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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[ 59905, 1923]*) (*NotebookOutlinePosition[ 84981, 2832]*) (* CellTagsIndexPosition[ 84685, 2819]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["COMPLEX ANALYSIS: ", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Mathematica ", FontSize->18, FontSlant->"Italic", FontColor->RGBColor[1, 0, 1]], StyleBox[ "3.0 Notebooks\n(c) John H. Mathews, and\nRussell W. Howell, 1998", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["\n", FontColor->RGBColor[1, 0, 0]], StyleBox["Complimentary software to accompany our textbook:", FontColor->RGBColor[0, 1, 0]] }], "Text", Editable->False, Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "COMPLEX ANALYSIS: for Mathematics and Engineering, \n3rd Edition, 1997, \ ISBN: 0-7637-0270-6", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox["\n", FontSize->18, FontColor->RGBColor[0, 1, 1]], StyleBox[ "Jones & Bartlett Publishers, Inc.\n40 Tall Pine Drive, Sudbury, MA 01776\n\ Tele. (800) 832-0034, FAX: (508) 443-8000\nE-mail: mkt@jbpub.com, \ http://www.jbpub.com/", FontSize->18, FontColor->RGBColor[0, 1, 0]], StyleBox["\n", FontSize->14], StyleBox[ "This free software is compliments of the authors.\nmathews@fullerton.edu, \ howell@westmont.edu", FontSize->14, FontColor->RGBColor[1, 0, 1]] }], "Text", Editable->False, Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True] }, Closed]], Cell[TextData[{ StyleBox["CHAPTER 1 ", FontSize->18], StyleBox["COMPLEX NUMBERS\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 1.1 ", FontSize->18], ButtonBox["The Origin of Complex Numbers", ButtonData:>"Section 1.1", ButtonStyle->"Hyperlink"], StyleBox["\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 1.2 ", FontSize->18], ButtonBox["The Algebra of Complex Numbers", ButtonData:>"Section 1.2", ButtonStyle->"Hyperlink"], StyleBox["\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 1.3 ", FontSize->18], ButtonBox["The Geometry of Complex Numbers", ButtonData:>"Section 1.3", ButtonStyle->"Hyperlink"], StyleBox["\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 1.4 ", FontSize->18], ButtonBox["The Geometry of Complex Numbers, Continued", ButtonData:>"Section 1.4", ButtonStyle->"Hyperlink"], StyleBox["\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 1.5 ", FontSize->18], ButtonBox["The Algebra of Complex Numbers, Revisited", ButtonData:>"Section 1.5", ButtonStyle->"Hyperlink"], StyleBox["\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 1.6 ", FontSize->18], ButtonBox["The Topology of Complex Numbers\n", ButtonData:>"Section 1.6", ButtonStyle->"Hyperlink"], StyleBox["GoTo Chapter", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], StyleBox[" ", FontSize->18], ButtonBox["1", ButtonData:>{"C1.nb", "CHAPTER 1"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["2", ButtonData:>{"C2.nb", "CHAPTER 2"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["3", ButtonData:>{"C3.nb", "CHAPTER 3"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["4", ButtonData:>{"C4.nb", "CHAPTER 4"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["5", ButtonData:>{"C5.nb", "CHAPTER 5"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["6", ButtonData:>{"C6.nb", "CHAPTER 6"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["7", ButtonData:>{"C7.nb", "CHAPTER 7"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["8", ButtonData:>{"C8.nb", "CHAPTER 8"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["9", ButtonData:>{"C9.nb", "CHAPTER 9"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["10", ButtonData:>{"C10.nb", "CHAPTER 10"}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->18], ButtonBox["11", ButtonData:>{"C11.nb", "CHAPTER 11"}, ButtonStyle->"Hyperlink"], StyleBox[".\n", FontSize->18], StyleBox["GoTo ", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], ButtonBox["Contents", ButtonData:>{"Contents.nb", "CONTENTS"}, ButtonStyle->"Hyperlink"] }], "Text", CellTags->"CHAPTER"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["CHAPTER 1 ", FontSize->18], StyleBox["COMPLEX NUMBERS", FontSize->18, FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"CHAPTER 1"], Cell[BoxData[{ \(<< Graphics`Arrow`\), \(\(Clear[b, bdd, c, eqn, ext, int, k, list, P, r1, r2, set, solset, t, \[Theta], \[Theta]1, \[Theta]2, values, x, x0, x1, x2, x3, xx, y0, y1, y2, y3, yy, z, z1, z2, z3, z4, Z]; 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\n v2\ = \ Graphics[{RGBColor[0, 1, 0], Arrow[{Re[z1], Im[z1]}, {Re[w1], Im[w1]}]}]; \n v3\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {Re[w1], Im[w1]}]}]; \n Show[v1, v2, v3, PlotRange \[Rule] {{0, 8}, {0, 7}}, Ticks \[Rule] {Range[0, 8, 1], Range[0, 7, 1]}, AspectRatio -> 7\/8, Axes \[Rule] True]; \nPrint[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\<\!