(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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[ 14435046, 739671]*) (*NotebookOutlinePosition[ 14437668, 739744]*) (* CellTagsIndexPosition[ 14437260, 739731]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Visualizing Complex Functions with the ", StyleBox["Cardano3", "TI"], " Application" }], "Title"], Cell["\<\ Murray Eisenberg David J. M. Park, Jr.\ \>", "Author"], Cell[TextData[{ "Visualization is an invaluable companion to symbolic computation in \ understanding the complex plane and complex-valued functions of a complex \ variable. The ", StyleBox["Cardano3", "SB"], " application, an add-on to ", StyleBox["Mathematica,", FontSlant->"Italic"], " provides a rich set of tools for assisting such visualization. This paper \ demonstrates some capabilities of the application, especially those relevant \ to an introduction to complex analysis, and it indicates some teaching and \ learning issues that arise from using the application. Included are examples \ of how complex functions map objects in the complex plane and on the Riemann \ sphere, and of how complex functions behave near singularities and at branch \ points." }], "Abstract", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[" Introduction", "SectionFirst"], Cell[TextData[{ "Visualization and symbolic computation are both essential to understanding \ how functions behave. Visualizing the behavior of a real-valued function of a \ real variable is often easy, because the function's graph may be plotted in \ the plane\[LongDash]a space with just two real dimensions. Visualizing the \ behavior of a complex-valued function of a complex variable, by contrast, is \ more difficult, because the graph lives in a space with four real dimensions. \ Whereas ", StyleBox["Mathematica", FontSlant->"Italic"], " is replete with resources for symbolic computation with complex \ functions, out-of-the-box it provides only a meager set of tools for \ visualizing such functions. To do much more than what is provided by the \ standard add-on package ", StyleBox["Graphics`ComplexMap`", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[LongDash]even to plot the image of a simple curve in the complex plane\ \[LongDash]requires the effort of constructing the image set and its \ graphical representation and then of combining graphics objects. The ", StyleBox["Cardano3", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " application [", ButtonBox[ CounterBox["Reference", "ParkCardano3package"], ButtonData:>"ParkCardano3package", ButtonStyle->"Hyperlink"], "], written by the second author, is an add-on to ", StyleBox["Mathematica", FontSlant->"Italic"], " that provides a rich set of tools for complex function plotting, along \ with some utilities for complex symbolic computation. ", StyleBox["Cardano3", "SR"], " includes full documentation, with individual HelpBrowser pages and \ examples for each command." }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ "In a ", StyleBox["Mathematica", FontSlant->"Italic"], "-enriched introductory complex analysis course, the first author prepared \ demonstration and exercise notebooks for", StyleBox[" ", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["Cardano3", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[" (see [", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], ButtonBox[ CounterBox["Reference", "EisenbergMath421website"], ButtonData:>"EisenbergMath421website", ButtonStyle->"Hyperlink"], StyleBox["]) ", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], "and experimented with students' using it; experience there led to some \ enhancement of the application. The course used the textbook by Mathews and \ Howell [", ButtonBox[ CounterBox["Reference", "Mathews&Howell"], ButtonData:>"Mathews&Howell", ButtonStyle->"Hyperlink"], "] which is fairly traditional in approach, although it does go a bit \ further than some introductions in emphasizing mapping properties of complex \ functions. ", StyleBox["Cardano3", "SR"], " would be especially suitable for use with a less traditional, more \ visually oriented text, such as the one by Needham [", ButtonBox[ CounterBox["Reference", "Needham"], ButtonData:>"Needham", ButtonStyle->"Hyperlink"], "], which inspired and helped guide development of the application. " }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ "In this paper we demonstrate some capabilities of the ", StyleBox["Cardano3", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " packages ", StyleBox["ComplexRoutines", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["ComplexGraphics", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", especially those relevant to an introduction to complex analysis. The \ examples include a geometric problem solved by complex means, representations \ of complex functions as mappings, depiction of singularities, and analysis of \ branch points. Typically the examples include embellishments that might be \ inappropriate for somebody to program who is a novice at both ", StyleBox["Mathematica", FontSlant->"Italic"], " and complex analysis, but these embellishments illustrate some of the \ advanced functionality of the packages. " }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ "Various graphical representations of complex functions are often needed to \ emphasize their various features. ", StyleBox["MappingGraphic", "MR"], " plots and ", StyleBox["TwoPanelPlot", "MR"], "s can be used to emphasize mapping properties. Riemann sphere plots can be \ used to better depict behavior near the point at infinity, or to emphasize \ periodic behavior. ", StyleBox["CodedDensity", "MR"], " (domain coloring) plots give an overall representation of a function in \ the plane that emphasizes singular points and branch lines. ", StyleBox["ComplexVector", "MR"], " plots emphasize how a function varies on a grid or along a path in the \ complex plane. Animation is especially effective in making various properties \ come alive. ", StyleBox["Cardano3", "SR"], " can give an overall picture or zoom in on particular portions of the \ domain. Multifunctions can be used with several of the plot types. All these \ techniques increase our ability to understand complex function behavior." }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ "Top-level ", StyleBox["Cardano3", "SR"], " functions ultimately call upon routines in the ", StyleBox["DrawGraphics", "SR"], " add-on application [", ButtonBox[ CounterBox["Reference", "ParkDrawGraphicspackage"], ButtonData:>"ParkDrawGraphicspackage", ButtonStyle->"Hyperlink"], "], which is discussed in Park [", ButtonBox[ CounterBox["Reference", "ParkMIERDrawGraphicsArticle"], ButtonData:>"ParkMIERDrawGraphicsArticle", ButtonStyle->"Hyperlink"], "]. To reproduce all the results in this notebook, you will therefore need \ not just ", StyleBox["Cardano3", "SR"], ", but ", StyleBox["DrawGraphics", "SR"], " as well; you will also need Ersek's ", StyleBox["RootSearch", "SR"], " package [", ButtonBox[ CounterBox["Reference", "RootSearchPackage"], ButtonData:>"RootSearchPackage", ButtonStyle->"Hyperlink"], "]. ", StyleBox["Cardano3", "SR"], " and ", StyleBox["DrawGraphics", "SR"], " are at present compatible with ", StyleBox["Mathematica", FontSlant->"Italic"], " versions 4.2\[Dash]5.2." }], "Text", TextAlignment->Left, TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell["Initialization", "Section"], Cell[TextData[{ "To begin, we initialize the two packages that constitute the ", StyleBox["Cardano3", "SR"], " application." }], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \(Needs["\"]\)}], "Input"], Cell[TextData[{ "The ", StyleBox["DrawGraphics", "SR"], " packages are automatically loaded by ", StyleBox["ComplexGraphics", "MR"], "." }], "Text", TextAlignment->Left, TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell["Geometry in the complex plane", "Section"], Cell[TextData[{ StyleBox["Cardano3", "SR"], " contains a complete set of graphics primitives that directly use complex \ numbers for points. Geometrical diagrams are an important part of \ mathematical discussions, yet students find it difficult to draw such \ diagrams. The power of complex algebra is that many such diagrams are more \ easily constructed and drawn in the complex plane. Such diagrams are not only \ an excellent introduction to ", StyleBox["Cardano3", "SR"], " and