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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 2 ", FontSize->18], StyleBox["COMPLEX FUNCTIONS\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 2.1 ", FontSize->18], ButtonBox["Functions of a Complex Variable", ButtonData:>"Section 2.1", ButtonStyle->"Hyperlink"], StyleBox["\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 2.2 ", FontSize->18], ButtonBox["Transformations and Linear Mappings", ButtonData:>"Section 2.2", ButtonStyle->"Hyperlink"], StyleBox["\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 2.3 ", FontSize->18], ButtonBox["The Mappings w = z^n and w = z^1/n", ButtonData:>"Section 2.3", ButtonStyle->"Hyperlink"], StyleBox["\nSection 2.4 ", FontSize->18], ButtonBox["Limits and Continuity", ButtonData:>"Section 2.4", ButtonStyle->"Hyperlink"], StyleBox["\n", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Section 2.5 ", FontSize->18], ButtonBox["Branches of Functions", ButtonData:>"Section 2.5", ButtonStyle->"Hyperlink"], StyleBox["\n", FontColor->RGBColor[1, 0, 1]], StyleBox["Section 2.6 ", FontSize->18], ButtonBox["The Reciprocal Transformation w = 1/z\n", ButtonData:>"Section 2.6", 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 2 ", FontSize->18], StyleBox["COMPLEX FUNCTIONS", FontSize->18, FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"CHAPTER 2"], Cell[BoxData[ \(SetOptions[Graphics, DefaultColor \[Rule] RGBColor[0, 0, 1]]; \n Get["\"]; \n Clear[a, ac, al, b, bc, bhc, bl, bverti, circlea, circleb, eq1, eq2, eq3, eq4, eq5, eqn, eqns, f, \ F, hc, hl, horiz, Iden, ineq, ineq1, ineq2, ineq3, ineq4, ineq5, ineq6, ineq7, ineq8, ineq9, \[Theta], r, u, u0, u1, u2, \ U, U0, v, vc, vl, verti, V, w, wplane, W, x, \ X, y, Y, z, Z, zplane]; \)], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Section 2.1 ", StyleBox["Functions of a Complex Variable", FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18, CellTags->"Section 2.1"], Cell[TextData[{ "A ", StyleBox["function", FontColor->RGBColor[1, 0, 1]], " f(z) of the ", StyleBox["complex variable", FontColor->RGBColor[1, 0, 1]], " z can be written:\n f(z) = f(x + i y) = u(x,y) + i v(x,y).\n\ The ", StyleBox["polar coordinate form", FontColor->RGBColor[1, 0, 1]], " of a complex function is:\n f(", Cell[BoxData[ FormBox[ StyleBox["z", FontSize->14], TraditionalForm]]], ") = f(", Cell[BoxData[ FormBox[ StyleBox["r", FontSize->14], TraditionalForm]]], " ", Cell[BoxData[ \(TraditionalForm\`e\^i\[Theta]\)], FontSize->14], ") = u(r, ", Cell[BoxData[ FormBox[ StyleBox["\[Theta]", FontSize->14], TraditionalForm]]], StyleBox[" ", FontSize->14], ") + i v(r, ", Cell[BoxData[ FormBox[ StyleBox["\[Theta]", FontSize->14], TraditionalForm]]], " )." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "These are the two approaches to defining a ", StyleBox["complex function", FontColor->RGBColor[1, 0, 1]], " in ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, u, v, x, y]; \nx /: Im[x] = 0; \ny /: Im[y] = 0; \n u /: Im[u[x, y]] = 0; \nv /: Im[v[x, y]] = 0; \nx /: Re[x] = x; \n y /: Re[y] = y; \nu /: Re[u[x, y]] = u[x, y]; \n v /: Re[v[x, y]] = v[x, y]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Method 1.", FontColor->RGBColor[1, 0, 1]], " Make f a function of two real variables (x,y)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Clear[f, u, v, x, y]; \n f[x_, y_]\ := \ u[x, y]\ + \ \[ImaginaryI]\ v[x, y]\), \(Print["\", f[x, y]]; \nPrint["\", f[2, 1]]; \n Print["\", ComplexExpand[Re[f[x, y]]]]; \n Print["\", ComplexExpand[Im[f[x, y]]]]; \)}], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Method 2.", FontColor->RGBColor[1, 0, 1]], " Make F a function of z." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[F, x, y, X, Y, u, v, U, V, Z]; \n F[Z_]\ := \ Module[{X = Re[Z], Y = Im[Z]}, Return[u[X, Y]\ + \ \[ImaginaryI]\ v[X, Y]]\ ]; \n Print["\< F[2+\[ImaginaryI]] = \>", F[2 + \[ImaginaryI]]]; \n Print["\", F[x + \[ImaginaryI]\ y]]; \n Print["\", MapAll[ComplexExpand, \ F[x + \[ImaginaryI]\ y]]]; \)], "Input", AspectRatioFixed->True], Cell["\<\ Remark: It was necessary to use the power of MapAll to achieve \ the last result.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Print["\", MapAll[ComplexExpand, \ Re[F[x + \[ImaginaryI]\ y]]]]\), \(\(Print["\", MapAll[ComplexExpand, \ Im[F[x + \[ImaginaryI]\ y]]]]; \)\)}], "Input"], Cell[TextData[{ "\n", StyleBox["Example 2.1, Page 38.", FontWeight->"Bold"], " Write ", Cell[BoxData[ FormBox[ RowBox[{\(f(z)\), "=", " ", StyleBox[\(z\^4\), FontSize->14]}], TraditionalForm]]], " in the f(z) = u(x,y) + i v(x,y) form." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, x, y]; \n f[x_, y_]\ = \ ComplexExpand[\((x\ + \ \[ImaginaryI]\ y)\)\^4]; \n Print["\", f[x, y]]; \nPrint["\", f[1, 2]]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Clear[F, x, y, z]; \nx /: Im[x] = 0; \ny /: Im[y] = 0; \n x /: Re[x] = x; \ny /: Re[y] = y; \n F[z_]\ := \ Module[{x = Re[z], y = Im[z]}, Return[Expand[\((x\ + \ \[ImaginaryI]\ y)\)\^4]]\ ]; \n Print["\", F[x + \[ImaginaryI]\ y]]; \n Print["\< F[1+2\[ImaginaryI]] = \>", F[1 + 2 \[ImaginaryI]]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.2, Page 39.", FontWeight->"Bold"], " Write f(z) = ", Cell[BoxData[ \(TraditionalForm\`z\&_\)], FontSize->14], " Re(z) + ", Cell[BoxData[ \(TraditionalForm\`z\^2\)], FontSize->14], " + Im(z) in the form\n f(z) = u(x,y) + i v(x,y)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, F, x, y]; \n f[x_, y_]\ = \ ComplexExpand[ Conjugate[x\ + \ \[ImaginaryI]\ y] Re[x\ + \ \[ImaginaryI]\ y]\ + \ \((x\ + \ \[ImaginaryI]\ y)\)\^2\ + \ Im[x\ + \ \[ImaginaryI]\ y]]; \nPrint["\", f[x, y]]; \n Print["\", f[3, 2]]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(x /: Im[x] = 0; \ny /: Im[y] = 0; \nx /: Re[x] = x; \ny /: Re[y] = y; \n F[z_]\ := \ Module[{x = Re[z], y = Im[z]}, Return[ComplexExpand[ Conjugate[x\ + \ \[ImaginaryI]\ y] Re[x\ + \ \[ImaginaryI]\ y]\ + \ \((x\ + \ \[ImaginaryI]\ y)\)\^2\ + \ Im[x\ + \ \[ImaginaryI]\ y]]]\ ]; \n Print["\", F[x + \[ImaginaryI]\ y]]; \n Print["\", F[3 + 2 \[ImaginaryI]]]; \)], "Input",\ AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.3, Page 39.", FontWeight->"Bold"], " Express ", Cell[BoxData[ FormBox[ RowBox[{\(f(z)\), " ", "=", " ", RowBox[{ RowBox[{"4", StyleBox[\(x\^2\), FontSize->14]}], "+", " ", RowBox[{"i", " ", "4", StyleBox[\(y\^2\), FontSize->14]}]}]}], TraditionalForm]]], " with