\(z\_1\) + \!\(z\_2\) = \>\""\ , \ z1\ + \ z2]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["(b) ", FontWeight->"Bold"], "Find ", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ StyleBox[\(z\_2\), FontSize->14], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[v1, v2, v3, w2, z, z1, z2]; \nz1 = \ 3\ + \ 7 \[ImaginaryI]; \n z2\ = \ 5\ - \ 6 \[ImaginaryI]; \nw2\ = \ z1 - z2; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z1], Im[z1]}]}]; 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\n v2\ = \ Graphics[{RGBColor[0, 1, 0], Arrow[{0, 0}, {Re[z2], Im[z2]}]}]; \nv3\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {Re[w1], Im[w1]}]}]; \n Show[v1, v2, v3, PlotRange \[Rule] {{0, 57}, {\(-6\), 17}}, Ticks \[Rule] {Range[0, 57, 5], Range[\(-6\), 17, 1]}, AspectRatio -> 1, Axes \[Rule] True]; \n Print[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\<\!\(z\_1\) \!\(z\_2\) = \>\""\ , \ z1\ z2]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example 1.3, Page 7.", FontWeight->"Bold"], " Find ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[ RowBox[{" ", FormBox[\(z\_1\), "TraditionalForm"]}], FontSize->14], StyleBox[\(z\_2\), FontSize->14]], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[v1, v2, v3, w1, z, z1, z2]; \nz1\ = \ 3\ + \ 7 \[ImaginaryI]; \nz2\ = \ 5\ - \ 6 \[ImaginaryI]; \nw1 = \(\ z1\)\/z2; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z1], Im[z1]}]}]; \n v2\ = \ Graphics[{RGBColor[0, 1, 0], Arrow[{0, 0}, {Re[z2], Im[z2]}]}]; \nv3\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {Re[w1], Im[w1]}]}]; \n Show[v1, v2, v3, PlotRange \[Rule] {{\(-1\), 5}, {\(-6\), 7}}, Ticks \[Rule] {Range[\(-1\), 5, 1], Range[\(-6\), 7, 1]}, AspectRatio -> 13\/6, Axes \[Rule] True]; \n Print[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\<\!\(z\_1\/z\_2\) = \>\""\ , \ z1\/z2]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Definition for Multiplication, Page 6.", FontWeight->"Bold"], " ", StyleBox["Formula (4),", FontColor->RGBColor[1, 0, 1]], " in general we can derive:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[x, x1, x2, y, y1, y2, z, z1, z2]; \n z1\ = \ x\_1\ + \ \[ImaginaryI]\ y\_1; \n z2\ = \ x\_2\ + \ \[ImaginaryI]\ y\_2; \n Print[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\<\!\(z\_1\) \!\(z\_2\) = \>\""\ , \ z1\ z2]; \n Print[\*"\"\<\!\(z\_1\) \!\(z\_2\) = \>\""\ , \ ComplexExpand[z1\ z2]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Definition for Division, Page 7.", FontWeight->"Bold"], " ", StyleBox["Formula (5),", FontColor->RGBColor[1, 0, 1]], " in general we can derive:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[x, x1, x2, y, y1, y2, z, z1, z2]; \n z1\ = \ x\_1\ + \ \[ImaginaryI]\ y\_1; \n z2\ = \ x\_2\ + \ \[ImaginaryI]\ y\_2; \n Print[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\<\!\(z\_1\/z\_2\) = \>\""\ , \ z1\/z2]; \n Print[\*"\"\<\!\(z\_1\/z\_2\) = \>\""\ , Expand[\((z1*\((x\_2 - \[ImaginaryI]\ y\_2)\))\)]\/Expand[ \((z2*\((x\_2 - \[ImaginaryI]\ y\_2)\))\)]]; \)], "Input"], Cell[TextData[{ StyleBox["Example 1.4, Page 10.", FontWeight->"Bold"], " Find Re(", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], ") and Re(", Cell[BoxData[ FormBox[ FormBox[\(z\_2\), "TraditionalForm"], TraditionalForm]], FontSize->14], ")." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[v1, v2, z, z1]; \nz1\ = \ \(-3\)\ + \ 7 \[ImaginaryI]; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z1], Im[z1]}]}]; \n v2\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {Re[z1], 0}]}]; \n Show[v1, v2, PlotRange \[Rule] {{\(-3\), 0}, {0, 7}}, Ticks \[Rule] {Range[\(-3\), 0, 1], Range[0, 7, 1]}, AspectRatio -> 7\/3, Axes \[Rule] True]; \n Print[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\\""\ , \ Re[z1]]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Clear[v1, v2, z, z2]; \nz2\ = \ \ 9\ + \ 4 \[ImaginaryI]; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z2], Im[z2]}]}]; \n v2\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {Re[z2], 0}]}]; \n Show[v1, v2, PlotRange \[Rule] {{0, 9}, {0, 4}}, Ticks \[Rule] {Range[0, 9, 1], Range[0, 4, 1]}, AspectRatio -> 4\/9, Axes \[Rule] True]; \nPrint[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\\""\ , \ Re[z2]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example 1.5, Page 10.", FontWeight->"Bold"], " Find Im(", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], ") and Im(", Cell[BoxData[ FormBox[ StyleBox[\(z\_2\), FontSize->14], TraditionalForm]]], ")." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[z, z1]; \nz1\ = \ \(-3\)\ + \ 7 \[ImaginaryI]; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z1], Im[z1]}]}]; \n v2\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {0, Im[z1]}]}]; \n Show[v1, v2, PlotRange \[Rule] {{\(-3\), 0.1}, {0, 7}}, Ticks \[Rule] {Range[\(-3\), 0, 1], Range[0, 7, 1]}, AspectRatio -> 7\/3, Axes \[Rule] True]; \n Print[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\\""\ , \ Im[z1]]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Clear[z, z2]; \nz2\ = \ \ 9\ + \ 4 \[ImaginaryI]; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z2], Im[z2]}]}]; \n v2\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {0, Im[z2]}]}]; \n Show[v1, v2, PlotRange \[Rule] {{0, 9}, {0, 4}}, Ticks \[Rule] {Range[0, 9, 1], Range[0, 4, 1]}, AspectRatio -> 4\/9, Axes \[Rule] True]; \nPrint[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\\""\ , \ Im[z2]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example 1.6, Page 10.", FontWeight->"Bold"], " Find ", Cell[BoxData[ \(TraditionalForm\`\(z\_1\)\&_\)], FontSize->14], " and ", Cell[BoxData[ \(TraditionalForm\`\(z\_2\)\&_\)], FontSize->14], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[v1, v2, v3, z, z1]; \nz1\ = \ \(-3\)\ + \ 7 \[ImaginaryI]; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z1], Im[z1]}]}]; \n v2\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {Re[z1], \(-Im[z1]\)}]}]; \n Show[v1, v2, PlotRange \[Rule] {{\(-3\), 0}, {\(-7\), 7}}, Ticks \[Rule] {Range[\(-3\), 0, 1], Range[\(-7\), 7, 1]}, AspectRatio -> 7\/3, Axes \[Rule] True]; \n Print[\*"\"\<\!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\<\!\(\(z\_1\)\&_\) = \>\""\ , Conjugate[z1]\ ]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Clear[v1, v2, v3, z, z2]; \nz2\ = \ \ 9\ + \ 4 \[ImaginaryI]; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z2], Im[z2]}]}]; \n v2\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {Re[z2], \(-Im[z2]\)}]}]; \n Show[v1, v2, PlotRange \[Rule] {{0, 9}, {\(-4\), 4}}, AspectRatio -> 8\/9, Axes \[Rule] True]; \n Print[\*"\"\<\!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\<\!\(\(z\_2\)\&_\) = \>\""\ , \ Conjugate[z2]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["The Commutative Law for Addition,", FontWeight->"Bold"], " ", StyleBox["Property (P1), Page 9. \n ", FontColor->RGBColor[1, 0, 1]], "In general we can derive:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[x, x1, x2, y, y1, y2, z, z1, z2]; \n z1\ = \ x\_1\ + \ \[ImaginaryI]\ y\_1; \n z2\ = \ x\_2\ + \ \[ImaginaryI]\ y\_2; \n Print[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\<\!\(z\_1\) + \!\(z\_2\) = \>\""\ , \ z1 + z2]; \n Print[\*"\"\<\!\(z\_2\) + \!\(z\_1\) = \>\""\ , \ z2 + z1]; \n Print[\*"\"\\"", z1 + z2\ == \ z1 + z2]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["The Associative Law for Multiplication,", FontWeight->"Bold"], " ", StyleBox["Property (P6), Page 10.", FontColor->RGBColor[1, 0, 1]], " \n In general we can derive:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[x, x1, x2, x3, y, y1, y2, y3, z, z1, z2, z3]; \n z1\ = \ x\_1\ + \ \[ImaginaryI]\ y\_1; \n z2\ = \ x\_2\ + \ \[ImaginaryI]\ y\_2; \n z3\ = \ x\_3\ + \ \[ImaginaryI]\ y\_3; \n Print[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\< \!\(z\_3\) = \>\""\ , \ z3]; \nPrint["\<\>"]; \n Print[\*"\"\< \!\(z\_1\)(\!\(z\_2\) + \!\(z\_3\)) = \>\""\ , \ z1 \((z2\ + \ z3)\)]; \n Print[\*"\"\<\!\(z\_1\) \!\(z\_2\) + \!\(z\_1\) \!\(z\_3\) = \>\""\ , \ z1\ z2\ + \ z1\ z3]; \nPrint["\<\>"]; \n Print[\*"\"\< \!\(z\_1\)(\!\(z\_2\) + \!\(z\_3\)) = \>\""\ , \ Expand[z1 \((z2\ + \ z3)\)]]; \n Print[\*"\"\<\!\(z\_1\) \!\(z\_2\) + \!\(z\_1\) \!\(z\_3\) = \>\""\ , \ Expand[z1\ z2\ + \ z1\ z3]]; \nPrint["\<\>"]; \n Print[\*"\"\\"", Expand[z1 \((z2\ + \ z3)\)]\ \ == \ \ Expand[z1\ z2\ + \ z1\ z3]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 1", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Section 1.3 ", FontSize->18], StyleBox["The Geometry of Complex Numbers", FontSize->18, FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"Section 1.3"], Cell[TextData[{ "The ", StyleBox["modulus", FontColor->RGBColor[1, 0, 1]], " of z is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[VerticalSeparator]", RowBox[{ StyleBox["z", FontSize->14], "\[VerticalSeparator]"}]}], "=", StyleBox[ SqrtBox[ RowBox[{ FormBox[\(x\^2\), "TraditionalForm"], " ", "+", " ", FormBox[\(y\^2\), "TraditionalForm"]}]], FontSize->14]}], TraditionalForm]]], ".