(************** Content-type: application/mathematica ************** 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[ 160065, 5436]*) (*NotebookOutlinePosition[ 160987, 5467]*) (* CellTagsIndexPosition[ 160943, 5463]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ RowBox[{ StyleBox["3.5", FontSize->18], StyleBox[" ", FontSize->18], RowBox[{ StyleBox["Mathematica", FontSize->18, FontSlant->"Italic"], StyleBox[":", FontSize->18], StyleBox["\:30e9\:30d7\:30e9\:30b9\:5909\:63db", FontSize->18]}]}]], "Input"], Cell[BoxData[""], "Input"], Cell[BoxData[ \(\:57fa\:672c\:7684\:306a\:95a2\:6570\:306e\:30e9\:30d7\:30e9\:30b9\:5909\ \:63db\)], "Input", CellDingbat->"\[FilledSquare]", 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\[ExponentialE]\^\(-t\)\ \((2 + t - \[ExponentialE]\^\(\(3\ t\)/2\)\ \((2 + t)\)\ Cos[\(\@3\ t\)\/2] - \@3\ \[ExponentialE]\^\(\(3\ t\)/2\ \)\ \((\(-2\) + t)\)\ Sin[\(\@3\ t\)\/2])\)\)], "Output"] }, Open ]], Cell[BoxData[{ \(\(\(s\^n + 1 = 0\ \:306e\:89e3\:3092\ \[Alpha]\_1, \[Alpha]\_2, \[Alpha]\_3, \ \[CenterEllipsis], \[Alpha]\_n\ \ \:3068\:304a\:304f\:3068\:ff0c 1\/\(s\^n + 1\) = \(-\(1\/n\)\) \(\[Sum]\+\(k = 1\)\%n \[Alpha]\_k\ \/\(s - \[Alpha]\_k\)\)\[IndentingNewLine] \:304c\:6210\:308a\:7acb\:3064\:3053\:3068\:3092\:8abf\:3079\:307e\:3059\ \:ff0e\:3053\:306e\:7d50\:679c\:3092\:5229\:7528\:3059\:308b\:3068\:9006\:5909\ \:63db\:3092\:77ed\:6642\:9593\:306b\:6c42\:3081\:3089\:308c\:307e\:3059\:ff0e\ \[IndentingNewLine] \:8a3c\:660e\)\(\:ff09\)\)\), "\[IndentingNewLine]", \(\:3000s\^n + 1 = 0 \:3000\:306b\:3064\:3044\:3066\:ff0c\:89e3\:3068\:4fc2\:6570\:306e\ \:95a2\:4fc2\:304c\:ff0c\), "\[IndentingNewLine]", \(\[Alpha]\_1 + \[Alpha]\_2 + \[Alpha]\_3 + \[CenterEllipsis] + \ \[Alpha]\_n = 0, \ \ \ \ \ \ \ \[Sum]\+\(i \[NotEqual] j\)\%n\( \[Alpha]\_i\) \ \[Alpha]\_j = 0, \[Sum]\+\(i \[NotEqual] j, j \[NotEqual] k, k \[NotEqual] i\)\%n\( \ \[Alpha]\_i\) \(\[Alpha]\_j\) \[Alpha]\_k = 0\), "\[IndentingNewLine]", \(\ \ \ \ \(\(,\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\[CenterEllipsis]\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(,\)\(\ \)\(\(\[Alpha]\_1\) \(\[Alpha]\_2\) \(\ \[Alpha]\_3\) \[CenterEllipsis]\[Alpha]\_n = \((\(-1\))\)\^n\)\)\), "\ \[IndentingNewLine]", \(\:3067\:3042\:308b\:3053\:3068\:3092\:7528\:3044\:308b\:3068\:ff0c\), "\ \[IndentingNewLine]", \(\(\[Alpha]\_1\) \((s - \[Alpha]\_2)\) \((s - \[Alpha]\_3)\) \ \[CenterEllipsis] \((s - \[Alpha]\_n)\) + \(\[Alpha]\_2\) \((s - \[Alpha]\_1)\ \) \((s - \[Alpha]\_3)\) \[CenterEllipsis] \((s - \[Alpha]\_n)\) + \ \[CenterEllipsis] + \(\[Alpha]\_k\) \((s - \[Alpha]\_1)\) \((s - \[Alpha]\_2)\ \) \[CenterEllipsis] \((s - \[Alpha]\_\(k - 1\))\) \((s - \[Alpha]\_\(k + \ 1\))\) \[CenterEllipsis] \((s - \[Alpha]\_n)\) + \ \[IndentingNewLine]\(\[Alpha]\_n\) \((s - \[Alpha]\_1)\) \((s - \ \[Alpha]\_2)\) \[CenterEllipsis] \((s - \[Alpha]\_\(n - 1\))\)\), "\n", \(\(\(=\)\(\((\[Alpha]\_1 + \[Alpha]\_2 + \[Alpha]\_3 + \[CenterEllipsis] \ + \[Alpha]\_n)\) s\^\(n - 1\) + \ \[Sum]\+\(i \[NotEqual] j\)\%n\( \[Alpha]\_i\) \ \(\[Alpha]\_j\) s\^\(n - 2\) + \[CenterEllipsis] + n \(\((\(-1\))\)\^\(n - 1\)\) \(\[Alpha]\_1\) \(\[Alpha]\_2\) \(\[Alpha]\_3\) \ \[CenterEllipsis]\[Alpha]\_n = \(-n\:3000\:ff0e\)\)\)\), \ "\[IndentingNewLine]", \(\:306a\:304a\:ff0c \((s - \[Alpha]\_2)\) \((s - \[Alpha]\_3)\) \ \[CenterEllipsis] \((s - \[Alpha]\_n)\) + \((s - \[Alpha]\_1)\) \((s - \ \[Alpha]\_3)\) \[CenterEllipsis] \((s - \[Alpha]\_n)\) + \[CenterEllipsis] + \ \((s - \[Alpha]\_1)\) \((s - \[Alpha]\_2)\) \[CenterEllipsis] \((s - \[Alpha]\ \_\(k - 1\))\) \((s - \[Alpha]\_\(k + 1\))\) \[CenterEllipsis] \((s - \ \[Alpha]\_n)\) + \[IndentingNewLine]\((s - \[Alpha]\_1)\) \((s - \[Alpha]\_2)\ \) \[CenterEllipsis] \((s - \[Alpha]\_\(n - 1\))\)\), "\[IndentingNewLine]", \(\(\(=\)\(ns\^\(n - 1\) + \[CenterEllipsis] + \[Sum]\+k\%\(n - 2\)\(\ \[Alpha]\_1\) \(\[Alpha]\_2\) \(\[CenterEllipsis]\[Alpha]\_k\) \ \(\[Alpha]\_\(k + 2\)\) \[CenterEllipsis]\[Alpha]\_n = \(ns\^\(n - 1\)\) \:3000\:3068\:306a\:308a\:307e\:3059\:ff0e\)\)\)}], \ "Input", CellDingbat->"\[FilledSquare]", Background->RGBColor[0.613291, 0.964858, 0.851575]], Cell["\<\ \:7b49\:5f0f\:306e\:78ba\:8a8d\ \>", "Input"], Cell[BoxData[ \(nb[n_Integer] := \((aa = Solve[x\^n + 1 \[Equal] 0, x]; \[Alpha] = Table[aa[\([k, 1, 2]\)], {k, 1, n}]; \[IndentingNewLine]\(\(-\(1\/n\)\) \(\[Sum]\+\(k = 1\)\%n \ \[Alpha][\([k]\)]\/\(s - \[Alpha][\([k]\)]\)\) // ExpToTrig\) // Simplify)\)\)], "Input"], Cell[CellGroupData[{ Cell["Table[nb[k],{k,1,5}]", "Input"], Cell[BoxData[ \({1\/\(1 + s\), 1\/\(1 + s\^2\), 1\/\(1 + s\^3\), 1\/\(1 + s\^4\), 1\/\(1 + s\^5\)}\)], "Output"] }, Open ]], Cell["", "Input"], Cell["(*\:3000\:4f8b\:ff14\:3000*)", "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(n = 3; aa = Solve[x\^n + 1 \[Equal] 0, x]; \[Alpha] = Table[aa[\([k, 1, 2]\)], {k, 1, n}];\), "\[IndentingNewLine]", \(bb = \(-\(1\/n\)\) \(\[Sum]\+\(k = 1\)\%n \[Alpha][\([k]\)]\/\(s - \ \[Alpha][\([k]\)]\)\)\)}], "Input"], Cell[BoxData[ \(1\/3\ \((1\/\(1 + s\) - \((\(-1\))\)\^\(1/3\)\/\(\(-\((\(-1\))\)\^\(1/3\ \)\) + s\) + \((\(-1\))\)\^\(2/3\)\/\(\((\(-1\))\)\^\(2/3\) + s\))\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell["InverseLaplaceTransform[bb,s,x]//Simplify", "Input"], Cell[BoxData[ \(1\/3\ \((\[ExponentialE]\^\(-x\) - \[ExponentialE]\^\(x/2\)\ Cos[\(\@3\ \ x\)\/2] + \@3\ \[ExponentialE]\^\(x/2\)\ Sin[\(\@3\ x\)\/2])\)\)], "Output"] }, Open ]], Cell["", "Input"], Cell[BoxData[{ \(s\^n + 1 = 0\ \:306e\:89e3\:3092\ \[Alpha]\_1, \[Alpha]\_2, \[Alpha]\_3, \ \[CenterEllipsis], \[Alpha]\_n\ \ \:3068\:304a\:304f\:3068\:ff0c\), "\n", \(1\/\((s\^n + 1)\)\^2 = \[Sum]\+\(k = 1\)\%n{A\_k\/\(s - \[Alpha]\_k\) + B\_k\/\((s - \[Alpha]\_k)\)\^2}, \ \ A\_k = \(\(1 - n\)\/n\^2\) \[Alpha]\_k, \ B\_k = \(1\/n\^2\) \[Alpha]\_k\^2\), "\n", \(\:304c\:6210\:308a\:7acb\:3061\:307e\:3059\:ff0e\:3053\:306e\:7d50\:679c\ \:3092\:5229\:7528\:3059\:308b\:3068\:9006\:5909\:63db\:3092\:77ed\:6642\:9593\ \:306b\:6c42\:3081\:3089\:308c\:307e\:3059\:ff0e\)}], "Input", CellDingbat->"\[FilledSquare]", Background->RGBColor[0.613291, 0.964858, 0.851575]], Cell["(*\:3000\:4f8b\:ff15\:3000*)", "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(n = 3; aa = Solve[x\^n + 1 \[Equal] 0, x]; \[Alpha] = Table[aa[\([k, 1, 2]\)], {k, 1, n}];\), "\[IndentingNewLine]", \(ww = \[Sum]\+\(k = 1\)\%n\((\(\((1 - n)\)\ \ \[Alpha][\([k]\)]\)\/\(\(n\^2\) \((s - \[Alpha][\([k]\)])\)\) + \ \[Alpha][\([k]\)]\^2\/\(\(n\^2\) \((s - \[Alpha][\([k]\)])\)\^2\))\)\)}], \ "Input"], Cell[BoxData[ \(1\/\(9\ \((1 + s)\)\^2\) + 2\/\(9\ \((1 + s)\)\) + \((\(-1\))\)\^\(2/3\)\/\(9\ \ \((\(-\((\(-1\))\)\^\(1/3\)\) + s)\)\^2\) - \(2\ \((\(-1\))\)\^\(1/3\)\)\/\(9\ \ \((\(-\((\(-1\))\)\^\(1/3\)\) + s)\)\) - \((\(-1\))\)\^\(1/3\)\/\(9\ \((\((\ \(-1\))\)\^\(2/3\) + s)\)\^2\) + \(2\ \((\(-1\))\)\^\(2/3\)\)\/\(9\ \ \((\((\(-1\))\)\^\(2/3\) + s)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["InverseLaplaceTransform[ww,s,x]//Simplify", "Input"], Cell[BoxData[ \(1\/9\ \[ExponentialE]\^\(-x\)\ \((2 + x - \[ExponentialE]\^\(\(3\ x\)/2\)\ \((2 + x)\)\ Cos[\(\@3\ x\)\/2] - \@3\ \[ExponentialE]\^\(\(3\ x\)/2\ \)\ \((\(-2\) + x)\)\ Sin[\(\@3\ x\)\/2])\)\)], "Output"] }, Open ]], Cell["", "Input"], Cell[BoxData[ \(\:5fae\:5206\:65b9\:7a0b\:5f0f\:3078\:306e\:5fdc\:7528\:3000\:3000\)], \ "Input", CellDingbat->"\[FilledSquare]", Background->RGBColor[0.613291, 0.964858, 0.851575]], Cell["(*\:3000\:4f8b\:ff11\:3000*)", "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(n =. ; ba = LaplaceTransform[\[Integral]\_0\%x\( y[ t]\/\@\(x - t\)\) \[DifferentialD]t - x\^n, x, s] /. {LaplaceTransform[y[x], x, s] \[Rule] u, y[0] \[Rule] 0}\)], "Input"], Cell[BoxData[ \(\(\@\[Pi]\ u\)\/\@s - s\^\(\(-1\) - n\)\ 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