a formula involving z and ", Cell[BoxData[ \(TraditionalForm\`z\&_\)], FontSize->14], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, w, x, y, z]; \nw\ = \ 4 x\^2\ + \ 4 y\^2; \n f[z_]\ = \ ReplaceAll[w \ , {x \[Rule] \(z\ + \ Conjugate[z]\)\/2, y \[Rule] \(z\ - \ Conjugate[z]\)\/\(2 \[ImaginaryI]\)}]; \n Print["\< w = \>", w]; \nPrint["\< f[z] = \>", f[z]]; \n Print["\< f[z] = \>", Expand[f[z]]]; \n Print["\", f[x + \[ImaginaryI]\ y]]; \n Print["\", ComplexExpand[f[x + \[ImaginaryI]\ y]]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.4, Page 39.", FontWeight->"Bold"], " Express f(z) = ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["z", FontSize->14], "5"], TraditionalForm]]], "+ 4 ", Cell[BoxData[ FormBox[ StyleBox[\(z\^2\), FontSize->14], TraditionalForm]]], " - 6 in polar coordinate form." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, r, \[Theta], z]; \nf[z_]\ = \ z\^5\ + \ 4\ z\^2\ - \ 6; \n F[r, \[Theta]]\ = \ ReplaceAll[ f[z]\ , { z\^2 \[Rule] r\^2\ Cos[2 \[Theta]]\ + \ \[ImaginaryI]\ r\^2\ Sin[2 \[Theta]], z\^5 \[Rule] r\^5\ Cos[5 \[Theta]]\ + \ \[ImaginaryI]\ r\^5\ Sin[5 \[Theta]]}]; \n Print["\< f[z] = \>", f[z]]; \n Print["\", F[r, \[Theta]]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 2", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Section 2.2 ", StyleBox["Transformations and Linear Mappings", FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18, CellTags->"Section 2.2"], Cell[TextData[{ " In the study of complex analysis the elementary transformation of the \ form ", Cell[BoxData[ \(w = \(W \((z)\) = A\ z + B\)\)]], " is called a ", StyleBox["linear transformation", FontColor->RGBColor[1, 0, 1]], " and it is a one-to-one mapping of the complex z-plane onto the complex \ w-plane. Any geometric object is mapped onto an object that is similar to \ the original object: hence linear transformations can be called similarity \ mappings.\n Notice that the usage of the word \"", StyleBox["linear", FontColor->RGBColor[1, 0, 1]], "\" is different than that used in linear algebra." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Load ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ComplexMap package. Make sure this is done only ONCE during a ", StyleBox["Mathematica", FontSlant->"Italic"], " session." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(Get["\"]; \)\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.5, Page 42.", FontWeight->"Bold"], " Show that the function f(z) = i z maps \nthe line y = x + 1 onto \ the line v = -u - 1." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Clear[eq1, eq2, eq3, F, solset, u, v, x, y, z]; \nx /: Im[x] = 0; \n y /: Im[y] = 0; \nx /: Re[x] = x; \ny /: Re[y] = y; \n F[z_]\ := \ \[ImaginaryI]\ z\), \(Print["\< F[z] = \>", F[z]]; \n Print["\", F[\ x + \[ImaginaryI]\ y]]; \n Print["\", Expand[F[\ x + \[ImaginaryI]\ y]]]; \)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(eq1\ = \ u\ == \ ComplexExpand[Re[F[x + \[ImaginaryI]\ y]]]; \n eq2\ = \ v\ == \ ComplexExpand[Im[F[x + \[ImaginaryI]\ y]]]; \n eq3\ = \ y\ == \ x\ + \ 1; \n solset\ = \ Eliminate[{eq1, eq2, eq3}, {x, y}]; \nPrint[eq3]; \n Print[solset]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.6, Page 44.", FontWeight->"Bold"], " Show that the linear transformation w = i z + i maps \nthe right \ half plane Re(z) > 1 onto the upper half plane Im(w) > 2." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[eq1, eq2, f, ineq, u, v, w, x, y, z, zz, Z]; \nx /: Im[x] = 0; \n y /: Im[y] = 0; \nx /: Re[x] = x; \ny /: Re[y] = y; \n f[z_]\ := \ \[ImaginaryI]\ z\ + \ \[ImaginaryI]; \n Print["\< f[z] = \>", f[z]]; \n Print["\", f[x + \[ImaginaryI]\ y]]; \n Print["\", Expand[f[x + \[ImaginaryI]\ y]]]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(solset\ = \ Solve[w\ == \ \[ImaginaryI]\ z\ + \ \[ImaginaryI], \ z]; \nzz\ = \ solset\[LeftDoubleBracket]1, 1, 2\[RightDoubleBracket]; \n Z\ = \ ReplaceAll[zz\ , w \[Rule] u\ + \ \[ImaginaryI]\ v]; \n Print["\", f[z]]; \n Print["\", f[x\ + \ \[ImaginaryI]\ y]]; \n Print["\", Expand[f[x + \[ImaginaryI]\ y]]]; \n Print["\", ComplexExpand[Re[f[x + \[ImaginaryI]\ y]]]]; \n Print["\", ComplexExpand[Im[f[x + \[ImaginaryI]\ y]]]]; \n Print["\< \>"]; \nPrint[solset]; \nPrint["\< \>"]; \n Print["\", zz]; \nPrint["\", Z]; \n Print["\", Expand[Z]]; \nPrint["\", ComplexExpand[Re[Z]]]; \nPrint["\", ComplexExpand[Im[Z]]]; \)], "Input", AspectRatioFixed->True], Cell["To express x > 1, we use the computations.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Print[ineq\ = \ ComplexExpand[Re[Z]]\ > \ 1]; \n Print[Distribute[ineq\ + \ 1, \ Greater]]; \)], "Input", AspectRatioFixed->True], Cell["This solution is the upper half plane Im(w) > 2.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, wplane, z, zplane]; \nIden[z_]\ = \ z; \n f[z_]\ = \ \[ImaginaryI]\ z\ + \ \[ImaginaryI]; \n zplane\ = \ CartesianMap[Iden, \ {1, 5, 0.5}, {\(-6\), 4, 0.5}, PlotRange \[Rule] {{\(-1\), 6}, {\(-3\), 4}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-1\), 6, 1], Range[\(-3\), 4, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \ \n wplane\ = \ CartesianMap[f, \ {1, 5, 0.5}, {\(-6\), 4, 0.5}, PlotRange \[Rule] {{\(-1\), 6}, {\(-1\), 6}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-1\), 6, 1], Range[\(-1\), 6, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \ \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \n Print["\< The mapping w = \>", f[z]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.7, Page 45.", FontWeight->"Bold"], " Show that the image of the open disk |z + 1 + i| < 1 \nunder the \ transformation w = (3 - 4 i) z + 6 + 2 i is the open disk |w + 1 - 3 i| \ < 5." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, ineq, solset, term, w, z, Z]; \n f[z_]\ = \ \((3\ - \ 4 \[ImaginaryI])\) z\ + \ 6\ + \ 2 \[ImaginaryI]; \n Print["\< f[z] = \>", f[z]]; \n Print["\", f[x + \[ImaginaryI]\ y]]; \n Print["\", ComplexExpand[f[x + \[ImaginaryI]\ y]]]; \)], "Input", AspectRatioFixed->True], Cell["Solve for z in terms of w.", "Text"], Cell[BoxData[ \(solset\ = \ Solve[w\ == \ f[z], z]; \n Z\ = \ Factor[solset\[LeftDoubleBracket]1, 1, 2\[RightDoubleBracket]]; \n term\ = \ Together[Z + 1 + \[ImaginaryI]]; \nPrint[solset]; \n Print["\< z = \>", Z]; \n Print[z + 1 + \[ImaginaryI], "\< = \>", Z + 1 + \[ImaginaryI]]; \n Print[z + 1 + \[ImaginaryI], "\< = \>", term]; \)], "Input", AspectRatioFixed->True], Cell["Solve for |z| < 1.", "Text"], Cell[BoxData[ \(Print[Abs[z + 1 + \[ImaginaryI]]\ < \ 1]; \n Print[ineq\ = \ Abs[term[\([1]\)]]\ Abs[term[\([2]\)]]\ < \ 1]; \n Print[Distribute[5\ ineq, \ Less]]; \)], "Input", AspectRatioFixed->True], Cell["Which is the disk |w + 1 - 3i| < 5 in the w-plane.