complex algebra but a valuable technique for mathematical work. " }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ "The problem considered here is from a posting on MathGroup [", ButtonBox[ CounterBox["Reference", "MathGroupGeometryProblem"], ButtonData:>"MathGroupGeometryProblem", ButtonStyle->"Hyperlink"], "] (with Greek letters used for the angle variables of the original): " }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ "\"I am trying to find the angle \[Beta] corresponding to the points on the \ circumference of the circle ", Cell[BoxData[ \(TraditionalForm\`\((5 + 8 Cos[\[Beta]], 7 + 8 Sin[\[Beta]])\)\)]], " that are a distance of 7 units from a point on the circumference of the \ circle ", Cell[BoxData[ \(TraditionalForm\`\((13 Cos[\[Alpha]], 13 Sin[\[Alpha]])\)\)]], " for angles ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\)]], " in the first quadrant.\"" }], "Quote", TextAlignment->Left, TextJustification->1], Cell["The following creates a diagram for the problem.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(With[{c = 5 + 7 \[ImaginaryI], r1 = 8, r2 = 13, \[Beta]1 = 60 \[Degree], \[Alpha]1 = 25 \[Degree]}, \[IndentingNewLine]ComplexGraphics[\ \[IndentingNewLine]{ComplexCurve[ c + r1\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Beta]\), {\[Beta], 0, 2 \[Pi]}], Red, ComplexCurve[ r2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\), {\[Alpha], 0, \[Pi]/2}], \[IndentingNewLine]Gray, ComplexLine[{c, c + r1}], ComplexLine[{c, c + r1\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Beta]1\)}], ComplexLine[{0, r2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]1\)}], \ \[IndentingNewLine]Blue, ComplexLine[{r2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]1\), c + r1\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Beta]1\)}], \ \[IndentingNewLine]Black, ComplexText["\<\[Beta]\>", c + r1/5\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Beta]1/2\)], ComplexText["\<\[Alpha]\>", r2/5\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]1/2\)], \ \[IndentingNewLine]ComplexCirclePoint[c, 3, Black, Linen], 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To solve this problem, we simplify things by writing the ", StyleBox["complex", FontSlant->"Italic"], " equation ", Cell[BoxData[ \(Abs[\((5 + 7 \[ImaginaryI])\) + 8\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Beta]\) - 13\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\)] \[Equal] 7\)]], " for ", Cell[BoxData[ \(\[Beta]\)]], " with ", Cell[BoxData[ \(\[Alpha]\)]], " as parameter." }], "Text", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ \(\[Beta]eqn[\[Alpha]_]\ = \ ComplexExpand[ Abs[\((5 + 7 \[ImaginaryI])\) + 8\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Beta]\) - 13\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Alpha]\)]\ \[Equal] \ 7]\)], "Input"], Cell[BoxData[ \(\@\(\((5 - 13\ Cos[\[Alpha]] + 8\ Cos[\[Beta]])\)\^2 + \((7 - 13\ Sin[\ \[Alpha]] + 8\ Sin[\[Beta]])\)\^2\) \[Equal] 7\)], "Output"] }, Open ]], Cell[TextData[{ "An additional geometrical diagram (not shown) indicates that the equation \ for ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\)]], " always has two roots. The roots are most easily found by using Ersek's ", StyleBox["RootSearch", "SR"], " package [", ButtonBox[ CounterBox["Reference", "RootSearchPackage"], ButtonData:>"RootSearchPackage", ButtonStyle->"Hyperlink"], "]. " }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(Needs["\"]\)], "Input"], Cell[TextData[{ "For example, evaluating", StyleBox[" ", "TR"], StyleBox["RootSearch[\[Beta]eqn[\[Pi]/4],{\[Beta],0,2\[Pi]}]", "MR"], StyleBox[" ", "TR"], "gives the result ", StyleBox["{{\[Beta]\[Rule]1.53612},{\[Beta]\[Rule]5.71073}}", "MR"], StyleBox[". Then t", "TR"], "he following animation shows the solutions as ", Cell[BoxData[ \(\[Alpha]\)]], " is varied from ", Cell[BoxData[ \(0\)]], " to ", Cell[BoxData[ \(\[Pi]/2\)]], "." }], "Text", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[{ \(\(frame2[\[Alpha]_] := Module[\ {c = 5 + 7 \[ImaginaryI], r1 = 8, r2 = 13, \[Beta]1, \[Beta]2, \[Theta]}, {\[Beta]1, \[Beta]2} = Flatten[\[Beta] /. 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