(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 139241, 4349]*) (*NotebookOutlinePosition[ 163517, 5225]*) (* CellTagsIndexPosition[ 163473, 5221]*) (*WindowFrame->Normal*) Notebook[{ Cell["Complex 2: Functions", "Title", Evaluatable->False, AspectRatioFixed->True], Cell["R.G. Palmer", "Subsubtitle", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Background and Copyright", "Subsection"], Cell[TextData[{ "This ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook was written by Richard G. Palmer (Physics Department, Duke \ University) for use in a course he taught. As of 1999, it has been made \ available for general non-profit use under the following copyright \ provision." }], "Text"], Cell[TextData[{ StyleBox["This Mathematica Notebook is Copyright Richard G. Palmer, 1997", FontWeight->"Bold"], ". It may be freely used by individuals, and by classes at academic \ institutions, provided:\n1. Credit is given to Richard Palmer as the original \ author; and\n2. It is not bought or sold or exchanged for profit, or \ incorporated into material that is bought or sold or exchanged for profit.\n\ Any other use requires the written permission of Richard Palmer, Dept. of \ Physics, Box 90305, Duke University, Durham, NC 27708, USA. ", "See ", StyleBox["http://www.phy.duke.edu/~palmer", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for the email address." }], "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Version: 4.\nDate: 7/21/00." }], "Text"], Cell[TextData[{ "See ", StyleBox["http://www.phy.duke.edu/~palmer/notebooks/", FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], " for other ", StyleBox["Mathematica", FontSlant->"Italic"], " noteboks by Richard Palmer." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Preface"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ This notebook is mainly about Mathematica techniques for dealing \ with complex functions and for plotting various representations of complex \ numbers and functions. It also examines a number of elementary functions of \ a complex variable, trying to make them familiar through pictures. And it \ introduces several new Mathematica operators and functions.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["There are 3 problems embedded in the text.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Note that you'll need to have the Mathematica book with you for \ some of this.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Pictures of Complex Functions -- The Exponential"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Introduction"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "It's often valuable to make graphical pictures of functions of complex \ variables. Since a complex function f(z) is equivalent to two real functions \ of two real variables we can't readily plot the whole thing -- we'd need to \ make a four dimensional plot. But Mathematica can make 3D plots, so we can \ use those to look at the real part, the imaginary part, the modulus (absolute \ value) and the argument (phase). "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Besides 3D surface plots, we can also use contour plots and density plots to \ examine those same functions (the real part, etc)."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Finally we can make plots of mappings, in which we see how a grid in the z \ plane is mapped into the w plane, where w= f(z)."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "In this section we'll explore each of those plots mainly for one function, \ the exponential. Then, in subsequent sections, we'll use selected plots for \ several other functions."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "As we explore the various plots, we'll try several variations and options to \ make \"prettier\" plots. The aim is not just aesthetic, but is intended to \ make clear the properties of the function we're examining. It'll also be \ useful practice in using Mathematica's plotting functions. You'll need to \ pay close attention to these issues, because you'll need the expertise for \ some of the problems.", Evaluatable->False, AspectRatioFixed->True]], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Surface plots for Re[Exp[z]]"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Let's start with using ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Plot3D[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " to look at the real part of our function, the exponential. 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Here's ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Re[Exp[z]]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" in the unit square:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Plot3D[Re[Exp[x + I y]], {x,-1,1},{y,-1,1}]", "Input", AspectRatioFixed->True], Cell[TextData["That's not very exciting. Let's try a bigger range:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Plot3D[Re[Exp[x + I y]], {x,-5,5}, {y,-5,5}]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Woah! Time to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["think", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["! What's the function we're plotting? Let's check", Evaluatable->False, AspectRatioFixed->True], StyleBox[":", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["ComplexExpand[Exp[x + I y]]", "Input", AspectRatioFixed->True], Cell["%[[1]]\t\t(* real part *)", "Input", AspectRatioFixed->True], Cell[TextData[ "Of course; we didn't really need Mathematica to tell us that."