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, wplane, W, z, zplane, Z]; \nIden[z_]\ = \ z; \n f[z_]\ = \ \((3\ - \ 4 \[ImaginaryI])\) z\ + \ 6\ + \ 2 \[ImaginaryI]; \n Z[z_]\ = \ Iden[z - 1 - \[ImaginaryI]]; \n W[z_]\ = \ f[z - 1 - \[ImaginaryI]]; \n zplane\ = \ PolarMap[Z\ , {0, 1, 0.2}, {0, 2 \[Pi], \[Pi]\/12}, PlotRange \[Rule] {{\(-6\), 4}, {\(-2\), 8}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-6\), 4, 1], Range[\(-2\), 8, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n wplane\ = \ PolarMap[W\ , {0, 1, 0.2}, {0, 2 \[Pi], \[Pi]\/12}, PlotRange \[Rule] {{\(-6\), 4}, {\(-2\), 8}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-6\), 4, 1], Range[\(-2\), 8, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \n Print["\< The mapping w = \>", f[z]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.8, Page 46.", FontWeight->"Bold"], " Show that the image of the right half plane Re(z) > 1 \nunder the \ linear transformation w = (-1 + i) z - 2 + 3 i is the half plane v > u \ + 7." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, ineq1, ineq2, ineq3, x, y, z]; \nx /: Im[x] = 0; \n y /: Im[y] = 0; \nx /: Re[x] = x; \ny /: Re[y] = y; \n f[z_]\ := \ \((\(-1\)\ + \ \[ImaginaryI])\) z\ - \ 2\ + \ 3 \[ImaginaryI]; \n Print["\< f[z] = \>", f[z]]; \n Print["\", f[\ x + \[ImaginaryI]\ y]]; \n Print["\", ComplexExpand[f[\ x + \[ImaginaryI]\ y]]]; \)], "Input", AspectRatioFixed->True], Cell["Solve for z in terms of w.", "Text"], Cell[BoxData[ \(solset\ = \ Solve[w\ == \ \((\(-1\)\ + \ \[ImaginaryI])\) z\ - \ 2\ + \ 3 \[ImaginaryI], \ z]; \nzz\ = \ solset\[LeftDoubleBracket]1, 1, 2\[RightDoubleBracket]; \n Z\ = \ ReplaceAll[zz\ , w \[Rule] u\ + \ \[ImaginaryI]\ v]; \n Print["\", f[z]]; \n Print["\", f[x\ + \ \[ImaginaryI]\ y]]; \n Print["\", Expand[f[x + \[ImaginaryI]\ y]]]; \n Print["\", ComplexExpand[Re[f[x + \[ImaginaryI]\ y]]]]; \n Print["\", ComplexExpand[Im[f[x + \[ImaginaryI]\ y]]]]; \n Print["\< \>"]; \nPrint[solset]; \nPrint["\< \>"]; \n Print["\", zz]; \nPrint["\", Z]; \n Print["\", Expand[Z]]; \nPrint["\", ComplexExpand[Re[Z]]]; \nPrint["\", ComplexExpand[Im[Z]]]; \)], "Input", AspectRatioFixed->True], Cell["Solve for Re(z) > 1.", "Text"], Cell[BoxData[ \(Print[Re[z]\ > \ 1]; \n Print[ineq1\ = \ ComplexExpand[Re[Z]]\ > \ 1]; \n Print[ineq2\ = \ MapAll[Together, ineq1]]; \n Print[ineq3\ = \ Distribute[2\ ineq2, \ Greater]]; \n Print[Distribute[ineq3\ + \ u\ + \ 5, \ Greater]]; \)], "Input", AspectRatioFixed->True], Cell["Which is the right half plane v > 7 + u in the w-plane.