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The first concept is that of a curve. Intuitively, we \ think of a curve as a piece of string placed on a flat surface in some type \ of meandering pattern. More Formally, we define a curve to be the range of a \ continuous complex-valued function z(t) defined on the interval [a,b]. \ That is, a curve C is the range of a function given by ", Cell[BoxData[ \(z \((t)\) = \(\((x \((t)\), y \((t)\))\) = x \((t)\) + \[ImaginaryI]\ y \((t)\)\)\)]], " for ", Cell[BoxData[ \(\(\(\ \)\(a \[LessEqual] t \[LessEqual] b\)\)\)]], ", where both x(t) and y(t) are continuous real valued functions. If \ both x(t) and y(t) are differentiable, we say that the curve is smooth. 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If we had another function whose range was the same set of points as \ z(t) but whose initial and final points were reversed, we would indicate the \ curve this function defines by -C." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["\nExample 1.22, Page 40.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Find parametrizations for C and -C, where C is the straight line segment \ beginning at ", Cell[BoxData[ \(z\_0 = x\_0 + \[ImaginaryI]\ y\_0\)]], " and ending at ", Cell[BoxData[ \(z\_1 = x\_1 + \[ImaginaryI]\ y\_1\)]], ". 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" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution for Page 40.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[{ "Enter the initial point ", Cell[BoxData[ \(z\_0 = \(-3\) + 2 \[ImaginaryI]\)], AspectRatioFixed->True], " and the terminal point ", Cell[BoxData[ \(z\_1 = \ 1\ + \ \[ImaginaryI]\)], AspectRatioFixed->True], " and construct the line segment." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[t, x, y, z];\)\ \), "\n", \(\(z\_0 = \(-3\) + 2 \[ImaginaryI];\)\ \), "\n", \(\(z\_1 = \ 1\ + \ \[ImaginaryI];\)\ \), "\n", \(x\_0 = Re[z\_0]; \ \ y\_0 = Im[z\_0]; \ \ x\_1 = Re[z\_1]; \ \ y\_1 = Im[z\_1];\ \), "\n", \(\(z[t_]\ = \ x\_0 + \((x\_1 - x\_0)\) t + \[ImaginaryI] \((y\_0 + \((y\_1 - y\_0)\) t)\);\)\ \), "\[IndentingNewLine]", \(\(dot0 = Graphics[{PointSize[0.03], Green, Point[{Re[z\_0], Im[z\_0]}]}];\)\ \), "\[IndentingNewLine]", \(\(dot1 = Graphics[{PointSize[0.03], Red, Point[{Re[z\_1], Im[z\_1]}]}];\)\ \), "\n", \(\(gr = ParametricPlot[{Re[z[t]], Im[z[t]]}, {t, 0, 1}, \[IndentingNewLine]PlotRange \[Rule] {{\(-3\), 1}, {0, 2}}, \[IndentingNewLine]Ticks \[Rule] {Range[\(-3\), 1, 1], Range[0, 2, 1]}, \[IndentingNewLine]AspectRatio \[Rule] 1\/2, PlotStyle \[Rule] Magenta, AxesLabel \[Rule] {"\", "\"}, \ \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(Show[gr, dot0, dot1, DisplayFunction \[Rule] $DisplayFunction];\)\ \), "\n", \(\(Print["\"];\)\ \), "\n", \(\(Print["\", z[t], "\< for 0 \[LessEqual] t \[LessEqual] 1\>"];\)\ \), "\n", \(\(Print["\", \ z[0]];\)\ \), "\n", \(\(Print["\", \ z[1]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[TextData[{ StyleBox["\nExample 1.23, Page 41.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Show that the circle C with center ", Cell[BoxData[ \(z\_0 = x\_0 + \[ImaginaryI]\ y\_0\)]], " and radius ", Cell[BoxData[ \(r\_0\)]], " can be parameterized to form a simple closed curve." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 1.23.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[{ "Enter ", Cell[BoxData[ \(z\_0\ \ and\ \ r\_0\)]], " and construct the standard parameterization for the circle.