(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 517297, 17860]*) (*NotebookOutlinePosition[ 517944, 17882]*) (* CellTagsIndexPosition[ 517900, 17878]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ GVLongterm13 \ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[\[Alpha], \ Num, \ Dst, \ k]\), "\[IndentingNewLine]", \(\(\[Alpha]\ = \ 1\/2;\)\), "\[IndentingNewLine]", \(\(Dst\ = 2;\)\), "\[IndentingNewLine]", \(\(Num\ = \ 11;\)\), "\[IndentingNewLine]", \(Plot[{Abs[FApprox[Num, \ \[Lambda], \ Dst, \ \[Alpha]]], Abs[FActual[Num, \ \[Lambda], \ Dst, \ \[Alpha]]\ ]}, \ 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KeU_F@1AKeT005=_F@03001_FFmI03U_F@03001_FFmI03U_F@03001_FFmI059_F@00E6mI00<006mI KeT0>6mI00<006mIKeT0>6mI0P00E6mI001EKeT00`00KeU_F@0gKeT00`00KeU_F@0gKeT2001EKeT0 05I_F@03001_FFmI03I_F@03001_FFmI03I_F@8005I_F@00EfmI00<006mIKeT0=FmI00<006mIKeT0 =FmI0P00EfmI001HKeT00`00KeU_F@0dKeT2000eKeT2001HKeT005U_F@03001_FFmI03=_F@03001_ FFmI03=_F@8005U_F@00FVmI00<006mIKeT0KeT00`00KeU_F@0:KeT6001nKeT0081_F@L000Q_ F@03001_FFmI00I_F@D008=_F@00QVmI4`00QfmI002?KeT00`00KeU_F@2>KeT008m_F@03001_FFmI 08i_F@00SfmI00<006mIKeT0SVmI002?KeT00`00KeU_F@2>KeT00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-3.30686, 17.6544, \ 0.0230443, 0.0141664}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ So the first-order convergence looks passable. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Normal[\ Series[Log[FActual[Num, \[Lambda], 1, \ \[Alpha]]], \ {\[Lambda], 0, 2}]]\)], "Input"], Cell[BoxData[ \(Log[ 1\/\(Num\ \[Alpha]\ Beta[Num\ \[Alpha], 1 + Num - Num\ \[Alpha]]\) - \ \(2\ Gamma[1 + Num]\ Gamma[1 - Num + Num\ \[Alpha]]\ Hypergeometric2F1[1, 1 - \ Num + Num\ \[Alpha], 2 + Num\ \[Alpha], \(-1\)]\ Sin[\[Pi]\ \((1 + Num - Num\ \ \[Alpha])\)]\)\/\(\[Pi]\ Gamma[2 + Num\ \[Alpha]]\)] + \((\[Lambda]\^2\ Gamma[ 1 + Num]\ Gamma[ 1 - Num + Num\ \[Alpha]]\ \((1\/2\ Hypergeometric2F1[1, 1 - Num + Num\ \[Alpha], 2 + Num\ \[Alpha], \(-1\)] + 1\/2\ Num\ Hypergeometric2F1[1, 1 - Num + Num\ \[Alpha], 2 + Num\ \[Alpha], \(-1\)] - 1\/2\ Num\ \[Alpha]\ Hypergeometric2F1[1, 1 - Num + Num\ \[Alpha], 2 + Num\ \[Alpha], \(-1\)] - Hypergeometric2F1[2, 2 - Num + Num\ \[Alpha], 3 + Num\ \ \[Alpha], \(-1\)]\/\(2\ \((2 + Num\ \[Alpha])\)\) + \(Num\^2\ \ Hypergeometric2F1[2, 2 - Num + Num\ \[Alpha], 3 + Num\ \[Alpha], \ \(-1\)]\)\/\(2\ \((2 + Num\ \[Alpha])\)\) + \(Num\ \[Alpha]\ \ Hypergeometric2F1[2, 2 - Num + Num\ \[Alpha], 3 + Num\ \[Alpha], \ \(-1\)]\)\/\(2\ \((2 + Num\ \[Alpha])\)\) - \(3\ Num\^2\ \[Alpha]\ \ Hypergeometric2F1[2, 2 - Num + Num\ \[Alpha], 3 + Num\ \[Alpha], \ \(-1\)]\)\/\(2\ \((2 + Num\ \[Alpha])\)\) + \(Num\^2\ \[Alpha]\^2\ \ Hypergeometric2F1[2, 2 - Num + Num\ \[Alpha], 3 + Num\ \[Alpha], \ \(-1\)]\)\/\(2 + Num\ \[Alpha]\))\)\ Sin[\[Pi]\ \((1 + Num - Num\ \[Alpha])\)])\)/\((\[Pi]\ Gamma[ 2 + Num\ \[Alpha]]\ \((1\/\(Num\ \[Alpha]\ Beta[Num\ \[Alpha], \ 1 + Num - Num\ \[Alpha]]\) - \(2\ Gamma[1 + Num]\ Gamma[1 - Num + Num\ \ \[Alpha]]\ Hypergeometric2F1[1, 1 - Num + Num\ \[Alpha], 2 + Num\ \[Alpha], \ \(-1\)]\ Sin[\[Pi]\ \((1 + Num - Num\ \[Alpha])\)]\)\/\(\[Pi]\ Gamma[2 + Num\ \ \[Alpha]]\))\))\)\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Normal[\ Series[Log[FActual[Num, \[Lambda], 2, \ \[Alpha]]], \ {\[Lambda], 0, 2}]]\)], "Input"], Cell[BoxData[ \(Log[ 1\/\(Num\ \[Alpha]\ Beta[Num\ \[Alpha], 1 + Num - Num\ \[Alpha]]\) + \ \(2\ Gamma[1 + Num]\ Gamma[2 - Num + Num\ \[Alpha]]\ HypergeometricPFQ[{1, 1 \ - Num\/2 + \(Num\ \[Alpha]\)\/2, 3\/2 - Num\/2 + \(Num\ \[Alpha]\)\/2}, {3\/2 \ + \(Num\ \[Alpha]\)\/2, 2 + \(Num\ \[Alpha]\)\/2}, 1]\ Sin[\[Pi]\ \((1 + Num \ - Num\ \[Alpha])\)]\)\/\(\[Pi]\ Gamma[3 + Num\ \[Alpha]]\)] + \((\[Lambda]\^2\ \ Gamma[1 + Num]\ Gamma[ 2 - Num + Num\ \[Alpha]]\ \((\(-HypergeometricPFQ[{1, 1 - Num\/2 + \(Num\ \[Alpha]\)\/2, 3\/2 - Num\/2 + \(Num\ \[Alpha]\)\/2}, {3\/2 + \(Num\ \ \[Alpha]\)\/2, 2 + \(Num\ \[Alpha]\)\/2}, 1]\) - \(3\ \((1 - Num\/2 + \(Num\ \[Alpha]\)\/2)\)\ \((3\ \/2 - Num\/2 + \(Num\ \[Alpha]\)\/2)\)\ HypergeometricPFQ[{2, 2 - Num\/2 + \ \(Num\ \[Alpha]\)\/2, 5\/2 - Num\/2 + \(Num\ \[Alpha]\)\/2}, {5\/2 + \(Num\ \ \[Alpha]\)\/2, 3 + \(Num\ \[Alpha]\)\/2}, 1]\)\/\(\((3\/2 + \(Num\ \[Alpha]\)\ \/2)\)\ \((2 + \(Num\ \[Alpha]\)\/2)\)\) - \(2\ \((1 - Num\/2 + \(Num\ \ \[Alpha]\)\/2)\)\ \((3\/2 - Num\/2 + \(Num\ \[Alpha]\)\/2)\)\ \((2 - Num\/2 + \ \(Num\ \[Alpha]\)\/2)\)\ \((5\/2 - Num\/2 + \(Num\ \[Alpha]\)\/2)\)\ \ HypergeometricPFQ[{3, 3 - Num\/2 + \(Num\ \[Alpha]\)\/2, 7\/2 - Num\/2 + \ \(Num\ \[Alpha]\)\/2}, {7\/2 + \(Num\ \[Alpha]\)\/2, 4 + \(Num\ \ \[Alpha]\)\/2}, 1]\)\/\(\((3\/2 + \(Num\ \[Alpha]\)\/2)\)\ \((2 + \(Num\ \ \[Alpha]\)\/2)\)\ \((5\/2 + \(Num\ \[Alpha]\)\/2)\)\ \((3 + \(Num\ \[Alpha]\)\ \/2)\)\))\)\ Sin[\[Pi]\ \((1 + Num - Num\ \[Alpha])\)])\)/\((\[Pi]\ Gamma[ 3 + Num\ \[Alpha]]\ \((1\/\(Num\ \[Alpha]\ Beta[Num\ \[Alpha], \ 1 + Num - Num\ \[Alpha]]\) + \(2\ Gamma[1 + Num]\ Gamma[2 - Num + Num\ \ \[Alpha]]\ HypergeometricPFQ[{1, 1 - Num\/2 + \(Num\ \[Alpha]\)\/2, 3\/2 - \ Num\/2 + \(Num\ \[Alpha]\)\/2}, {3\/2 + \(Num\ \[Alpha]\)\/2, 2 + \(Num\ \ \[Alpha]\)\/2}, 1]\ Sin[\[Pi]\ \((1 + Num - Num\ \[Alpha])\)]\)\/\(\[Pi]\ \ Gamma[3 + Num\ \[Alpha]]\))\))\)\)], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Normal[\ Series[Log[FActual[Num, \[Lambda], 3, \ \[Alpha]]], \ {\[Lambda], 0, 2}]]\)], "Input"], Cell[BoxData[ \(Log[ 1\/\(Num\ \[Alpha]\ Beta[Num\ \[Alpha], 1 + Num - Num\ \[Alpha]]\) - \ \(2\ Gamma[1 + Num]\ Gamma[3 - Num + Num\ \[Alpha]]\ HypergeometricPFQ[{1, 1 \ - Num\/3 + \(Num\ \[Alpha]\)\/3, 4\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3, 5\/3 - \ Num\/3 + \(Num\ \[Alpha]\)\/3}, {4\/3 + \(Num\ \[Alpha]\)\/3, 5\/3 + \(Num\ \ \[Alpha]\)\/3, 2 + \(Num\ \[Alpha]\)\/3}, \(-1\)]\ Sin[\[Pi]\ \((1 + Num - \ Num\ \[Alpha])\)]\)\/\(\[Pi]\ Gamma[4 + Num\ \[Alpha]]\)] + \((\[Lambda]\^2\ \ Gamma[1 + Num]\ Gamma[ 3 - Num + Num\ \[Alpha]]\ \((HypergeometricPFQ[{1, 1 - Num\/3 + \(Num\ \[Alpha]\)\/3, 4\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3, 5\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3}, {4\/3 + \(Num\ \ \[Alpha]\)\/3, 5\/3 + \(Num\ \[Alpha]\)\/3, 2 + \(Num\ \[Alpha]\)\/3}, \(-1\)] - \(3\ \((1 - Num\/3 + \ \(Num\ \[Alpha]\)\/3)\)\ \((4\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3)\)\ \((5\/3 \ - Num\/3 + \(Num\ \[Alpha]\)\/3)\)\ HypergeometricPFQ[{2, 2 - Num\/3 + \(Num\ \ \[Alpha]\)\/3, 7\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3, 8\/3 - Num\/3 + \(Num\ \ \[Alpha]\)\/3}, {7\/3 + \(Num\ \[Alpha]\)\/3, 8\/3 + \(Num\ \[Alpha]\)\/3, 3 \ + \(Num\ \[Alpha]\)\/3}, \(-1\)]\)\/\(\((4\/3 + \(Num\ \[Alpha]\)\/3)\)\ \((5\ \/3 + \(Num\ \[Alpha]\)\/3)\)\ \((2 + \(Num\ \[Alpha]\)\/3)\)\) + \((2\ \((1 \ - Num\/3 + \(Num\ \[Alpha]\)\/3)\)\ \((4\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3)\)\ \((5\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3)\)\ \((2 - Num\/3 + \(Num\ \[Alpha]\)\/3)\)\ \((7\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3)\)\ \((8\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3)\)\ \ HypergeometricPFQ[{3, 3 - Num\/3 + \(Num\ \[Alpha]\)\/3, 10\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3, 11\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3}, {10\/3 + \(Num\ \ \[Alpha]\)\/3, 11\/3 + \(Num\ \[Alpha]\)\/3, 