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["So in the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" (real) direction it's blowing up exponentially, while in the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " (imaginary) direction it's oscillating. In the first plot we didn't have \ a wide enough y range to see a whole oscillation. In the second plot the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["exp", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "losion in x led to clipping as the function hit top and bottom. We can \ stop the clipping with a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotRange->All", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" option:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Plot3D[Re[Exp[x + I y]], {x,-5,5}, {y,-5,5}, PlotRange->All]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Better, but it's not a work of art. Notice the vertical range. So it \ looks as though we should choose a smaller range in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", and perhaps a bigger range in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[". Here goes:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Plot3D[Re[Exp[x + I y]], {x,-1,1}, {y,-10,10}]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Much better, but now it's a bit coarse. We could reduce the y range a \ bit, or increase the number of subdivisions with the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" option. The default ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" is 15 for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Plot3D", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", so let's try doubling it:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Plot3D[Re[Exp[x + I y]], {x,-1,1}, {y,-10,10}, \t\t\tPlotPoints->30]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Alright! Of course that used 4 times as many points as the previous one, \ and took appreciably longer. Usually it's best to keep ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " fairly small while adjusting the range and other options, and then \ increase it for a final \"production\" plot.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "There are lots of other things we could do to fancy up the above plot. We \ might change the viewing angle (", Evaluatable->False, AspectRatioFixed->True], StyleBox["ViewPoint", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ") with the help of the 3D ViewPoint Selector. We could change the \ relative scaling of x, y, and the vertical. We could add labels. We could \ change the way the axes are shown, or which edges have the axes. We could \ change the lighting and/or shading.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Here's a view looking down the x axis, with labels added for the axes:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Plot3D[ \tRe[Exp[x + I y]], \t{x,-1,1}, {y,-10,10}, \tPlotPoints->30, \tViewPoint->{-3, 0, 1}, \tAxesLabel->{\"x\",\"y\",\"exp(x+iy)\"} ]\ \>", "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[ "Surface plots for Im[Exp[z]], Abs[Exp[z]], and Arg[Exp[z]]"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Let's quickly check what the other standard functions give, using the same \ range and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" as above. Here's the imaginary part: ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Plot3D[Im[Exp[x + I y]], {x,-1,1}, {y,-10,10}, \t\tPlotPoints->30]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "That's basically the same as the real part apart from a 90 degree change \ of phase of the oscillations. Note that the result is 0 on the real axis, \ though that lies halfway between two mesh lines. It would have been better \ to have used an ", Evaluatable->False, AspectRatioFixed->True], StyleBox["odd", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" value for ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" so that the center value lay on a grid line.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "For the absolute value and the phase, you should be able to anticipate what \ you'll see before actually trying it, using"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(e\^\(x + iy\) = \ \(e\^x\) e\^iy\)], "DisplayFormula"], Cell[TextData["Think before evaluating!"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Plot3D[Abs[Exp[x + I y]], {x,-1,1}, {y,-10,10}, \t\t\tPlotPoints->30]\ \>", "Input", AspectRatioFixed->True], Cell["\<\ Plot3D[Arg[Exp[x + I y]], {x,-1,1}, {y,-10,10}, \t\t\tPlotPoints->30]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Surprised? Recall that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Arg", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" is defined to be between -", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Pi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Pi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Contour plots and density plots for Re[Exp[z]]"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Let's go back to the real part of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Exp[z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ", and try using some other types of plots. Here's a contour plot:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ ContourPlot[Re[Exp[x + I y]], {x,-1,1}, {y,-10,10}, \t\t\t\t\t\t\t\t\t\tPlotPoints->30]\ \>", "Input", AspectRatioFixed->True], Cell["\<\ Note that light is high, dark is low (I always think of snow on \ mountain tops). Color plots are pretty, but the default implementation uses \ red color for both high and low points:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ ContourPlot[Re[Exp[x + I y]], {x,-1,1}, {y,-10,10}, \t\t\t\t\t\tPlotPoints->30, ColorFunction->Hue]\ \>", "Input", AspectRatioFixed->True], Cell["Density plots are pretty similar (and very quick):", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ DensityPlot[Re[Exp[x + I y]], {x,-1,1}, {y,-10,10}, \t\t\t\t\t\t\t\t\t\tPlotPoints->30]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[ "Again white is high. I generally prefer them without the mesh though:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ DensityPlot[Re[Exp[x + I y]], {x,-1,1}, {y,-10,10}, \t\t\t\t\t\t\tPlotPoints->30, Mesh->False]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[ "Now you can see more clearly the graying out towards the left."