\n\nThe ", StyleBox["distance", FontColor->RGBColor[1, 0, 1]], " between ", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox[\(z\_2\), FontSize->14], TraditionalForm]]], " is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[VerticalSeparator]", RowBox[{ StyleBox[\(z\_1 - z\_2\), FontSize->14], "\[VerticalSeparator]"}]}], "=", StyleBox[ SqrtBox[ RowBox[{ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox[ StyleBox["x", FontSize->14], "1"], StyleBox["-", FontSize->14], SubscriptBox[ StyleBox["x", FontSize->14], "2"]}], ")"}], "2"], "TraditionalForm"], " ", "+", " ", FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox[ StyleBox["y", FontSize->14], "1"], StyleBox["-", FontSize->14], SubscriptBox[ StyleBox["y", FontSize->14], "2"]}], ")"}], "2"], "TraditionalForm"]}]], FontSize->14]}], TraditionalForm]]], ". \n\nThe ", StyleBox["triangle inequality", FontColor->RGBColor[1, 0, 1]], " is |", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ FormBox[\(z\_2\), "TraditionalForm"], TraditionalForm]], FontSize->14], " | \[LessEqual] |", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], "| + |", Cell[BoxData[ FormBox[ StyleBox[\(z\_2\), FontSize->14], TraditionalForm]]], "|.\n\nThe ", StyleBox["product", FontColor->RGBColor[1, 0, 1]], " obeys ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[VerticalSeparator]", RowBox[{ StyleBox[\(\(z\_1\) z\_2\), FontSize->14], "\[VerticalSeparator]"}]}], "=", RowBox[{"\[VerticalSeparator]", RowBox[{ StyleBox[\(z\_1\), FontSize->14], "\[VerticalSeparator]", RowBox[{"\[VerticalSeparator]", RowBox[{ StyleBox[\(z\_2\), FontSize->14], "\[VerticalSeparator]"}]}]}]}]}], TraditionalForm]]], ".\n\nThe ", StyleBox["quotient", FontColor->RGBColor[1, 0, 1]], " obeys ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[VerticalSeparator]", RowBox[{ StyleBox[ FractionBox[ StyleBox[\(z\_1\), FontSize->14], StyleBox[\(z\_2\), FontSize->14]], FontSize->18], "\[VerticalSeparator]"}]}], "=", FractionBox[ RowBox[{"\[VerticalSeparator]", " ", RowBox[{ StyleBox[\(z\_1\), FontSize->14], StyleBox[" ", FontSize->14], "\[VerticalSeparator]"}]}], RowBox[{"\[VerticalSeparator]", " ", RowBox[{ StyleBox[\(z\_2\), FontSize->14], StyleBox[" ", FontSize->14], "\[VerticalSeparator]"}]}]]}], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example 1.7, Page 15.", FontWeight->"Bold"], " Verify the Triangle Inequality for ", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox[\(z\_2\), FontSize->14], TraditionalForm]]], " .\n |", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ StyleBox[\(z\_2\), FontSize->14], TraditionalForm]]], " | \[LessEqual] |", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], "| + |", Cell[BoxData[ FormBox[ StyleBox[\(z\_2\), FontSize->14], TraditionalForm]]], "|." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[z, z1, z2]; \nz1\ = \ 7\ + \ \[ImaginaryI]; \n z2\ = \ 3\ + \ 5 \[ImaginaryI]; \nw1\ = \ z1 + z2; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z1], Im[z1]}]}]; \n v2\ = \ Graphics[{RGBColor[0, 1, 0], Arrow[{Re[z1], Im[z1]}, {Re[w1], Im[w1]}]}]; \n v3\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {Re[w1], Im[w1]}]}]; \n Show[v1, v2, v3, PlotRange \[Rule] {{0, 10}, {0, 6}}, Ticks \[Rule] {Range[0, 10, 1], Range[0, 6, 1]}, AspectRatio -> 6\/10, Axes \[Rule] True]; \n Print[\*"\"\< \!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\< \!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\< \!\(z\_1\) + \!\(z\_2\) = \>\""\ , \ z1\ + \ z2]; \n Print["\<\>"]; \n Print[\*"\"\< |\!\(z\_1\) + \!\(z\_2\)| = \>\"", \ Abs[z1\ + \ z2]]; \n Print[\*"\"\< |\!\(z\_1\)| + |\!\(z\_2\)| = \>\"", \ Abs[z1] + Abs[z2]]; \nPrint[\*"\"\\"", Abs[z1\ + \ z2]\ <= \ Abs[z1]\ + \ Abs[z2]]; \n Print["\<\>"]; \n Print[\*"\"\< |\!\(z\_1\) + \!\(z\_2\)| = \>\"", \ N[Abs[z1\ + \ z2]]]; \n Print[\*"\"\< |\!\(z\_1\)| + |\!\(z\_2\)| = \>\"", \ N[Abs[z1] + Abs[z2]]]; \n Print[\*"\"\\"", Abs[z1\ + \ z2]\ <= \ Abs[z1]\ + \ Abs[z2]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 1.8, Page 16.", FontWeight->"Bold"], " Verify that |", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["z", StyleBox["1", FontSize->10]], FontSize->15], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["z", StyleBox["2", FontSize->10]], FontSize->15], TraditionalForm]]], " | = |", StyleBox["z", FontSize->14], StyleBox["1", FontSize->10, FontVariations->{"CompatibilityType"->"Subscript"}], "|", StyleBox[" ", FontSize->9], StyleBox["|", FontSize->13], Cell[BoxData[ FormBox[ FormBox[\(z\_2\), "TraditionalForm"], TraditionalForm]], FontSize->14], "|." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[z, z1, z2, z3]; \nz1\ = \ 1\ + \ 2 \[ImaginaryI]; \n z2\ = \ 3\ + \ 2 \[ImaginaryI]; \nz3\ = \ z1\ z2; \n v1\ = \ Graphics[{RGBColor[1, 0, 0], Arrow[{0, \ 0}, {Re[z1], Im[z1]}]}]; \n v2\ = \ Graphics[{RGBColor[0, 1, 0], Arrow[{0, 0}, {Re[z2], Im[z2]}]}]; \nv3\ = \ Graphics[{RGBColor[0, 0, 1], Arrow[{0, \ 0}, {Re[z3], Im[z3]}]}]; \n Show[v1, v2, v3, PlotRange \[Rule] {{\(-1\), 3}, {0, 8}}, Ticks \[Rule] {Range[\(-1\), 3, 1], Range[0, 8, 1]}, AspectRatio -> 2, Axes \[Rule] True]; \nPrint[\*"\"\<\!