4 + \(Num\ \[Alpha]\)\/3}, \(-1\)])\)/\((\((4\/3 + \ \(Num\ \[Alpha]\)\/3)\)\ \((5\/3 + \(Num\ \[Alpha]\)\/3)\)\ \((2 + \(Num\ \ \[Alpha]\)\/3)\)\ \((7\/3 + \(Num\ \[Alpha]\)\/3)\)\ \((8\/3 + \(Num\ \ \[Alpha]\)\/3)\)\ \((3 + \(Num\ \[Alpha]\)\/3)\))\))\)\ Sin[\[Pi]\ \((1 + Num - Num\ \[Alpha])\)])\)/\((\[Pi]\ Gamma[ 4 + Num\ \[Alpha]]\ \((1\/\(Num\ \[Alpha]\ Beta[Num\ \[Alpha], \ 1 + Num - Num\ \[Alpha]]\) - \(2\ Gamma[1 + Num]\ Gamma[3 - Num + Num\ \ \[Alpha]]\ HypergeometricPFQ[{1, 1 - Num\/3 + \(Num\ \[Alpha]\)\/3, 4\/3 - \ Num\/3 + \(Num\ \[Alpha]\)\/3, 5\/3 - Num\/3 + \(Num\ \[Alpha]\)\/3}, {4\/3 + \ \(Num\ \[Alpha]\)\/3, 5\/3 + \(Num\ \[Alpha]\)\/3, 2 + \(Num\ \[Alpha]\)\/3}, \ \(-1\)]\ Sin[\[Pi]\ \((1 + Num - Num\ \[Alpha])\)]\)\/\(\[Pi]\ Gamma[4 + Num\ \ \[Alpha]]\))\))\)\)], "Output"] }, Closed]], Cell["\<\ So I can certainly integrate this approximation: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_\(-\ \[Pi]\)\%\[Pi] Log[ A\ + B\ \[Lambda]\^2]\ \[DifferentialD]\[Lambda]\)], "Input"], Cell[BoxData[ \(If[Re[A] > 0 && \((Im[A\/B] \[NotEqual] 0 || Re[A\/B] \[GreaterEqual] 0)\) || Re[A] > 0 && \((Im[\@\(-\(A\/B\)\)] \[NotEqual] 0 || Re[\@\(-\(A\/B\)\)] > \[Pi])\) && \((Im[A\/B] \[NotEqual] 0 || \[Pi]\^2 + Re[A\/B] < 0)\), 2\ \((\(2\ \@A\ ArcTan[\(\@B\ \[Pi]\)\/\@A]\)\/\@B + \[Pi]\ \((\(-2\) + Log[A] + Log[1 + \(B\ \[Pi]\^2\)\/A])\))\), Integrate[Log[A + B\ \[Lambda]\^2], {\[Lambda], \(-\[Pi]\), \[Pi]}, Assumptions \[Rule] \(! \((Re[A] > 0 && \((Im[A\/B] \[NotEqual] 0 || Re[A\/B] \[GreaterEqual] 0)\) || Re[A] > 0 && \((Im[\@\(-\(A\/B\)\)] \[NotEqual] 0 || Re[\@\(-\(A\/B\)\)] > \[Pi])\) && \((Im[ A\/B] \[NotEqual] 0 || \[Pi]\^2 + Re[A\/B] < 0)\))\)\)]]\)], "Output"] }, Closed]], Cell["\<\ Let me then translate this into the desired form and see how it works. I'll \ eventually need to take a limit as N grows.\ \>", "Text"], Cell[BoxData[{ \(\(A[Num_, \ \[Alpha]_, \ D_]\ := \ Binomial[ Num, \ \[Alpha]\ Num] + \(\(\((\(-1\))\)\^D\ 2\ Gamma[ 1\ + \ Num]\ Gamma[ D\ - \ Num\ + \ Num\ \[Alpha]]\ Sin[\[Pi] \((1\ + \ Num\ - \ Num\ \[Alpha])\)]\)\/\(\[Pi]\ Gamma[ D\ + \ 1\ + \ Num\ \[Alpha]]\)\) HypergeometricPFQ[ Union[{1}, \ Table[Array1[i, \ D, \ Num, \ \[Alpha]], \ {i, \ 1, \ D}]], \ Union[Table[ Array2[i, \ D, \ Num, \ \[Alpha]], \ {i, \ 1, \ D}]], \ \((\(-1\))\)\^D\ ];\)\), "\[IndentingNewLine]", \(\(B[Num_, \ \[Alpha]_, \ D_]\ := \ Gamma[1 + Num]\ Gamma[ D - Num + Num\ \[Alpha]]\ Sin[\[Pi]\ \((1 + Num - Num\ \[Alpha])\)] \((\(\((\(-1\))\)\^\(D - 1\)\) HypergeometricPFQ[ Union[{1}, \ Table[Array1[i, \ D, \ Num, \ \[Alpha]], \ {i, \ 1, \ D}]], \ Union[Table[ Array2[i, \ D, \ Num, \ \[Alpha]], \ {i, \ 1, \ D}]], \ \((\(-1\))\)\^D\ ] - \(\(3 \((\ \ \[Product]\+\(i = 0\)\%\(D - 1\)\((1 + i\/D - Num\/D + \(Num\ \[Alpha]\)\/D)\))\)\)\/\((\ \ \[Product]\+\(i = 1\)\%D\((1 + i\/D + \(Num\ \[Alpha]\)\/D)\))\)\) HypergeometricPFQ[ Union[{2}, \ Table[Array1[i, \ D, \ Num, \ \[Alpha]] + 1, \ {i, \ 1, \ D}]], \ Union[Table[ Array2[i, \ D, \ Num, \ \[Alpha]] + 1, \ {i, \ 1, \ D}]], \ \((\(-1\))\)\^D\ ] + \(\((\(-1\))\)\^\(D - 1\)\) \(\(2 \((\ \[Product]\+\(i = 0\)\%\(2 D - \ 1\)\((1 + i\/D - Num\/D + \(Num\ \[Alpha]\)\/D)\))\)\)\/\((\ \ \[Product]\+\(i = 1\)\%\(2 D\)\((1 + i\/D + \(Num\ \[Alpha]\)\/D)\))\)\) \ \((HypergeometricPFQ[ Union[{3}, \ Table[Array1[i, \ D, \ Num, \ \[Alpha]] + 2, \ {i, \ 1, \ D}]], \ Union[Table[ Array2[i, \ D, \ Num, \ \[Alpha]] + 2, \ {i, \ 1, \ D}]], \ \((\(-1\))\)\^D\ ])\)\ )\)/\((\[Pi]\ \ Gamma[D + 1 + Num\ \[Alpha]]\ \((A[Num, \ \[Alpha], \ D]\ )\))\);\)\)}], "Input"], Cell["\<\ Despite the apparent complexity, these are functions of N, alpha and D only, \ and from these I should be able to complete the integral. After that, I want \ to take 1/N the limit of it the above expression to get the long-term \ area-wise function of distance D only. Let me first type the result and see \ if it works.\ \>", "Text"], Cell[BoxData[ \(Approximate[N_, \ \[Alpha]_, \ D_]\ := 2 \((\(\(2\ \@A[N, \[Alpha], D]\)\/\(\(\ \)\(\@B[N, \[Alpha], D]\)\)\) ArcTan[\(\@B[N, \[Alpha], D]\/\@A[N, \[Alpha], D]\) \[Pi]] + \ \ \[Pi]\ \((\(-2\)\ + \ Log[A[N, \[Alpha], D]]\ + \ Log[1 + \(B[N, \[Alpha], D]\ \[Pi]\^2\)\/A[N, \[Alpha], \ D]])\))\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(Clear[\[Alpha], \ Num, \ Dst, \ k, \ Val]\[IndentingNewLine]\n \(Dst\ = 2;\)\[IndentingNewLine]\n \(Val[\[Lambda]_, \ Num_, \ \[Alpha]_]\ = Normal[\ Series[ Log[FActual[Num, \[Lambda], \ Dst, \ \[Alpha]]], \ {\[Lambda], 0, 2}]];\)\n \(\[Alpha]\ = \ 1\/2;\)\n \(Num = 49;\)\n Plot[{Val[\[Lambda], 49, 1\/2]\ , Log[Abs[FActual[Num, \ \[Lambda], \ Dst, \ \[Alpha]]\ ]]}, \ {\[Lambda], \ \(-\ \ \[Pi]\), \ \ \ \[Pi]}]\n \(Clear[\[Alpha], \ Num, \ Dst, \ k, \ A, \ B];\)\[IndentingNewLine] \)\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! 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