(************** 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[ 37254, 786]*) (*NotebookOutlinePosition[ 37899, 808]*) (* CellTagsIndexPosition[ 37855, 804]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\( (*\ We\ specify\ the\ relation\ between\ the\ notation\ of\ the\ article\ \ and\ the\ notation\ used\ in\ this\ file, \ using\ latex\ notation\ *) \)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\( (*\ This\ routine\ verifies\ that\ G_n\ achieves\ its\ absolute\ minimum\ \ at\ {t_n, \(-t_n\)}, \ when\ n = \(7\ and\ \\alpha_ 7 = \(-3.215\)\)*) \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\ \)\( (*\ LegendreP[a, c, t] = P_ {a}^{c} \((t)\)\ *) \)\(\[IndentingNewLine]\)\( (*\ LegendreQ[a, c, t] = Q_ {a}^{c} \((t)\)\ *) \)\(\[IndentingNewLine]\)\( (*\ b\ denotes\ the\ parameter\ \\alpha\ *) \)\(\[IndentingNewLine]\)\(P[ b_, n_, t_] := LegendreP[b + \((n - 3)\)/2, \(-\((n - 3)\)\)/2, t]\[IndentingNewLine] p[b_, n_, t_] := LegendreP[b + \((n - 3)\)/2, \ \(-\((n - 1)\)\)/2, t]\[IndentingNewLine] Q[n_, t_] := LegendreQ[\((n - 1)\)/2, \((n - 3)\)/2, t]\[IndentingNewLine] q[n_, t_] := LegendreQ[\((n - 1)\)/2, \((n - 1)\)/2, t]\[IndentingNewLine]\[IndentingNewLine] (*\ En[b, n, t]\ denotes\ E_n \((t)\)\ *) \[IndentingNewLine] En[b_, n_, t_] := \((P[b, n, t] + P[b, n, \(-t\)])\)\[IndentingNewLine]\[IndentingNewLine] (*\ Z[b, n, t]\ denotes\ Z_n \((t)\)\ *) \[IndentingNewLine] Z[b_, n_, t_] := \((p[b, n, t] - p[b, n, \(-t\)])\)\[IndentingNewLine]\[IndentingNewLine] (*\ G[b, n, t]\ denotes\ G_n \((t)\)\ *) \[IndentingNewLine] G[b_, n_, t_] := \((1 - t^2)\)^\((\(-\((n - 3)\)\)/ 2)\)\ {\((q[n, t] + b \((b + n - 2)\) Q[n, t]\ Z[b, n, t]/En[b, n, t])\)^2\ + \((1 - b)\)^2\ \((Q[n, t])\)^2}\[IndentingNewLine]\[IndentingNewLine] (*\ The\ following\ command\ plots\ the\ graph\ of\ G_n \((t)\)\ on\ the\ \ interval\ \((\(-t_n\), \ t_n)\), for\ n = \(7\ and\ alpha_ 7 = \(-3.215\)\)\ \ *) \[IndentingNewLine]\[IndentingNewLine] Plot[G[\(-3.215\), 7, t], {t, \(-0.5173305417372707\), 0.5173305417372707}]\[IndentingNewLine]\[IndentingNewLine] (*\ The\ following\ command\ evaluates\ G_ 7 \((0)\) - G_ 7 \((t_ 7)\)\ for\ \\alpha_ 7 = \(-3.215\)\ \ *) \[IndentingNewLine]\[IndentingNewLine] G[\(-3.215\), 7, 0] - G[\(-3.215\), 7, 0.5173305417372707]\)\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier 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