\nUse the \ complex exponential form ", Cell[BoxData[ \(z[t] = z\_0 + r\_0\ \[ExponentialE]\^\(\[ImaginaryI]\ t\)\)]], ". " }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[t, x, y, z];\)\ \ \), "\[IndentingNewLine]", \(\(z\_0 = 2 + \[ImaginaryI];\)\ \ \), "\[IndentingNewLine]", \(\(r\_0 = 2;\)\ \ \), "\[IndentingNewLine]", \(\(z[t_]\ = \ z\_0 + r\_0\ \[ExponentialE]\^\(\[ImaginaryI]\ t\);\)\ \ \), "\ \[IndentingNewLine]", \(\(x[t_] = ComplexExpand[Re[z[t]]];\)\ \ \), "\[IndentingNewLine]", \(\(y[t_] = ComplexExpand[Im[z[t]]];\)\ \ \), "\[IndentingNewLine]", \(\(ParametricPlot[{x[t], y[t]}, \[IndentingNewLine]{t, 0, 2 \[Pi]}, PlotRange \[Rule] {{0.0, 5.0}, {\(-1.5\), 3.5}}, \[IndentingNewLine]Ticks \[Rule] {Range[0.0, 5.0, 1.0], Range[\(-1.0\), 3.0, 1.0]}, \[IndentingNewLine]AspectRatio \[Rule] 1, Prolog \[Rule] {Thickness[0.01], Green}, AxesLabel \[Rule] {"\", "\"}];\)\ \ \), \ "\[IndentingNewLine]", \(\(Print["\"];\)\ \ \), "\ \[IndentingNewLine]", \(\(Print[\*"\"\\""];\)\ \ \ \), "\n", \(\(Print["\", z[t]];\)\ \ \), "\[IndentingNewLine]", \(\(Print["\< for 0 \[LessEqual] t \[LessEqual] 2\[Pi]\>"];\)\ \ \ \), "\[IndentingNewLine]", \(\(Print["\", x[t]];\)\ \ \), "\[IndentingNewLine]", \(\(Print["\", \ y[t], "\< for 0 \[LessEqual] t \[LessEqual] 2\[Pi]\>"];\)\ \ \), "\ \[IndentingNewLine]", \(\)}], "Input"] }, Closed]], Cell[TextData[{ StyleBox["\nExample for Page 41.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " The curve ", Cell[BoxData[ \(x[t] = Sin[2 t] Cos[t], \ \ y[t] = Sin[2 t] Sin[t]\)]], " for ", Cell[BoxData[ \(0 \[LessEqual] t \[LessEqual] 2 \[Pi]\)]], ". \n", StyleBox["Remark.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], " The curve looks like a \"four leafed rose\"." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution for Page 41.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->False], Cell["Enter the curve and plot it.", "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[t, x, y];\)\ \), "\[IndentingNewLine]", \(\(x[t_] = Sin[2 t] Cos[t];\)\ \), "\[IndentingNewLine]", \(\(y[t_] = Sin[2 t] Sin[t];\)\ \), "\[IndentingNewLine]", \(\(ParametricPlot[{x[t], y[t]}, \[IndentingNewLine]{t, 0, 2 \[Pi]}, PlotRange \[Rule] {{\(-0.8\), 0.8}, {\(-0.8\), 0.8}}, \[IndentingNewLine]Ticks \[Rule] {Range[\(-0.8\), 0.8, 0.8], Range[\(-0.8\), 0.8, 0.8]}, \[IndentingNewLine]AspectRatio \[Rule] 1, Prolog \[Rule] {Thickness[0.01], Green}, AxesLabel \[Rule] {"\", "\"}];\)\ \), "\[IndentingNewLine]", \(\(Print["\", x[t]];\)\ \), "\[IndentingNewLine]", \(\(Print["\", \ y[t], "\< for 0 \[LessEqual] t \[LessEqual] 2\[Pi]\>"];\)\ \), "\ \[IndentingNewLine]", \(\)}], "Input"] }, Closed]], Cell[TextData[{ "\n", StyleBox["Example 1.24, Page 42.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " The solution sets of the inequalities ", Cell[BoxData[ \(\(\(|\)\(z\)\(|\)\(\(<\)\(1\)\)\)\)]], ", ", Cell[BoxData[ \(\(\(|\)\(z - \[ImaginaryI]\)\(|\)\(\(<\)\(2\)\)\)\)]], ", and ", Cell[BoxData[ \(\(\(|\)\(z + 1 + 2 \[ImaginaryI]\)\(|\)\(\(<\)\(3\)\)\)\)]], " are neighborhoods of the points ", Cell[BoxData[ RowBox[{"0", ",", "\[ImaginaryI]", ",", RowBox[{ StyleBox["and", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], " ", "-", "1", "-", \(2 \[ImaginaryI]\)}]}]]], ", with radius 1,2, and 3, respectively. They can also be expressed as ", Cell[BoxData[ \(\(D\_1\) \((0)\)\)]], ", ", Cell[BoxData[ \(\(D\_2\) \((\[ImaginaryI])\)\)]], ", and ", Cell[BoxData[ \(\(D\_3\) \((\(-1\) - 2 \[ImaginaryI])\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Definitions (Interior Point, Exterior Point, Boundary Point), \ Page 42.