(*********************************************************************** 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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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[ 493232, 13670]*) (*NotebookOutlinePosition[ 518607, 14561]*) (* CellTagsIndexPosition[ 518531, 14555]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Appendix C*", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["\n\n", FontSize->16, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["A BRIEF (AND NECESSARILY INCOMPLETE) EXCURSION INTO FOURIER \ TRANSFORMS", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]] }], "Text", PageWidth->PaperWidth, TextAlignment->Center], Cell[TextData[{ StyleBox["*The ", FontSize->9, FontColor->GrayLevel[0]], StyleBox["Mathematica", FontSize->9, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" Notebooks in this dissertation benefited from the substantial \ assistance of Marlyn L. Hicks", FontSize->9, FontColor->GrayLevel[0]] }], "Text", PageWidth->PaperWidth], Cell[TextData[{ "This is the first of four ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks included in the Appendices of this dissertation. The first \ two, Appendices C and D, deal with Fourier and Inverse Fourier Transforms, \ the third, contained in Appendix E, describes a method to model optimal \ holding period, and the fourth, Appendix G, contains a random number \ generator that simulates \"efficient\" frontiers using stable random \ variates.\n\nAll of the notebooks provided were produced with ", StyleBox["Mathematica", FontSlant->"Italic"], " version 4.0 and should be executable in ", StyleBox["Mathematica", FontSlant->"Italic"], " version 3.0 with few or no modifications. They all may be viewed with ", StyleBox["MathReader", FontSlant->"Italic"], ", a free software produced by Wolfram Research and downloadable from \ http://www.Wolfram.com/", StyleBox["MathReader.", FontSlant->"Italic"], " The electronic version of this dissertation on CD-Rom contains ", StyleBox["MathReader", FontSlant->"Italic"], ". Opening the notebook in ", StyleBox["MathReader", FontSlant->"Italic"], " is recommended.\n\nViewing or printing in color considerably enhances the \ graphics created in these notebooks.\n\nThis is the first of two Appendices \ intended to supplement the main text in the area of Fourier Transforms. \ Here, Appendix C expands on the general nature of Inverse Fourier Transforms \ (IFT) briefly discussed in Chapter 2. Appendix D is intended to supplement \ Chapter 5, covering the more complex application of the IFT to stable \ distributions and showing how many of the illustrations in this dissertation \ are produced.\n\nTo provide context, I begin by replicating the \ illustration, also shown in Chapter 2, from Farlow(1993), depicting the \ reason Fourier Transforms are so helpful" }], "Text", PageWidth->PaperWidth], Cell[TextData[{ StyleBox[Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@00020020 4P0000`000010000DP0004`100010000c_ooo`0000000000000009010000000014004U@0J@1]06D0 L`0P04h0I@1g0200DP1_06d0H@1^0000000000000000000000000000000000000000000000000901 00020PH31@@50P<4QgX080000808000000000?l100000000E01Y06d0I@1c0200CP1U07L0801B06l0 K@1Q06h0002ZTb]0000000000000P03`nGcjolcS4P000000003HV0M000000000083n?`4000010000 g=hB00000034XPP`a=hB0040003LgQ80000000P4Q`0L05P2/>DB06PhE`;Oai@`oooooooooooooal0 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Farlow, \ ", FontSize->9, FontColor->GrayLevel[0]], StyleBox["Partial Differential Equations for Scientists and Engineers", FontSize->9, FontColor->GrayLevel[0], FontVariations->{"Underline"->True}], StyleBox[", 1993, Dover Publications, New York, p. 73.", FontSize->9, FontColor->GrayLevel[0]], StyleBox["\n\n", FontColor->GrayLevel[0]], StyleBox[ "In our case, we wish to illustrate how two parameters of the stable \ distribution, its shape parameter (\[Alpha]) and its skewness parameter (\ \[Beta]) relate to one another. Doing so requires graphing the utility of \ different risk choices as modeled by different probability distributions. The \ \"obstacle\" overcome by the Fourier Transform is the fact that the stable \ distribution does not have a probability distribution function (pdf). \ Chapters 10-12 of Farlow constitute a good resource for Fourier review. The \ ch.f. and the pdf of the stable distribution constitute a Fourier transform \ pair as described in Farlow.", FontFamily->"Times New Roman", FontSize->12, FontColor->GrayLevel[0]] }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(CleanSlate[]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(CleanSlate[]\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[BoxData[{ \(<< Calculus`\), "\[IndentingNewLine]", \(<< Calculus`FourierTransform`\), "\n", \(<< Statistics`\), "\n", \(<< Graphics`\)}], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(\(Off[General::spell1];\)\n \(Off[General::spell];\)\), FontFamily->"Courier New"]], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(Null\^2\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "Below is an example of a Fourier Series, an expansion series in the same \ general family as a Taylor series. 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