}, WindowMargins->{{38, Automatic}, {Automatic, 58}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "home", \ "furnstah", "Research", "papers", "nuclear_matter_paper", "Mathematica"}, \ "CR_check.nb.ps", CharacterEncoding -> "ISO8859-1"], "Magnification"->1} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1739, 51, 166, 4, 83, "Title", Evaluatable->False], Cell[1908, 57, 840, 22, 212, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[2773, 83, 80, 2, 54, "Section", Evaluatable->False], Cell[2856, 87, 240, 6, 50, "Text", Evaluatable->False], Cell[3099, 95, 80, 2, 27, "Input"], Cell[3182, 99, 148, 5, 32, "Text", Evaluatable->False], Cell[3333, 106, 136, 3, 27, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[3506, 114, 71, 2, 54, "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[3602, 120, 259, 5, 59, "Input"], Cell[3864, 127, 370, 7, 23, "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[4283, 140, 98, 2, 54, "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[4406, 146, 34, 0, 45, "Subsection"], Cell[4443, 148, 96, 2, 32, "Text", Evaluatable->False], Cell[4542, 152, 72, 1, 27, "Input"], Cell[4617, 155, 140, 3, 27, "Input"], Cell[4760, 160, 122, 3, 27, "Input"], Cell[CellGroupData[{ Cell[4907, 167, 63, 1, 27, "Input"], Cell[4973, 170, 35, 1, 27, "Output"] }, Open ]], Cell[5023, 174, 114, 2, 32, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[5162, 180, 64, 1, 27, "Input"], Cell[5229, 183, 38, 1, 27, "Output"] }, Open ]], Cell[5282, 187, 306, 8, 117, "Input"], Cell[5591, 197, 465, 10, 123, "Input"], Cell[CellGroupData[{ Cell[6081, 211, 72, 1, 27, "Input"], Cell[6156, 214, 442, 13, 46, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6635, 232, 72, 1, 27, "Input"], Cell[6710, 235, 449, 13, 46, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7196, 253, 72, 1, 27, "Input"], Cell[7271, 256, 476, 14, 46, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7784, 275, 72, 1, 27, "Input"], Cell[7859, 278, 478, 14, 46, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8374, 297, 170, 3, 43, "Input"], Cell[8547, 302, 532, 16, 46, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[9116, 323, 170, 3, 43, "Input"], Cell[9289, 328, 695, 21, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10021, 354, 170, 3, 43, "Input"], Cell[10194, 359, 509, 16, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10740, 380, 170, 3, 43, "Input"], Cell[10913, 385, 518, 16, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11468, 406, 170, 3, 43, "Input"], Cell[11641, 411, 509, 16, 54, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[12187, 432, 170, 3, 43, "Input"], Cell[12360, 437, 512, 16, 50, "Output"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)