(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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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This is tricky though because as we will learn later, the \ function has to be periodic. We can make a hat-like function out of sines or \ cosines depending on whether we want the periodic extension of the function \ to look like a sawtooth or a blocky sine wave. We will go for the blocky \ sine wave. In the interval of interest [0,1] it will look like a hat.\ \>", "Text", FontSize->18], Cell[BoxData[ \(<< Calculus`FourierTransform`\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(h[x_]\ = \ If[x\ < \ .5, \ x, \ \((1 - x)\)]\), \(g[x_]\ = \ If[x\ > \ \(-.5\), \ x, \ \(-\((x + 1)\)\)]\), \(f[x_]\ = \ If[x\ <= \ 0, \ g[x], h[x]]\)}], "Input"], Cell[BoxData[ RowBox[{"If", "[", RowBox[{ RowBox[{"x", "<", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1]}], ",", "x", ",", \(1 - x\)}], "]"}]], "Output"], Cell[BoxData[ RowBox[{"If", "[", RowBox[{ RowBox[{"x", ">", RowBox[{"-", StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1]}]}], ",", "x", ",", \(-\((x + 1)\)\)}], "]"}]], "Output"], Cell[BoxData[ \(If[x \[LessEqual] 0, g[x], h[x]]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[f[x], {x, \(-1\), 1}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform 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You can safely \ ignore the error messages about accuracty. The result is accurate enough for \ our purposes.\ \>", "Text", FontSize->18], Cell[CellGroupData[{ Cell[BoxData[ \(\(fs[x_]\ = NFourierTrigSeries[f[x], {x, \(-1\), 1}, 10]\n\)\)], "Input"], Cell[BoxData[ \(\(0.`\[InvisibleSpace]\) + 0.`\ Cos[\[Pi]\ x] + 0.`\ Cos[2\ \[Pi]\ x] + 0.`\ Cos[3\ \[Pi]\ x] + 0.`\ Cos[4\ \[Pi]\ x] + 0.`\ Cos[5\ \[Pi]\ x] + 0.`\ Cos[6\ \[Pi]\ x] + 0.`\ Cos[7\ \[Pi]\ x] + 0.`\ Cos[8\ \[Pi]\ x] + 0.`\ Cos[9\ \[Pi]\ x] + 0.`\ Cos[10\ \[Pi]\ x] + 0.405284734569351101`\ Sin[\[Pi]\ x] + 2.82875123065046132`*^-17\ Sin[2\ \[Pi]\ x] - 0.0450316371743723919`\ Sin[3\ \[Pi]\ x] + 4.78281955863590102`*^-17\ Sin[4\ \[Pi]\ x] + 0.0162113893827740351`\ Sin[5\ \[Pi]\ x] - 6.78784265630653127`*^-17\ Sin[6\ \[Pi]\ x] - 0.00827111703202757908`\ Sin[7\ \[Pi]\ x] + 1.07278632826787978`*^-16\ Sin[8\ \[Pi]\ x] + 0.00500351524155240845`\ Sin[9\ \[Pi]\ x] - 1.27202649197381156`*^-16\ Sin[10\ \[Pi]\ x]\)], "Output"] }, Open ]], Cell["\<\ The Mathematica chop function is designed to get 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