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["And ...", "Text"], Cell["\<\ DensityPlot[Re[Exp[x + I y]], {x,-1,1}, {y,-10,10}, \t\t\t\tPlotPoints->30, Mesh->False, ColorFunction->Hue]\ \>", "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Options"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Note that I've actually been doing a lot of re-evaluating when I could \ have just used ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Show[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ". That was to focus on one issue at a time, but now let's play a bit with \ ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Show[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ". The most common use is to change the options used in a previous plot. \ For example, I can remove the frame and restore the mesh with:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Show[%, Frame->False, Mesh->True, ColorFunction->GrayLevel]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "What happens here is that Mathematica uses the same values at the grid \ points (i.e. the 15 x 15 values of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Re[Exp[z]]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ") as before; nothing is recalculated. But it redoes the conversion from \ its internal ", Evaluatable->False, AspectRatioFixed->True], StyleBox["graphics objects", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " to postscript, and renders the new postscript. Remember that the \ graphics objects are what Mathematica actually saves from each calculation, \ as indicated by the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["-DensityGraphics-", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12], StyleBox[" as the result of the above calculation.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Of course we can only use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Show[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " to change options that don't require re-evaluation of the function \ itself. We couldn't change ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" for example.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "If you want to know which options you can and cannot apply, either in ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Show[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" or in the original function like ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Plot3D[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" or ", Evaluatable->False, AspectRatioFixed->True], StyleBox["DensityPlot[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", you need to learn how to make efficient use of the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica Reference Guide", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" in the back of the Mathematica book. ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Look up some the following entries", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["For options like ", Evaluatable->False, AspectRatioFixed->True], StyleBox["PlotPoints", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " that can only be used in the original function, you must look up that \ function, like ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Plot[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Plot3D[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["DensityPlot[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ContourPlot[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[", etc. But for options that you can also change later with ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Show[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ", you must look up the particular type of graphics object concerned. 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The ", Evaluatable->False, AspectRatioFixed->True], StyleBox["CartesianMap[]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" function expects a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["pure function", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" as its first argument, so we can just use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Exp", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ " without any arguments. We'll discuss pure functions in more detail \ another time. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["CartesianMap[Exp, {-1,1}, {-1,1}]", "Input", AspectRatioFixed->True], Cell[TextData[ "Get it? 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Mostly we'll look at 3D plots \ of the real and imaginary parts."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Sin[z]"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Here are the real and imaginary parts of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Sin[z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Plot3D[Re[Sin[x + I y]], {x,-10,10}, {y,-3,3}, \t\tPlotPoints->30, AxesLabel->{\"x\",\"y\",\"\"}]\ \>", "Input", AspectRatioFixed->True], Cell["\<\ Plot3D[Im[Sin[x + I y]], {x,-10,10}, {y,-3,3}, \t\tPlotPoints->30, AxesLabel->{\"x\",\"y\",\"\"}]\ \>", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Note that on the real axis we just have ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Sin[x]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ ", with imaginary part 0. 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So the real and imaginary parts are the product of an oscillatory \ sinusoidal function in x and something that blows up as |y| increases.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Here's the absolute value (on a different scale). 