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " The point ", Cell[BoxData[ \(z\_0\)]], " is said to be an ", StyleBox["interior point", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of the set S provided that there exists an ", Cell[BoxData[ \(\[Epsilon]\)]], "-neighborhood of ", Cell[BoxData[ \(z\_0\)]], " that contains only points of S", "; ", " ", Cell[BoxData[ \(z\_0\)]], " is called an ", StyleBox["exterior point", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of S if there exists an ", Cell[BoxData[ \(\[Epsilon]\)]], "-neighborhood of ", Cell[BoxData[ \(z\_0\)]], " that contains no points of S. If ", Cell[BoxData[ \(z\_0\)]], " is neither an interior point nor an exterior point of S, then it is \ called a ", StyleBox["boundary point", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of S and has the property that each ", Cell[BoxData[ \(\[Epsilon]\)]], "-neighborhood of ", Cell[BoxData[ \(z\_0\)]], " contains both points in S and points not in S." }], "Text"], Cell[TextData[{ "\n", StyleBox["Example 1.25, Page 43.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Let ", Cell[BoxData[ \(S = {z : \(\(|\)\(z\)\(|\)\(\(<\)\(1\)\)\)}\)]], ". ", StyleBox["(a)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Find the interior of S. ", StyleBox["(b)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Find the exterior of S. ", StyleBox["(c)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Find boundary of S." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Solution 1.25 (a).", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], " Find the interior of S." }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[{ "Let ", Cell[BoxData[ \(z\_0\)]], " be a point of S. Then ", Cell[BoxData[ \(\(\(|\)\(z\_0\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\)]], " so that we can choose ", Cell[BoxData[ \(\[Epsilon] = \(\(1\)\(-\)\) | z\_0 | \(\(>\)\(0\)\)\)]], ". If z lies in the disk ", Cell[BoxData[ \(\(\(|\)\(z - z\_0\)\(|\)\(\(<\)\(\ \)\(\[Epsilon]\)\)\)\)]], ", then \n\n", Cell[BoxData[ \(\(\(|\)\(z\)\(|\)\) = \(\(|\)\(z\_0 + z - z\_0\)\(|\)\(\(\[LessEqual]\)\(\ \)\(\(|\)\(z\_0\)\(|\)\(+\(\(|\)\ \(z - z\_0\)\(|\)\(\(<\)\(\ \)\(\(|\)\(z\_0\)\(|\)\(\ \)\(\(+\ \[Epsilon]\)\ \ < \ 1\)\)\)\)\)\)\)\)\)]], ". \n\nHence the ", Cell[BoxData[ \(\[Epsilon]\)]], "-neighborhood of ", Cell[BoxData[ \(z\_0\)]], " is contained in S, and ", Cell[BoxData[ \(z\_0\)]], " is an interior point of S. It follows that the interior of S is the \ open unit disk." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(dot = Graphics[{PointSize[0.03], Point[{0.3, 0.7}]}];\)\ \), "\[IndentingNewLine]", \(\(r = 1 - \@\(0.3\^2 + 0.7\^2\);\)\ \), "\[IndentingNewLine]", \(\(disk\ = \ Graphics[{Yellow, Disk[{0.3, 0.7}, r]}];\)\ \), "\[IndentingNewLine]", \(\(bdd\ = \ Graphics[{Circle[{0, 0}, 1]}];\)\ \), "\n", \(\(Show[bdd, disk, dot, \ PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1.5\), 1.5}}, Ticks \[Rule] {Range[\(-1.5\), 1.5, 0.5], Range[\(-1.5\), 1.5, 0.5]}, \n\ \ \ \ \ \ \ \ \ \ AspectRatio \[Rule] 1, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\\""];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(int\ = \ Graphics[{Yellow, Disk[{0, 0}, 1]}];\)\ \), "\n", \(\(Show[int, \ PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1.5\), 1.5}}, Ticks \[Rule] {Range[\(-1.5\), 1.5, 1.5], Range[\(-1.5\), 1.5, 1.5]}, \n\ \ \ \ \ \ \ \ \ \ AspectRatio \[Rule] 1, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Solution 1.25 (b). ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], " Find the exterior of S." }], "Text"], Cell[TextData[{ "If ", Cell[BoxData[ \(\(\(|\)\(z\_0\)\(|\)\(\(>\)\(\ \)\(1\)\)\)\)]], " then a similar argument will produce an ", Cell[BoxData[ \(\[Epsilon]\)]], "-neighborhood of ", Cell[BoxData[ \(z\_0\)]], " that lies entirely in the region ", Cell[BoxData[ \(\(\(|\)\(z\)\(|\)\(\(>\)\(\ \)\(1\)\)\)\)]], ", hence ", Cell[BoxData[ \(z\_0\)]], " is an exterior point of S. It follows that the exterior of S is the \ open region ", Cell[BoxData[ \(\(\(|\)\(z\)\(|\)\(\(>\)\(\ \)\(1\)\)\)\)]], ". " }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(dot = Graphics[{PointSize[0.03], Point[{0.4, 1.1}]}];\)\ \), "\[IndentingNewLine]", \(\(r = \@\(0.4\^2 + 1.1\^2\) - 1;\)\ \), "\[IndentingNewLine]", \(\(disk\ = \ Graphics[{Cyan, Disk[{0.4, 1.1}, r]}];\)\ \), "\[IndentingNewLine]", \(\(bdd\ = \ Graphics[{Circle[{0, 0}, 1]}];\)\ \), "\n", \(\(Show[bdd, disk, dot, \ PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1.5\), 1.5}}, Ticks \[Rule] {Range[\(-1.5\), 1.5, 0.5], Range[\(-1.5\), 1.5, 0.5]}, \n\ \ \ \ \ \ \ \ \ \ AspectRatio \[Rule] 1, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\ 1.\>\""];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(ext\ = \ Graphics[{{Cyan, Disk[{0, 0}, 3]}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {White, Disk[{0, 0}, 1]}}];\)\ \), "\n", \(\(Show[ext, \ PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1.5\), 1.5}}, \n\ \ \ \ \ \ \ \ \ \ AspectRatio \[Rule] 1, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}];\)\ \), "\[IndentingNewLine]", \(\(Print["\ \ 1}.\>"];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Solution 1.25 (c).", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], " Find the boundary of S." }], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[{ "If ", Cell[BoxData[ \(z\_0 = \[ExponentialE]\^\(\[ImaginaryI]\ \[Theta]\_0\)\)]], " is any point on the unit circle, then any ", Cell[BoxData[ \(\[Epsilon]\)]], "-neighborhood of ", Cell[BoxData[ \(z\_0\)]], " will contain the point ", Cell[BoxData[ \(\((1 - \[Epsilon]\/2)\) \[ExponentialE]\^\(\[ImaginaryI]\ \[Theta]\_0\ \)\)]], ", which belongs to S, and ", Cell[BoxData[ \(\((1 + \[Epsilon]\/2)\) \[ExponentialE]\^\(\[ImaginaryI]\ \[Theta]\_0\ \)\)]], " which does not belong to S. It follows that the boundary of S is the \ unit circle ", Cell[BoxData[ \(\(\(|\)\(z\)\(|\)\) = \ 1\)]], ". " }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(dot = Graphics[{PointSize[0.03], Point[{0.4, \@\(1 - 0.4\^2\)}]}];\)\ \), "\[IndentingNewLine]", \(\(r = 0.2;\)\ \), "\[IndentingNewLine]", \(\(disk\ = \ Graphics[{YellowBrown, Disk[{0.4, \@\(1 - 0.4\^2\)}, r]}];\)\ \), "\[IndentingNewLine]", \(\(bdd\ = \ Graphics[{Circle[{0, 0}, 1]}];\)\ \), "\n", \(\(Show[bdd, disk, dot, \ PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1.5\), 1.5}}, Ticks \[Rule] {Range[\(-1.5\), 1.5, 0.5], Range[\(-1.5\), 1.5, 0.5]}, \n\ \ \ \ \ \ \ \ \ \ AspectRatio \[Rule] 1, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\\""];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\ \[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(bdd\ = \ Graphics[{Brown, Thickness[0.01], Circle[{0, 0}, 1]}];\)\ \), "\n", \(\(Show[bdd, \ PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1.5\), 1.5}}, Ticks \[Rule] {Range[\(-1.5\), 1.5, 0.5], Range[\(-1.5\), 1.5, 0.5]}, \n\ \ \ \ \ \ \ \ \ \ AspectRatio \[Rule] 1, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\"}];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[TextData[{ StyleBox["\nTheorem 1.6 (", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ButtonBox["Jordan Curve Theorem", ButtonData:>{ URL[ "http://mathworld.wolfram.com/JordanCurveTheorem.html"], None}, ButtonStyle->"Hyperlink"], StyleBox["), Page 46.