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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[ 259589, 6937]*) (*NotebookOutlinePosition[ 260236, 6959]*) (* CellTagsIndexPosition[ 260192, 6955]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ The generating function summed. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[N, \ \[Alpha]\ N\ + \ k]\ Cos[ k\ \[Lambda]]\), "\[IndentingNewLine]", \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[N, \ \[Alpha]\ N\ + 2\ k]\ Cos[ k\ \[Lambda]]\), "\[IndentingNewLine]", \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[N, \ \[Alpha]\ N\ + 3\ k]\ Cos[ k\ \[Lambda]]\), 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F@00;6mI00<006mIKeT036mI00<006mIKeT0ofmIfVmI00<006mIKeT06VmI000/KeT00`00KeU_F@0< KeT00`00KeU_F@3oKeWJKeT00`00KeU_F@0JKeT002a_F@03001_FFmI00a_F@03001_FFmI0?m_FMY_ F@03001_FFmI01Y_F@00;6mI00<006mIKeT036mI00<006mIKeT0ofmIfVmI00<006mIKeT06VmI000/ KeT00`00KeU_F@0"], ImageRangeCache->{{{0, 563}, {347.438, 0}} -> {-0.547967, -5.54724*^29, \ 0.0124126, 2.61581*^28}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Further, it appears that as distance D increases, the function becomes less \ and less in danger of violating the conditions of the Toeplitz determinant \ formula. As N increases, the function becomes more bathtub-shaped and \ overall larger: it will diverge, but I will need this additional factor in \ order to cancel the overall area factor in my formula for entropy. Note that \ in the case I originally laid out, this formula does not converge as a zero \ times infinity appears. Thus, I must interpolate between interim values. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[\[Alpha], \ Num, \ Dst, \ k]\), "\[IndentingNewLine]", \(\(\[Alpha]\ = \ 1\/2;\)\), "\[IndentingNewLine]", \(\(Dst\ = 2;\)\), "\[IndentingNewLine]", \(\(Num\ = \ 11;\)\), "\[IndentingNewLine]", \(NIntegrate[\ \(1\/\(2\ \[Pi]\)\) Log[Binomial[Num, \ \[Alpha]\ Num] + \ 2\ \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[ Num, \ \[Alpha]\ Num\ + Dst\ k]\ Cos[ k\ \[Lambda]]\)], \ {\[Lambda], 0, 2\ \[Pi]}]\), "\[IndentingNewLine]", \(\(Num = 13;\)\), "\[IndentingNewLine]", \(NIntegrate[\ \(1\/\(2\ \[Pi]\)\) Log[Binomial[Num, \ \[Alpha]\ Num] + \ 2\ \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[ Num, \ \[Alpha]\ Num\ + Dst\ k]\ Cos[ k\ \[Lambda]]\)], \ {\[Lambda], 0, 2\ \[Pi]}]\), "\[IndentingNewLine]", \(\(Num = 15;\)\), "\[IndentingNewLine]", \(NIntegrate[\ \(1\/\(2\ \[Pi]\)\) Log[Binomial[Num, \ \[Alpha]\ Num] + \ 2\ \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[ Num, \ \[Alpha]\ Num\ + Dst\ k]\ Cos[ k\ \[Lambda]]\)], \ {\[Lambda], 0, 2\ \[Pi]}]\), "\[IndentingNewLine]", \(\(Num = 17;\)\), "\[IndentingNewLine]", \(NIntegrate[\ \(1\/\(2\ \[Pi]\)\) Log[Binomial[Num, \ \[Alpha]\ Num] + \ 2\ \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[ Num, \ \[Alpha]\ Num\ + Dst\ k]\ Cos[ k\ \[Lambda]]\)], \ {\[Lambda], 0, 2\ \[Pi]}]\), "\[IndentingNewLine]", \(\(Num = 19;\)\), "\[IndentingNewLine]", \(NIntegrate[\ \(1\/\(2\ \[Pi]\)\) Log[Binomial[Num, \ \[Alpha]\ Num] + \ 2\ \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[ Num, \ \[Alpha]\ Num\ + Dst\ k]\ Cos[ k\ \[Lambda]]\)], \ {\[Lambda], 0, 2\ \[Pi]}]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Clear[\[Alpha], \ Num, \ Dst, \ k]\)}], "Input"], Cell[BoxData[ \(5.768739723197077`\)], "Output"], Cell[BoxData[ \(6.927450889804874`\)], "Output"], Cell[BoxData[ \(8.088345252008146`\)], "Output"], Cell[BoxData[ \(9.250602254425877`\)], "Output"], Cell[BoxData[ \(10.4136506493629`\)], "Output"] }, Open ]], Cell[TextData[{ "This indicates that Log[", Cell[BoxData[ \(TraditionalForm\`Det[\ matrix\ ]\^\(1\/p\)\)]], "]. Let me now compare this to the actual value for such a matrix." }], "Text"], Cell[BoxData[{ \(\(GetStartPoint[index_, \ coord_, \ d_]\ := If\ [Mod[index, \ 2] \[Equal] 1, \ If[coord\ \[Equal] \ 1, \ \ d\ \((\(index - 1\)\/2)\), \ \(-\ d\)\ \((\(index - 1\)\/2\ )\)], \ If[coord\ \[Equal] \ 1, \(-\ \ d\)\ \((index\/2\ )\), \ \ d\ \((index\/2\ )\)]\ ];\)\ \), "\[IndentingNewLine]", \(\(GetEndPoint[index_, \ coord_, length_, \ d_, \ slope_]\ := \ GetStartPoint[index, \ coord, \ d]\ + \ If[coord \[Equal] 1, \ length\ slope, \ length\ \((1\ - \ slope)\)];\)\), "\[IndentingNewLine]", \(\(PathCount[i_, \ j_, \ length_, \ d_, \ slope_]\ := Binomial[ GetEndPoint[j, \ 1, length, \ d, \ slope] - GetStartPoint[i, \ 1, d] + GetEndPoint[j, 2, length, d, slope] - GetStartPoint[i, \ 2, d]\ , GetEndPoint[j, \ 1, length, d, slope] - GetStartPoint[i, \ 1, d]\ ];\)\), "\[IndentingNewLine]", \(\(AllConfigs[count_, length_, \ d_, \ slope_]\ := \ Table[\ PathCount[i, \ j, \ length, \ d, \ slope], \ {i, \ 1, \ count}, \ {j, \ 1, \ count}];\)\), "\[IndentingNewLine]", \(\(Actual[alpha_, \ d_, \ p_] := \ \(1\/p\) Log[Det[\ \((AllConfigs[p, \ d\ p, d, \ alpha\ ]\ )\)]];\)\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(N[Actual[1\/2, 2, 5]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 6]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 7]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 8]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 9]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 10]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 25]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 26]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 50]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 51]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 100]]\), "\[IndentingNewLine]", \(N[Actual[1\/2, 2, 101]]\)}], "Input"], Cell[BoxData[ \(5.280227195526091`\)], "Output"], Cell[BoxData[ \(6.463329405138627`\)], "Output"], Cell[BoxData[ \(7.648554369716921`\)], "Output"], Cell[BoxData[ \(8.835056980600077`\)], "Output"], Cell[BoxData[ \(10.02237925711519`\)], "Output"], Cell[BoxData[ \(11.210253892431894`\)], "Output"], Cell[BoxData[ \(29.047216714921724`\)], "Output"], Cell[BoxData[ \(30.236716603794292`\)], "Output"], Cell[BoxData[ \(58.78662867604101`\)], "Output"], Cell[BoxData[ \(59.97624016145591`\)], "Output"], Cell[BoxData[ \(118.26765603401527`\)], "Output"], Cell[BoxData[ \(119.45727947814876`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[N, \ \[Alpha]\ N\ + \ k]\ Cos[ k\ \[Lambda]]\), "\[IndentingNewLine]", \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[N, \ \[Alpha]\ N\ + 2\ k]\ Cos[ k\ \[Lambda]]\), "\[IndentingNewLine]", \(\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[N, \ \[Alpha]\ N\ + 3\ k]\ Cos[ k\ \[Lambda]]\), "\[IndentingNewLine]", \(Simplify[\[Sum]\+\(k = 1\)\%\[Infinity] Binomial[ N, \ \[Alpha]\ N\ + 4\ k]\ Cos[ k\ \[Lambda]]]\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[BoxData[ \(\(\[ImaginaryI]\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Pi]\ \((1 + N \ - N\ \[Alpha])\) - \[ImaginaryI]\ \[Lambda]\)\ \((\(-1\) + \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Pi]\ \((1 + N - N\ \[Alpha])\)\))\)\ \ Gamma[1 + N]\ Gamma[1 - N + N\ \[Alpha]]\ \((HypergeometricPFQ[{1, 1 - N + N\ \ \[Alpha]}, {2 + N\ \[Alpha]}, \(-\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \ \[Lambda]\)\)] + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Lambda]\)\ \ HypergeometricPFQ[{1, 1 - N + N\ \[Alpha]}, {2 + N\ \[Alpha]}, \(-\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Lambda]\)\)])\)\)\/\(4\ \[Pi]\ Gamma[2 + \ N\ \[Alpha]]\)\)], "Output"], Cell[BoxData[ \(\(-\(\(1\/\(4\ \[Pi]\ Gamma[ 3 + N\ \[Alpha]]\)\) \((\[ImaginaryI]\ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \[Pi]\ \((1 + N - N\ \[Alpha])\) - \[ImaginaryI]\ \ \[Lambda]\)\ \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Pi]\ \((1 + N \ - N\ \[Alpha])\)\))\)\ Gamma[1 + N]\ Gamma[ 2 - N + N\ \[Alpha]]\ \((HypergeometricPFQ[{1, 1 - N\/2 + \(N\ \[Alpha]\)\/2, 3\/2 - N\/2 + \(N\ \[Alpha]\)\/2}, {3\/2 + \(N\ \ \[Alpha]\)\/2, 2 + \(N\ \[Alpha]\)\/2}, \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \[Lambda]\)] + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Lambda]\)\ HypergeometricPFQ[{1, 1 - N\/2 + \(N\ \[Alpha]\)\/2, 3\/2 - N\/2 + \(N\ \[Alpha]\)\/2}, {3\/2 + \(N\ \ \[Alpha]\)\/2, 2 + \(N\ \[Alpha]\)\/2}, \[ExponentialE]\^\(\ \[ImaginaryI]\ \[Lambda]\)])\))\)\)\)\)], "Output"], Cell[BoxData[ \(\(1\/\(4\ \[Pi]\ Gamma[ 4 + N\ \[Alpha]]\)\) \((\[ImaginaryI]\ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \[Pi]\ \((1 + N - N\ \[Alpha])\) - \[ImaginaryI]\ \ \[Lambda]\)\ \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Pi]\ \((1 + N \ - N\ \[Alpha])\)\))\)\ Gamma[1 + N]\ Gamma[ 3 - N + N\ \[Alpha]]\ \((HypergeometricPFQ[{1, 1 - N\/3 + \(N\ \[Alpha]\)\/3, 4\/3 - N\/3 + \(N\ \[Alpha]\)\/3, 5\/3 - N\/3 + \(N\ \[Alpha]\)\/3}, {4\/3 + \(N\ \ \[Alpha]\)\/3, 5\/3 + \(N\ \[Alpha]\)\/3, 2 + \(N\ \[Alpha]\)\/3}, \(-\[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \[Lambda]\)\)] + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Lambda]\)\ HypergeometricPFQ[{1, 1 - N\/3 + \(N\ \[Alpha]\)\/3, 4\/3 - N\/3 + \(N\ \[Alpha]\)\/3, 5\/3 - N\/3 + \(N\ \[Alpha]\)\/3}, {4\/3 + \(N\ \ \[Alpha]\)\/3, 5\/3 + \(N\ \[Alpha]\)\/3, 2 + \(N\ \[Alpha]\)\/3}, \(-\[ExponentialE]\^\(\ \[ImaginaryI]\ \[Lambda]\)\)])\))\)\)], "Output"], Cell[BoxData[ \(\(-\(\(1\/\(4\ \[Pi]\ Gamma[ 5 + N\ \[Alpha]]\)\) \((\[ImaginaryI]\ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \((\[Pi]\ \((1 + N - N\ \[Alpha])\) + \[Lambda])\)\)\ \ \((\(-1\) + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \[Pi]\ \((1 + N - N\ \ \[Alpha])\)\))\)\ Gamma[1 + N]\ Gamma[ 4 + N\ \((\(-1\) + \[Alpha])\)]\ \((HypergeometricPFQ[{1, 1 - N\/4 + \(N\ \[Alpha]\)\/4, 5\/4 - N\/4 + \(N\ \[Alpha]\)\/4, 3\/2 - N\/4 + \(N\ \[Alpha]\)\/4, 7\/4 - N\/4 + \(N\ \[Alpha]\)\/4}, {5\/4 + \(N\ \ \[Alpha]\)\/4, 3\/2 + \(N\ \[Alpha]\)\/4, 7\/4 + \(N\ \[Alpha]\)\/4, 2 + \(N\ \[Alpha]\)\/4}, \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \[Lambda]\)] + \[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Lambda]\)\ HypergeometricPFQ[{1, 1 - N\/4 + \(N\ \[Alpha]\)\/4, 5\/4 - N\/4 + \(N\ \[Alpha]\)\/4, 3\/2 - N\/4 + \(N\ \[Alpha]\)\/4, 7\/4 - N\/4 + \(N\ \[Alpha]\)\/4}, {5\/4 + \(N\ \ \[Alpha]\)\/4, 3\/2 + \(N\ \[Alpha]\)\/4, 7\/4 + \(N\ \[Alpha]\)\/4, 2 + \(N\ \[Alpha]\)\/4}, \[ExponentialE]\^\(\ \[ImaginaryI]\ \[Lambda]\)])\))\)\)\)\)], "Output"] }, Open ]], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(Array1[i_, \ D_, N_, \ \[Alpha]_]\ = \ \(D\ + \ i\ - \ 1\)\/D - \ N\/D + \ \(N\ \[Alpha]\)\/D;\)\[IndentingNewLine] \(Array2[i_, \ D_, N_, \ \[Alpha]_]\ = \ \(D\ + \ i\)\/D + \ \(N\ \[Alpha]\)\/D;\)\ \[IndentingNewLine] \(FActual[N_, \ \[Lambda]_, \ D_, \ \[Alpha]_]\ := \ Binomial[N, \ \[Alpha]\ N]\ - 4\ \(\((\(-1\))\)\^\(D\ - \ 1\)\) \((Sin[\[Pi]\ \((1\ + \ N\ - \ N\ \[Alpha])\)]\ \(\(Gamma[1\ + \ N]\ Gamma[ D\ - \ N\ + \ N\ \[Alpha]]\)\/\(4\ \[Pi]\ Gamma[ D\ + \ 1\ + \ N\ \[Alpha]]\)\) \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Lambda]\))\) \((HypergeometricPFQ[ Union[{1}, \ Table[Array1[i, \ D, \ N, \ \[Alpha]], \ {i, \ 1, \ D}]], \ Union[Table[ Array2[i, \ D, \ N, \ \[Alpha]], \ {i, \ 1, \ D}]], \ \((\(-1\))\)\^D\ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \[Lambda]\)] + \(\[ExponentialE]\^\(2\ \[ImaginaryI]\ \ \[Lambda]\)\) HypergeometricPFQ[ Union[{1}, \ Table[Array1[i, \ D, \ N, \ \[Alpha]], \ {i, \ 1, \ D}]], \ Union[Table[ Array2[i, \ D, \ N, \ \[Alpha]], \ {i, \ 1, \ D}]], \ \((\(-1\))\)\^D\ \[ExponentialE]\^\(\ \[ImaginaryI]\ \[Lambda]\)])\);\)\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[11, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[13, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[15, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[17, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[19, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[49, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[51, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[99, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[101, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[199, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FActual[201, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\)\)\)], "Input"], Cell[BoxData[ \(5.768357416605009`\)], "Output"], Cell[BoxData[ \(6.927446635320304`\)], "Output"], Cell[BoxData[ \(8.088411813033122`\)], "Output"], Cell[BoxData[ \(9.250602891081268`\)], "Output"], Cell[BoxData[ \(10.413638330099877`\)], "Output"] }, Open ]], Cell["\<\ The following values converged slowly under numerical integration.\ \>", "Text"], Cell[BoxData[ \(\(\(27.89050198959444`\)\(\[InvisibleSpace]\)\) - 1.064419380602652`*^-15\ \[ImaginaryI]\)], "Output"], Cell[BoxData[ \(\(\(29.056327138824273`\)\(\[InvisibleSpace]\)\) + 8.211011286866771`*^-14\ \[ImaginaryI]\)], "Output"], Cell[BoxData[ \(\(\(57.01194249300693`\)\(\[InvisibleSpace]\)\) + 0.004723546420950976`\ \[ImaginaryI]\)], "Output"], Cell[BoxData[ \(\(\(58.220487860560496`\)\(\[InvisibleSpace]\)\) - 2.3390829087770058`*^-6\ \[ImaginaryI]\)], "Output"], Cell[BoxData[ \(\(\(120.1853889030997`\)\(\[InvisibleSpace]\)\) + 0.10026971926471118`\ \[ImaginaryI]\)], "Output"], Cell[BoxData[ \(\(\(121.46479824864426`\)\(\[InvisibleSpace]\)\) - 0.02288453683864322`\ \[ImaginaryI]\)], "Output"], Cell["\<\ So already the desired convergent behavior and a degree of agreement is \ apparent, but again this formula is designed to be extended only to an \ infinite matrix. Indeed, at