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Note again that they always intersect at right angles. In fact \ we have constructed here another 2D orthogonal coordinate system. If you add \ a cartesian third dimension (perpendicular to the plane of the figure), you \ also have a new 3D orthogonal coordinate system. It's called the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["elliptic cylindrical", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" system, and is one of the 11 standard ones, like ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Cartesian", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["cylindrical", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["spherical polar", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ". Alternatively, we can add a third coordinate by rotating the above \ figure about either the horizontal or the vertical axis, in each case adding \ an angle ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Phi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " as the third coordinate; these both give 3D orthogonal coordinate \ systems, called the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["prolate spheroidal", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" system and the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["oblate spheroidal", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " system respectively. Finally, note that all these coordinate systems \ have two singular points, at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[PlusMinus]", Evaluatable->False, AspectRatioFixed->True], StyleBox["1 in the above diagram.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The above picture can also be interpreted as the equipotential and field \ lines of two equal point electric charges (at ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[PlusMinus]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ "1). Lines of constant x or y are automatically harmonic functions \ (solutions of Laplace's equation) in the w = f(z) plane, and vice-versa, for \ any analytic function.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Cos[z]"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["What about cos z? Well it's generally true that"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["sin z = cos(", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Pi]", Evaluatable->False, AspectRatioFixed->True], StyleBox["/2 - z) = cos(z - ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Pi]", Evaluatable->False, AspectRatioFixed->True], StyleBox["/2)", Evaluatable->False, AspectRatioFixed->True] }], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "(not just for real z), as you can verify by using the exponential \ representations for the trig functions, like"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[" iz -iz\ncos z = (e + e )/2"], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["So ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Sin[z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Cos[z]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" look exactly the same except for a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Pi]", Evaluatable->False, AspectRatioFixed->True, FontSize->12], StyleBox["/2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12], StyleBox[" shift along the real axis. Verify this if you wish. Their ", Evaluatable->False, AspectRatioFixed->True], StyleBox["CartesianMap", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ "'s will look exactly the same if you again take exactly one period.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Cosh[z], Sinh[z]"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The hyperbolic functions cosh z and sinh z are very similar except for a 90 \ degree rotation, because"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["cosh z = cos iz\nsinh z = -i sin iz"], "Print", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["For example:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Plot3D[Im[Sinh[x + I y]], {x,-3,3}, {y,-10,10}, \t\tPlotPoints->30, AxesLabel->{\"x\",\"y\",\"\"}]\ \>", "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Problem 1", "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Produce a pretty picture of three side-by-side surface plots (using ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Plot3D", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["GraphicsArray", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->12, FontWeight->"Bold"], StyleBox[ "), showing the real parts of cos z, exp iz and exp -iz. These should make \ it very plausible visually that cos z = (exp iz + exp -iz)/2 (at least for \ the real part). Choosing scales and options to produce a pleasing picture is \ the main challenge. You may want to turn off all the labels and axes to \ avoid clutter. Clipped plots are not acceptable.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Integer Powers"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Examples"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Here are the real and imaginary parts of z", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[":", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Plot3D[Re[(x + I y)^2],{x,-10,10},{y,-10,10}]", "Input", AspectRatioFixed->True], Cell["Plot3D[Im[(x + I y)^2],{x,-10,10},{y,-10,10}]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Of course you know what the absolute value |z", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["| looks like, since it's just |z|", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[". (If you don't know, plot it!).", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Now try z", Evaluatable->False, AspectRatioFixed->True], StyleBox["3", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" or z", Evaluatable->False, AspectRatioFixed->True], StyleBox["4", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[".", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "It's also instructive to make a contour plot of one of those (the real or \ imaginary part of z", Evaluatable->False, AspectRatioFixed->True], StyleBox["3", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" or z", Evaluatable->False, AspectRatioFixed->True], StyleBox["4", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["). Immediately after plotting one of those, just do:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Show[ContourGraphics[%]]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Can you now guess what you'd see for z", Evaluatable->False, AspectRatioFixed->True], StyleBox["n", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ " for any positive n? Hint: how is the number of maxima and minima as you \ walk (say) around a unit circle related to n?", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Problem 2", "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Identify the following picture. Give a Mathematica expression that \ reproduces it exactly. 