\(z\_1\) = \>\""\ , \ z1]; \n Print[\*"\"\<\!\(z\_2\) = \>\""\ , \ z2]; \n Print[\*"\"\<\!\(z\_1\) \!\(z\_2\) = \>\""\ , \ z3]; \n Print[\*"\"\<|\!\(z\_1\)| = \>\""\ , \ Abs[z1]]; \n Print[\*"\"\<|\!\(z\_2\)| = \>\""\ , \ Abs[z2]]; \n Print[\*"\"\<|\!\(z\_1\) \!\(z\_2\)| = \>\""\ , \ Abs[z1\ z2]]; \n Print[\*"\"\\""]; \nPrint[Abs[z1\ z2], \ "\< = \>", \ Abs[z1]\ Abs[z2]]; \n Print["\<\>"]; \n Print[\*"\"\<|\!\(z\_1\) \!\(z\_2\)| = \>\"", \ N[Abs[z1\ z2]]]; \n Print[\*"\"\<|\!\(z\_1\)||\!\(z\_2\)| = \>\"", \ N[Abs[z1] Abs[z2]]]; \n Print[\*"\"\\"", \n\ \ Abs[z1\ \ z2]\ == \ Abs[z1]\ Abs[z2]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 1", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Section 1.4 ", FontSize->18], StyleBox["The Geometry of Complex Numbers, Continued", FontSize->18, FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"Section 1.4"], Cell[TextData[{ "An ", StyleBox["argument", FontColor->RGBColor[1, 0, 1]], " of z is ", StyleBox["\[Theta]", FontSize->14], " = arg(", StyleBox["z", FontSize->14], ") = arctan(", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["y", FontSize->14], StyleBox["x", FontSize->14]], TraditionalForm]]], ") if x \[NotEqual] 0, but we must be careful to specify the choice of \ arctan(", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["y", FontSize->14], StyleBox["x", FontSize->14]], TraditionalForm]]], ") so that the point z corresponding to r and ", StyleBox["\[Theta]", FontSize->14], " lies in the appropriate quadrant.\n\nThe ", StyleBox["principal value", FontColor->RGBColor[1, 0, 1]], " of arg(", StyleBox["z", FontSize->14], ") is denoted by Arg(", StyleBox["z", FontSize->14], ") = ", StyleBox["\[Theta] ", FontSize->14], " where ", StyleBox["- \[Pi] < \[Theta] \[LessEqual] \[Pi]", FontSize->14], ".\n\nA famous equation known as ", StyleBox["Euler's formula", FontColor->RGBColor[1, 0, 1]], " is ", Cell[BoxData[ \(e\^\[ImaginaryI]\[Theta] = \ cos\ \[Theta]\ + \ i\ sin\ \[Theta]\)]], ".\n\nThe ", StyleBox["exponential form", FontColor->RGBColor[1, 0, 1]], " of z is z = r ", Cell[BoxData[ FormBox[ StyleBox[\(e\^i\[Theta]\), FontSize->14], TraditionalForm]], FontSize->18], ", where r = |z| and ", StyleBox["\[Theta]", FontSize->14], " = arg(z).\n\nThe ", StyleBox["product", FontColor->RGBColor[1, 0, 1]], " obeys the rule ", Cell[BoxData[ \(arg \((\(z\_1\) z\_2)\) = \ arg \((z\_1)\) + arg \((z\_2)\)\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example 1.9, Page 19.", FontWeight->"Bold"], " Find several polar forms of z." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[z, z1, z2, z3]; \nz1\ = \ \@3\ + \ \[ImaginaryI]; \n z2\ = \ 2\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Pi]/6\); \n z3\ = \ 2\ \[ExponentialE]\^\(\[ImaginaryI]\ 13\ \[Pi]/6\); \n z4\ = \ 2\ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ 11\ \[Pi]/6\); \n Print[\*"\"\<\!\(z\_1\) = 2 \!\(\[ExponentialE]\^\(\[ImaginaryI]\\\ \ \[Pi]/6\)\) = \>\"", z1]; \n Print[\*"\"\<\!\(z\_2\) = 2 \!\(\[ExponentialE]\^\(\[ImaginaryI]\\\ \ \[Pi]/6\)\) = \>\"", z2, "\< = \>", ComplexExpand[z2]]; \n Print[\*"\"\<\!\(z\_3\) = 2 \!\(\[ExponentialE]\^\(\[ImaginaryI]\\\ 13\\\ \ \[Pi]/6\)\) = \>\"", z3, "\< = \>", ComplexExpand[z3]]; \n Print[\*"\"\<\!\(z\_4\) = 2 \!\(\[ExponentialE]\^\(\(-\\\ \[ImaginaryI]\)\ \\\ 11\\\ \[Pi]/6\)\) = \>\"", z4, "\< = \>", ComplexExpand[z4]]; \)], "Input",\ AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 1.10, Page 19.", FontWeight->"Bold"], " Find the polar form of z, \nby computing |z| and ", StyleBox["\[Theta]", FontSize->14], " = Arg(z)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[\[Theta], \[Theta]1, \[Theta]2, r, z, z1, z2]; \n z\ = \ \(-\ \@3\)\ - \ \[ImaginaryI]; \nr\ = \ Abs[z]; \n \[Theta]1\ = \ ArcTan[Im[z]\/Re[z]] + \ \[Pi]; \n z1\ = \ r\ \((Cos[\[Theta]1]\ + \ \[ImaginaryI]\ Sin[\[Theta]1])\); \n \[Theta]2\ = \ ArcTan[Im[z]\/Re[z]] + \ 3 \[Pi]; \n z2\ = \ r\ \((Cos[\[Theta]2]\ + \ \[ImaginaryI]\ Sin[\[Theta]2])\); \n Print["\", z]; \nPrint["\", r]; \n Print[\*"\"\<\!\(\[Theta]\_1\) = Arg[z] = \>\"", \[Theta]1]; \n Print[\*"\"\<\!\(z\_1\) = \>\"", z1]; \n Print[\*"\"\<\!\(z\_1\) = \>\"", Expand[z1]]; \n Print[\*"\"\\"", z\ == \ Expand[z1]]; \n Print["\<\>"]; \nPrint["\", z]; \nPrint["\", r]; \n Print[\*"\"\<\!\(\[Theta]\_2\) = Arg[z] = \>\"", \[Theta]2]; \n Print[\*"\"\<\!\(z\_2\) = \>\"", z2]; \n Print[\*"\"\<\!\(z\_2\) = \>\"", ComplexExpand[z2]]; \n Print[\*"\"\\"", z\ == \ ComplexExpand[z2]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 1.11, Page 20.", FontWeight->"Bold"], " Write z in the r ", Cell[BoxData[ \(TraditionalForm\`e\^i\[Theta]\)], FontSize->14], " form." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[\[Theta], r, z, z1]; \nz\ = \ \(-\ \@3\)\ - \ \[ImaginaryI]; \n r\ = \ Abs[z]; \n\[Theta]\ = \ Arg[z]\ + \ 2 \[Pi]; \n z1\ = \ r\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Theta]\); \n Print["\", z]; \nPrint["\", r]; \n Print["\<\[Theta] = Arg[z] = \>", \[Theta]]; \n Print[\*"\"\<\!\(z\_1\) = 2 \!\(\[ExponentialE]\^\(\[ImaginaryI]7\[Pi]\/6\ \)\)\>\""]; \nPrint[\*"\"\<\!\(z\_1\) = \>\"", z1]; \n Print[\*"\"\<\!\(z\_1\) = \>\"", ComplexExpand[z1]]; \n Print[\*"\"\\"", z\ == \ ComplexExpand[z1]]; \)], "Input"], Cell[TextData[{ "\n", StyleBox["Example 1.12, Page 22.", FontWeight->"Bold"], " Given z, find |z| and ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox["z", FontSize->14]], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[r, z, z1, z2, z3]; \nz\ = \ 5\ + \ 12 \[ImaginaryI]; \n r\ = \ Abs[z]; \nz1\ = \ Conjugate[z]; \nz2\ = \ Conjugate[z]\/r; \n z3\ = \ 1\/r\ Conjugate[z]\/r; \nPrint["\", z]; \n Print["\", r]; \nPrint[\*"\"\<\!\(z\&_\) = \>\"", z1]; \n Print[\*"\"\<\!\(z\&_\/r\) = \>\"", z2]; \n Print[\*"\"\<\!\(1\/r\)\!\(z\&_\/r\) = \>\"", z3]; \n Print[\*"\"\<\!\(1\/z\) = \>\"", 1\/z]; \n Print[\*"\"\\"", 1\/z\ == \ 1\/r\ Conjugate[z]\/r]; \)], "Input"], Cell[TextData[{ "\n", StyleBox["Example 1.13, Page 22.", FontWeight->"Bold"], " Given ", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox[\(z\_2\), FontSize->14], TraditionalForm]]], ", compute ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[ FormBox[\(z\_1\), "TraditionalForm"], FontSize->14], StyleBox[ FormBox[\(z\_2\), "TraditionalForm"], FontSize->14]], TraditionalForm]]], " using polar computations." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[\[Theta], \[Theta]1, \[Theta]2, r1, r2, w1, w2, z, z1, z2]; \n z1\ = \ 8 \[ImaginaryI]; \nz2\ = \ 1\ + \ \[ImaginaryI]\ \@3; \n r1\ = \ Abs[z1]; \n\[Theta]1\ = \ Arg[z1]; \n Print[\*"\"\<\!\(z\_1\) = \>\"", z1]; \n Print[\*"\"\<\!\(r\_1\) = |\!\(z\_1\)| = \>\"", r1]; \n Print[\*"\"\<\!\(\[Theta]\_1\) = Arg[\!\(z\_1\)] = \>\"", \[Theta]1]; \n r2\ = \ Abs[z2]; \n\[Theta]2\ = \ Arg[z2]; \n Print[\*"\"\<\!\(z\_2\) = \>\"", z2]; \n Print[\*"\"\<\!\(r\_2\) = |\!\(z\_2\)| = \>\"", r2]; \n Print[\*"\"\<\!\(\[Theta]\_2\) = Arg[\!\(z\_2\)] = \>\"", \[Theta]2]; \n w1\ = \ z1\/z2; \n w2\ = \ r1\/r2\ \[ExponentialE]\^\(\[ImaginaryI] \((\[Theta]1\ - \ \[Theta]2)\)\); \n Print["\< \>"]; \nPrint[\*"\"\<\!\(z\_1\/z\_2\) = \>\"", w1]; \n Print[\*"\"\<\!\(r\_1\/r\_2\)\!\(\[ExponentialE]\^\(\[ImaginaryI] \((\ \[Theta]\_1 - \[Theta]\_2)\)\)\) = \>\"", w2]; \nPrint["\<\>"]; \n Print[\*"\"\<\!\(z\_1\/z\_2\) = \>\"", ComplexExpand[w1]]; \n Print[\*"\"\<\!\(r\_1\/r\_2\)\!\(\[ExponentialE]\^\(\[ImaginaryI] \((\ \[Theta]\_1 - \[Theta]\_2)\)\)\) = \>\"", ComplexExpand[w2]]; \n Print[\*"\"\\"", ComplexExpand[z1\/z2]\ == \ ComplexExpand[ r1\/r2\ \[ExponentialE]\^\(\[ImaginaryI] \((\[Theta]1\ - \ \[Theta]2)\)\)]]; \)], "Input"], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 1", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Section 1.5 ", FontSize->18], StyleBox["The Algebra of Complex Numbers, Revisited", FontSize->18, FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"Section 1.5"], Cell[TextData[{ "The ", StyleBox["nth power", FontColor->RGBColor[1, 0, 1]], " of z is ", Cell[BoxData[ FormBox[ StyleBox[\(z\^n\), FontSize->14], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[\(r\^n\), FontSize->14], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ StyleBox[\(e\^in\[Theta]\), FontSize->14], TraditionalForm]]], ". \n\nA famous formula known as ", StyleBox["De Moivre's formula", FontColor->RGBColor[1, 0, 1]], " is ", Cell[BoxData[ \(\((cos\ \[Theta]\ + i\ sin\ \[Theta])\)\^n = \ cos\ n\[Theta]\ + i\ sin\ n\[Theta]\)]], ".\n\n", StyleBox["Corollary 1.1, Page 25.", FontWeight->"Bold"], " ", StyleBox["(Corollary to the fundamental theorem of algebra)", FontColor->RGBColor[1, 0, 1]], " If P(z) is a polynomial of degree n (n > 0) with complex \ coefficients, then the equation P(z) = 0 has precisely n (not necessarily \ distinct) solutions.\n\nThe solutions to ", Cell[BoxData[ \(z\^n = 1\)]], " are called the ", StyleBox["nth roots of unity", FontColor->RGBColor[1, 0, 1]], " and can be expressed as ", Cell[BoxData[ RowBox[{ SubscriptBox["z", StyleBox["k", FontSize->10]], "=", SuperscriptBox["\[ExponentialE]", StyleBox[\(\[ImaginaryI]\ 2 \[Pi]\ k/n\), FontSize->10]]}]]], " for ", Cell[BoxData[ \(k = 0, 1, \[CenterEllipsis], n - 1\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example 1.14, Page 24.", FontWeight->"Bold"], " Show that ", Cell[BoxData[ FormBox[ SuperscriptBox[\((\(-\@3\) - i)\), StyleBox["3", FontSize->13]], TraditionalForm]]], " = - 8i in two ways." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[\[Theta], r, z, Z]; \nz\ = \ \(-\ \@3\)\ - \ \[ImaginaryI]; \n r\ = \ Abs[z]; \n\[Theta]\ = \ Arg[z]; \n Z\ = \ ComplexExpand[r\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Theta]\)]; \n Print["\", z]; \nPrint["\", r]; \n Print["\<\[Theta] = Arg[z] = \>", \[Theta]]; \n Print[\*"\"\\"", Z]; \nPrint[\*"\"\<\!\(z\^3\) = \>\"", \ Expand[z\^3]]; \n Print[\*"\"\<\!\(z\^3\) = \!\(r\^3\)\!\(\[ExponentialE]\^\[ImaginaryI]3\ \[Theta]\) = \>\"", \ r\^3\ \[ExponentialE]\^\(\[ImaginaryI]\ 3\ \[Theta]\)]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 1.15, Page 25.", FontWeight->"Bold"], " Evaluate ", Cell[BoxData[ FormBox[ SuperscriptBox[\((\(-\@3\) - i)\), StyleBox[\(-6\), FontSize->13]], TraditionalForm]]], " = ", Cell[BoxData[ FractionBox[ StyleBox[\(-1\), FontSize->14], StyleBox["64", FontSize->14]]]], " in two ways." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[\[Theta], r, z, Z]; \nz\ = \(-\ \@3\)\ - \ \[ImaginaryI]; \n r\ = \ Abs[z]; \n\[Theta]\ = \ Arg[z]; \n Z\ = \ ComplexExpand[r\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Theta]\)]; \n Print["\", z]; \nPrint["\", r]; \n Print["\<\[Theta] = Arg[z] = \>", \[Theta]]; \n Print[\*"\"\\"", Z]; \nPrint[\*"\"\<\!\(z\^\(-6\)\) = \>\"", \ Expand[z\^\(-6\)], \ "\< = \>"\ , \ ExpandAll[z\^\(-6\)]]; \n Print[\*"\"\<\!\(z\^\(-6\)\) = \!\(r\^\(-6\)\)\!\(\[ExponentialE]\^\(-\ \[ImaginaryI]6\[Theta]\)\) = \>\"", \ r\^\(-6\)\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 6\ \[Theta]\)]; \)], "Input"], Cell[TextData[{ "\n", StyleBox["Example 1.16, Page 25.", FontWeight->"Bold"], " Use Demoivre's formula to show that\n cos(5", StyleBox["\[Theta]", FontSize->14], ") = ", Cell[BoxData[ FormBox[ SuperscriptBox["cos", StyleBox["5", FontSize->13]], TraditionalForm]]], "(", StyleBox["\[Theta]", FontSize->14], ") - 10 ", Cell[BoxData[ FormBox[ SuperscriptBox["cos", StyleBox["3", FontSize->13]], TraditionalForm]]], "(", StyleBox["\[Theta]", FontSize->14], ")", Cell[BoxData[ FormBox[ SuperscriptBox["sin", StyleBox["2", FontSize->13]], TraditionalForm]]], "(", StyleBox["\[Theta]", FontSize->14], ") + 5 cos(", StyleBox["\[Theta]", FontSize->14], ") ", Cell[BoxData[ FormBox[ SuperscriptBox["sin", StyleBox["4", FontSize->13]], TraditionalForm]]], "(", StyleBox["\[Theta]", FontSize->14], ")." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[\[Theta], z1, z2, z3]; \n z1\ = \ Cos[5 \[Theta]]\ + \ \[ImaginaryI]\ Sin[5 \[Theta]]; \n z2\ = \ \((Cos[\[Theta]]\ + \ \[ImaginaryI]\ Sin[\[Theta]])\)\^5; \n z3\ = \ ComplexExpand[z2]; \nPrint[z1, "\< = \>", z2]; \nPrint["\< \>"]; \nPrint[z1, "\< = \>", z3]; \nPrint["\< \>"]; \n Print["\"]; \n Print[ComplexExpand[Re[z1]]\ == \ ComplexExpand[Re[z3]]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 1.17, Page 25.", FontWeight->"Bold"], " Factor the polynomial P(z) = ", Cell[BoxData[ \(TraditionalForm\`z\^3\)], FontSize->14], " + (2 - 2i)", Cell[BoxData[ FormBox[ StyleBox[\(z\^2\), FontSize->14], TraditionalForm]]], " + (-1-4i)z - 2." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[P, z]; \n P[z_]\ = \ z\^3\ + \ \((2 - 2 \[ImaginaryI])\)\ z\^2\ + \ \((\(-1\) - 4 \[ImaginaryI])\)\ z\ - \ 2; \n Print["\", P[z]]; \nPrint["\", Factor[P[z]]]; \n Print["\"]; \nPrint[Solve[P[z] == 0, z]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 1.18, Page 26.", FontWeight->"Bold"], " Find all the solutions to the equation ", Cell[BoxData[ FormBox[ StyleBox[\(z\^8\), FontSize->14], TraditionalForm]]], " = 1." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[k, list, set, solset, values, xx, yy, z]; \n solset\ = \ Solve[z\^8\ == \ 1\ , \ z]; \n values\ = \ solset[\([Table[k, {k, Length[solset]}], 1, 2]\)]; \n set\ = \ N[solset[\([Table[k, {k, Length[solset]}], 1, 2]\)]]; \n set\ = \ \ ReplaceAll[set, xx_Real \[Rule] {xx, 0}]; \n list\ = \ \ ReplaceAll[set, Complex[xx_, yy_] \[Rule] {xx, yy}]; \n Print[\*"\"\\""]; \nPrint[values]; \n Print["\< \>"]; \nPrint[ComplexExpand[values]]; \n ListPlot[list, Prolog \[Rule] {PointSize[0.03], RGBColor[1, 0, 0]}, AspectRatio \[Rule] 1, AxesLabel \[Rule] {"\", "\"}, Ticks \[Rule] {Range[\(-1\), 1, 1], Range[\(-1\), 1, 1]}, PlotRange \[Rule] {{\(-1.5\), 1.5}, {\(-1.5\), 1.5}}, Prolog \[Rule] PointSize[0.03]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 1.19, Page 28.", FontWeight->"Bold"], " Find all the cube roots of 8i,\ni.e. find all solutions to the equation \ ", Cell[BoxData[ FormBox[ StyleBox[\(z\^3\), FontSize->14], TraditionalForm]]], " = 8i." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[k, list, set, solset, values, xx, yy, z]; \n solset\ = \ Solve[z\^3\ == \ 8 \[ImaginaryI]\ , \ z]; \n values\ = \ solset[\([Table[k, {k, Length[solset]}], 1, 2]\)]; \n set\ = \ N[solset[\([Table[k, {k, Length[solset]}], 1, 2]\)]]; \n set\ = \ ReplaceAll[set, xx_Real \[Rule] {xx, 0}]; \n list\ = \ \ ReplaceAll[set, Complex[xx_, yy_] \[Rule] {xx, yy}]; \n Print["\"]; \nPrint[values]; \n Print["\< \>"]; \nPrint[ComplexExpand[values]]; \n ListPlot[list, Prolog \[Rule] {PointSize[0.03], RGBColor[1, 0, 0]}, AspectRatio \[Rule] 1, AxesLabel \[Rule] {"\", "\"}, Ticks \[Rule] {Range[\(-2\), 2, 1], Range[\(-2\), 2, 1]}, PlotRange \[Rule] {{\(-2.1\), 2.1}, {\(-2.1\), 2.1}}]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 1", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Section 1.6 ", FontSize->18], StyleBox["The Topology of Complex Numbers", FontSize->18, FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"Section 1.6"], Cell[TextData[{ " In this section we investigate some basic ideas concerning sets of \ points in the plane. The first concept is that of a curve.\n", StyleBox["Definition, Page 30.", FontWeight->"Bold"], " A ", StyleBox["curve", FontColor->RGBColor[1, 0, 1]], " in the complex plane is C: z(t) = x(t) + i y(t) for a \ \[LessEqual] t \[LessEqual] b." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example in the narrative, Page 30.", FontWeight->"Bold"], " If ", Cell[BoxData[ FormBox[ StyleBox[\(z\_0\), FontSize->14], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[\(x\_0\), FontSize->14], TraditionalForm]]], " + i ", Cell[BoxData[ FormBox[ StyleBox[\(y\_0\), FontSize->14], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[\(x\_1\), FontSize->14], TraditionalForm]]], " + i ", Cell[BoxData[ FormBox[ StyleBox[\(y\_1\), FontSize->14], TraditionalForm]]], " are two given points, then the straight line segment joining ", Cell[BoxData[ FormBox[ StyleBox[\(z\_0\), FontSize->14], TraditionalForm]]], " to ", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], " is:\nC: z(t) = ", Cell[BoxData[ FormBox[ StyleBox[\(x\_0\), FontSize->14], TraditionalForm]]], " + (", Cell[BoxData[ FormBox[ StyleBox[\(x\_1\), FontSize->14], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ StyleBox[\(x\_0\), FontSize->14], TraditionalForm]]], ") t + i ( ", Cell[BoxData[ FormBox[ StyleBox[\(y\_0\), FontSize->14], TraditionalForm]]], " + (", Cell[BoxData[ FormBox[ StyleBox[\(y\_1\), FontSize->14], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ StyleBox[\(y\_0\), FontSize->14], TraditionalForm]]], ")t ) for 0 \[LessEqual] t \[LessEqual] 1." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[t, x0, x1, y0, y1, z]; \n z[t_]\ = \ x0\ + \ \((x1 - x0)\) t\ + \ \[ImaginaryI] \((y0\ + \ \((y1 - y0)\) t)\); \n Print["\", z[t]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example for, Page 30.", FontWeight->"Bold"], " Find the equation of the line segment with \nthe initial point ", Cell[BoxData[ FormBox[ StyleBox[\(z\_0\), FontSize->14], TraditionalForm]]], " = - 3 + 2i and the terminal point ", Cell[BoxData[ FormBox[ StyleBox[\(z\_1\), FontSize->14], TraditionalForm]]], " = 1 + i ." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Options[ParametricPlot]\)], "Input"], Cell[BoxData[{ \(Clear[t, x0, x1, y0, y1, z]; \nz0\ = \ \(-\ 3\)\ + \ 2*\[ImaginaryI]; \nz1\ = \ 1\ + \ \[ImaginaryI]; \nx0\ = \ Re[z0]; \ y0\ = \ Im[z0]; \ x1\ = \ Re[z1]; \ y1\ = \ Im[z1]; \n z[t_]\ = \ x0\ + \ \((x1 - x0)\) t\ + \ \[ImaginaryI] \((y0\ + \ \((y1 - y0)\) t)\); \n ParametricPlot[{Re[z[t]], Im[z[t]]}, {t, 0, 1}, PlotRange \[Rule] {{\(-3\), 1}, {0, 2}}, Ticks \[Rule] {Range[\(-3\), 1, 1], Range[0, 2, 1]}, AspectRatio \[Rule] 1\/2, Prolog \[Rule] RGBColor[0, 0, 1]]; \n Print["\"]\), \(Print["\", z[t]]; \n Print["\", \ z[0]]; \n Print["\", \ z[1]]; \)}], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example for, Page 31.", FontWeight->"Bold"], " The curve x(t) = sin 2t cos t, y(t) = sin 2t sin t for 0 \ \[LessEqual] t \[LessEqual] 2\[Pi] forms a four-leafed rose." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(ParametricPlot[{Sin[2 t] Cos[t], Sin[2 t] Sin[t]}, {t, 0, 2 \[Pi]}, PlotRange \[Rule] {{\(-1\), 1}, {\(-1\), 1}}, Ticks \[Rule] {Range[\(-1\), 1, 1], Range[\(-1\), 2, 1]}, AspectRatio \[Rule] 1, Prolog \[Rule] RGBColor[0, 0, 1]]; \)\)], "Input"], Cell[TextData[{ "\n", StyleBox["Example 1.20, Page 33.", FontWeight->"Bold"], " Let S = {z: |z| < 1}. 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Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{12, 10}, {7, 7}}, LineSpacing->{1, 3}, CounterIncrements->"Text"], 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 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