the desired points it doesn't even converge, but \ this is a property of the generating function, not the actual values now. I have taken the first-order expansion near z = 0, and here are the results \ if this is used in place of the PFQ function: \ \>", "Text"], Cell[BoxData[ \(\(FApprox[N_, \ \[Lambda]_, \ D_, \ \[Alpha]_]\ := \ Binomial[N, \ \[Alpha]\ N]\ - 4\ \(\((\(-1\))\)\^\(D\ - \ 1\)\) \((Sin[\[Pi]\ \((1\ + \ N\ - \ N\ \[Alpha])\)]\ \(\(Gamma[1\ + \ N]\ Gamma[ D\ - \ N\ + \ N\ \[Alpha]]\)\/\(4\ \[Pi]\ Gamma[ D\ + \ 1\ + \ N\ \[Alpha]]\)\) \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Lambda]\))\) \((\((1\ + \ \(\[Product]\+\(j = 1\)\%D\((\(D\ + \ j\ - \ 1\)\/D - N\/D + \(N\ \ \[Alpha]\)\/D)\)\)\/\(\[Product]\+\(j = 1\)\%D\((\(\(D\)\(\ \)\(+\)\(\ \ \)\(j\)\(\ \)\)\/D + \(N\ \[Alpha]\)\/D)\)\)\ \(\((\(-1\))\)\^D\) \ \[ExponentialE]\^\(\[ImaginaryI]\ \[Lambda]\))\) + \(\[ExponentialE]\^\(2\ \ \[ImaginaryI]\ \[Lambda]\)\) \((1\ + \(\[Product]\+\(j = 1\)\%D\((\(D\ + \ \ j\ - \ 1\)\/D - N\/D + \(N\ \[Alpha]\)\/D)\)\)\/\(\[Product]\+\(j = \ 1\)\%D\((\(\(D\)\(\ \)\(+\)\(\ \)\(j\)\(\ \)\)\/D + \(N\ \[Alpha]\)\/D)\)\)\ \ \(\((\(-1\))\)\^D\) \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Lambda]\))\))\);\ \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FApprox[11, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FApprox[13, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FApprox[15, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FApprox[17, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FApprox[19, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FApprox[49, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FApprox[51, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FApprox[99, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine] NIntegrate[\(1\/\(2\ \[Pi]\)\) Log[FApprox[101, \ \[Lambda], \ 2, \ 1\/2]], \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\[IndentingNewLine]\[IndentingNewLine] \)\)\)], "Input"], Cell[BoxData[ \(5.9277771676224145`\)], "Output"], Cell[BoxData[ \(7.21838528624847`\)], "Output"], Cell[BoxData[ \(8.552958618566326`\)], "Output"], Cell[BoxData[ \(9.9272615127699`\)], "Output"], Cell[BoxData[ \(11.325884695642632`\)], "Output"], Cell[BoxData[ \(32.255720763431135`\)], "Output"], Cell[BoxData[ \(33.640285421367736`\)], "Output"], Cell[BoxData[ \(66.80500649353489`\)], "Output"], Cell[BoxData[ \(68.18620085723754`\)], "Output"] }, Open ]], Cell["\<\ This does not appear to be converging to the desired result as N grows unlike \ the actual result. Here is a picture of result versus actual--this first \ term approximation is not working well.\ \>", "Text"], 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]]\ ]}, \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\), "\[IndentingNewLine]", \(\(\[Alpha]\ = \ 1\/2;\)\), "\[IndentingNewLine]", \(\(Dst\ = 2;\)\), "\[IndentingNewLine]", \(\(Num\ = 49;\)\), "\[IndentingNewLine]", \(Plot[{Abs[FApprox[Num, \ \[Lambda], \ Dst, \ \[Alpha]]], Abs[FActual[Num, \ \[Lambda], \ Dst, \ \[Alpha]]\ ]}, \ {\[Lambda], \ 0, \ 2\ \[Pi]}]\), "\[IndentingNewLine]", \(\(\[Alpha]\ = \ 1\/2;\)\), "\[IndentingNewLine]", \(\(Dst\ = 2;\)\), "\[IndentingNewLine]", \(\(Num\ = 101;\)\), "\[IndentingNewLine]", \(Plot[{Abs[FApprox[Num, \ 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