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LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Subsubtitle", "Condensed"], CellMargins->{{8, 10}, {8, 8}}, FontSize->12], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {12, 8}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[FilledSquare]", CellMargins->{{25, Inherited}, {8, 24}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Section", "Presentation"], CellMargins->{{40, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Section", "Condensed"], CellMargins->{{18, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Section", "Printout"], CellMargins->{{13, 0}, {7, 22}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 20}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[StyleData["Subsection", "Presentation"], CellMargins->{{36, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->22], Cell[StyleData["Subsection", "Condensed"], CellMargins->{{16, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Subsection", "Printout"], CellMargins->{{9, 0}, {7, 22}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 18}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, CounterIncrements->"Subsubsection", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], Cell[StyleData["Subsubsection", "Presentation"], CellMargins->{{34, 10}, {11, 26}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Subsubsection", "Condensed"], CellMargins->{{17, Inherited}, {6, 12}}, FontSize->10], Cell[StyleData["Subsubsection", "Printout"], CellMargins->{{9, 0}, {7, 14}}, FontSize->11] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{12, 10}, {7, 7}}, LineSpacing->{1, 3}, CounterIncrements->"Text", FontSize->14], Cell[StyleData["Text", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["Text", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{12, 10}, {6, 6}}, LineSpacing->{1, 3}, CounterIncrements->"SmallText", FontFamily->"Helvetica", FontSize->9], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names. Some attributes for these styles are actually set in FormatType Styles \ (in the last section of this stylesheet). \ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{45, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold"], Cell[StyleData["Input", "Presentation"], CellMargins->{{72, Inherited}, {8, 10}}, LineSpacing->{1, 0}], Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {2, 3}}], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, FontSize->9] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", StyleMenuListing->None, FontWeight->"Bold"], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{47, 10}, {7, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Output"], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}], Cell[StyleData["Output", "Printout"], CellMargins->{{39, 0}, {6, 4}}, FontSize->9] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Print", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Condensed"], ImageMargins->{{38, Inherited}, {Inherited, 0}}, Magnification->0.6], Cell[StyleData["Graphics", "Printout"], ImageMargins->{{30, Inherited}, {Inherited, 0}}, FontSize->9, Magnification->0.8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "Condensed"], FontSize->9], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, ScriptLevel->1, SingleLetterItalics->True], Cell[StyleData["InlineFormula", "Presentation"], CellMargins->{{24, 10}, {10, 10}}, LineSpacing->{1, 5}], Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}], Cell[StyleData["DisplayFormula", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->PrimaryPlaceholder, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1754, 51, 85, 2, 110, "Title", Evaluatable->False], Cell[1842, 55, 107, 3, 53, "Subsubtitle", Evaluatable->False], Cell[CellGroupData[{ Cell[1974, 62, 46, 0, 45, "Subsection"], Cell[2023, 64, 320, 8, 71, "Text"], Cell[2346, 74, 725, 15, 166, "Text"], Cell[3074, 91, 112, 4, 52, "Text"], Cell[3189, 97, 266, 10, 52, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[3492, 112, 87, 2, 29, "Subsection", Evaluatable->False], Cell[3582, 116, 431, 8, 90, "Text", Evaluatable->False], Cell[4016, 126, 106, 2, 33, "Text", Evaluatable->False], Cell[4125, 130, 151, 5, 33, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[4313, 140, 125, 2, 35, "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[4463, 146, 92, 2, 45, 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Cell[11269, 381, 972, 35, 71, "Text", Evaluatable->False], Cell[12244, 418, 119, 4, 42, "Input"], Cell[12366, 424, 609, 19, 71, "Text", Evaluatable->False], Cell[12978, 445, 730, 21, 90, "Text", Evaluatable->False], Cell[13711, 468, 148, 4, 33, "Text", Evaluatable->False], Cell[13862, 474, 188, 9, 117, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[14087, 488, 139, 3, 29, "Subsection", Evaluatable->False], Cell[14229, 493, 486, 17, 52, "Text", Evaluatable->False], Cell[14718, 512, 117, 4, 42, "Input"], Cell[14838, 518, 826, 26, 71, "Text", Evaluatable->False], Cell[15667, 546, 196, 4, 52, "Text", Evaluatable->False], Cell[15866, 552, 74, 1, 25, "DisplayFormula"], Cell[15943, 555, 98, 2, 33, "Text", Evaluatable->False], Cell[16044, 559, 120, 4, 42, "Input"], Cell[16167, 565, 120, 4, 42, "Input"], Cell[16290, 571, 689, 27, 33, "Text", Evaluatable->False] }, Closed]], Cell[CellGroupData[{ Cell[17016, 603, 126, 2, 29, "Subsection", Evaluatable->False], Cell[17145, 607, 456, 16, 52, 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