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " The compliment of any simple closed curve ", StyleBox["C", FontWeight->"Bold"], " can be partitioned into two mutually exclusive domains ", StyleBox["I", FontWeight->"Bold"], " and ", StyleBox["E", FontWeight->"Bold"], " in such a way that ", StyleBox["I", FontWeight->"Bold"], " is bounded, ", StyleBox["E", FontWeight->"Bold"], " is unbounded, and ", StyleBox["C", FontWeight->"Bold"], " is the boundary for both ", StyleBox["I", FontWeight->"Bold"], " and ", StyleBox["E", FontWeight->"Bold"], ". In addition ", Cell[BoxData[ RowBox[{ StyleBox["I", FontWeight->"Bold"], "\[Union]", StyleBox["E", FontWeight->"Bold"], "\[Union]", StyleBox["C", FontWeight->"Bold"]}]]], " is the entire complex plane. \n\n", StyleBox["Definition (Interior, Exterior), Page 46.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " The domain ", StyleBox["I", FontWeight->"Bold"], " is called the interior of ", StyleBox["C", FontWeight->"Bold"], ", and the domain ", StyleBox["E", FontWeight->"Bold"], " is called the exterior of ", StyleBox["C", FontWeight->"Bold"], "." }], "Text"], Cell[TextData[{ "\n\tThe Jordan curve theorem is a classic example of a result in \ mathematics that seems obvious but is very hard to demonstrate. Its proof is \ beyond the scope of this book. ", ButtonBox["Jordan's", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Jordan.\ html"], None}, ButtonStyle->"Hyperlink"], " original argument, in fact, was inadequate, and it was not until 1905 \ that a correct version was finally given by the American topologist ", ButtonBox["Oswald Veblen", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Veblen.\ html"], None}, ButtonStyle->"Hyperlink"], ". The difficulty lies in describing the interior and exterior of a simple \ closed curve analytically, and in showing that they are connected sets. For \ example, in which domain (interior or exterior) do the two points depicted in \ Figure 1.28 lie? If they are in the same domain, how, specifically, can they \ be connected with a curve? If you appreciated the subtleties involved in \ showing the right half plane is connected, you can begin to appreciate the \ obstacles that Veblen had to navigate." }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Library Research Experience for Undergraduates", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.996109]]], "Text"], Cell[TextData[{ StyleBox["Project I. Write a paper on teaching complex analysis and discuss \ some of the new approaches.", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], "\n\n", StyleBox["1.", FontWeight->"Bold"], " Bannon, Thomas J., (1991), ''Fractals and Transformations,'' Math. \ Teach., V. 81, No. 3, pp. 178-185.\n\n", StyleBox["2.", FontWeight->"Bold"], " Barton, Ray, (1990), ''Chaos and Fractals,'' Math. Teach., V. 83, No. 7, \ pp. 524-529.\n\n", StyleBox["3.", FontWeight->"Bold"], " Boyd, James N., (1985), ''A Property of Inversion in Polar Coordinates, \ The Math. 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M., V. 98, No. 3, pp. 249-255.\n\n", StyleBox["9.", FontWeight->"Bold"], " Harding, R. D., (1974), ''Computer Aided Teaching in Applied Math.,'' \ Int. J. of Math. Ed. in Sci. and Tech., V. 5, No. 4, pp. 447-455.\n\n", StyleBox["10.", FontWeight->"Bold"], " Kern, Jane F. and Cherry C. Mauk, (1990), ''Exploring Fractals--a \ Problem-solving Adventure Using Math. and Logo,'' Math. Teach., V. 83, No. 3, \ pp. 179-185.\n\n", StyleBox["11.", FontWeight->"Bold"], " Kim, David and Robert Travers, (1982), ''Those elusive imaginary zeros,'' \ The Math. Teach., V. 75, pp. 62-64.\n\n", StyleBox["12.", FontWeight->"Bold"], " Kimberling, Clark, (1987), ''Power of Complex Numbers,'' The Math. \ Teach., V. 80, No. 1, pp. 63-67.\n\n", StyleBox["13.", FontWeight->"Bold"], " Kleiner, Israel,(1988), ''Thinking the Unthinkable: The Story of Complex \ Numbers (with a Moral),'' The Math. 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