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, wplane, z, zplane]; \nIden[z_]\ = \ z; \n f[z_]\ = \ \((\(-1\)\ + \ \[ImaginaryI])\) z\ - \ 2\ + \ 3 \[ImaginaryI]; \n zplane\ = \ CartesianMap[Iden, \ {1, 5, 0.5}, {\(-6\), 7, 0.5}, PlotRange \[Rule] {{\(-1\), 6}, {\(-1\), 6}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-1\), 6, 1], Range[\(-1\), 6, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \ \n wplane\ = \ CartesianMap[f, \ {1, 5, 0.5}, {\(-6\), 7, 0.5}, PlotRange \[Rule] {{\(-10\), 2}, {\(-2\), 10}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-10\), 2, 2], Range[2, 10, 2]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \ \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \n Print["\< The mapping w = \>", f[z]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 2", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Section 2.3 ", StyleBox["The Mappings w = ", FontColor->RGBColor[1, 0, 1]], Cell[BoxData[ FormBox[ StyleBox[\(z\^n\), FontSize->24, FontColor->RGBColor[1, 0, 1]], TraditionalForm]], FontColor->RGBColor[1, 0, 0]], StyleBox[" and w = ", FontColor->RGBColor[1, 0, 1]], Cell[BoxData[ FormBox[ StyleBox[\(z\^\(1/n\)\), FontSize->24, FontColor->RGBColor[1, 0, 1]], TraditionalForm]]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18, CellTags->"Section 2.3"], Cell[TextData[{ " The function ", Cell[BoxData[ RowBox[{ RowBox[{"w", " ", "=", " ", RowBox[{\(f \((z)\)\), " ", "=", " ", RowBox[{ FormBox[ StyleBox[\(z\^2\), FontSize->14], "TraditionalForm"], " ", "=", " ", RowBox[{ FormBox[ StyleBox[\(x\^2\), FontSize->14], "TraditionalForm"], " ", "-", " ", FormBox[ StyleBox[\(y\^2\), FontSize->14], "TraditionalForm"], " ", "+", " ", \(i\ 2 xy\)}]}]}]}], " "}]]], "can be expressed in polar coordinates by ", Cell[BoxData[ RowBox[{"w", " ", "=", " ", RowBox[{\(f \((z)\)\), " ", "=", RowBox[{ FormBox[ StyleBox[\(z\^2\), FontSize->14], "TraditionalForm"], " ", "=", " ", RowBox[{ FormBox[ StyleBox[\(\(r\^2\) e\^i2\[Theta]\), FontSize->14], "TraditionalForm"], FormBox["", "TraditionalForm"]}]}]}]}]]], ".\n\n The function ", Cell[BoxData[ RowBox[{"w", " ", "=", " ", RowBox[{\(f \((z)\)\), " ", "=", " ", FormBox[ StyleBox[\(z\^\(1/2\)\), FontSize->14], "TraditionalForm"]}]}]]], " can be expressed in polar coordinates by ", Cell[BoxData[ RowBox[{"w", " ", "=", " ", RowBox[{\(f \((z)\)\), "=", RowBox[{ FormBox[ RowBox[{ RowBox[{ StyleBox[\(z\^\(1/2\)\), FontSize->14], "="}], " "}], "TraditionalForm"], \(r\^\(1/2\)\), StyleBox[\(e\^\(i\[Theta]/2\)\), FontSize->14]}]}]}]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Load ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ComplexMap package. Make sure this is done only ONCE during a ", StyleBox["Mathematica", FontSlant->"Italic"], " session." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Get["\"]\)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example 2.9, Page 49.", FontWeight->"Bold"], " The transformation w = ", Cell[BoxData[ FormBox[ StyleBox[\(z\^2\), FontSize->14], TraditionalForm]]], " maps lines onto lines or parabolas." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Clear[a, b, eq1, eq2, eq3, eq4, x, y, u, u1, u2, v, U, V]; \n f[z_]\ := \ z\^2\), \(eq1\ = \ u\ == \ ComplexExpand[Re[f[x + \[ImaginaryI]\ y]]]; \n eq2\ = \ v\ == \ ComplexExpand[Im[f[x + \[ImaginaryI]\ y]]]; \n Print["\", f[z]]; \n Print["\", f[x + \[ImaginaryI]\ y]]; \n Print["\", Expand[f[x + \[ImaginaryI]\ y]]]; \n Print["\< \>", eq1]; \nPrint["\< \>", eq2]; \)}], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["(a)", FontWeight->"Bold"], " Find the image of the vertical line x = a. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Print[eqns\ = \ ReplaceAll[{eq1, eq2}, x \[Rule] a]]; \n Print[eq3\ = \ Eliminate[eqns, {y}]]; \n Print[solset1\ = \ Solve[eq3, u]]; \n u1[v_]\ = \ Expand[solset1\[LeftDoubleBracket]1, 1, 2\[RightDoubleBracket]]; \n Print["\", u1[v]]; \)], "Input", AspectRatioFixed->True], Cell["\<\ Hence, the image of the vertical line x = a is a parabola. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["(b)", FontWeight->"Bold"], " Find the image of the horizontal line y = b. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Print[eqns\ = \ ReplaceAll[{eq1, eq2}, y \[Rule] b]]\), \(Print[eq4\ = \ Eliminate[eqns, {x}]]\), \(Print[solset2\ = \ Solve[eq4, u]]\), \(u2[v_]\ = \ Expand[solset2\[LeftDoubleBracket]1, 1, 2\[RightDoubleBracket]]; \n Print["\", u2[v]]; \)}], "Input", AspectRatioFixed->True], Cell["\<\ Hence, the image of the horizontal line y = b is a parabola. \ \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, wplane, z, zplane]; \nIden[z_]\ = \ z; \n f[z_]\ = \ z\^2; \n zplane\ = \ CartesianMap[Iden, \ {0, 0.5, 0.1}, {0, 2, 0.1}, PlotRange \[Rule] {{\(-0.1\), 1}, {\(-0.2\), 2.2}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[0, 0.5, 0.5], Range[0, 2, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \ \n wplane\ = \ CartesianMap[f, \ {0, 0.5, 0.1}, {0, 2, 0.1}, PlotRange \[Rule] {{\(-4\), 0.5}, {\(-0.2\), 2.2}}, AspectRatio \[Rule] 2.4\/4.5, Ticks \[Rule] {Range[\(-4\), 0, 1], Range[0, 2, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \ \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \n Print["\< The mapping w = \>", f[z]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.10, Page 50.", FontWeight->"Bold"], " The transformation w = ", Cell[BoxData[ FormBox[ StyleBox[\(z\^\(1/2\)\), FontSize->14], TraditionalForm]]], " maps lines onto lines or hyperbolas." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, wplane, z, zplane]; \nIden[z_]\ = \ z; \n f[z_]\ = \ \@z; \n zplane\ = \ CartesianMap[Iden, \ {\(-9\), 9, 1}, {0, 9, 1}, PlotRange \[Rule] {{\(-9\), 10}, {\(-0.5\), 10}}, AspectRatio \[Rule] 1\/2, Ticks \[Rule] {Range[\(-9\), 9, 3], Range[0, 9, 3]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \ \n wplane\ = \ CartesianMap[f, \ {\(-9\), 9, 1}, {0, 9, 1}, PlotRange \[Rule] {{\(-0.3\), 3.3}, {\(-0.3\), 3.3}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[0, 3, 1], Range[0, 3, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \n Print["\< The mapping w = \>", f[z]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 2", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Section 2.4 ", StyleBox["Limits and Continuity", FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18, CellTags->"Section 2.4"], Cell[TextData[{ " Let u = u(x,y) be a real-valued function of the two real variables \ x and y. We say that u has the ", StyleBox["limit", FontColor->RGBColor[1, 0, 1]], " ", Cell[BoxData[ FormBox[ StyleBox[\(u\_0\), FontSize->14], TraditionalForm]]], " as (x,y) approaches (", Cell[BoxData[ FormBox[ StyleBox[\(x\_0\), FontSize->14], TraditionalForm]]], ",", Cell[BoxData[ FormBox[ StyleBox[\(y\_0\), FontSize->14], TraditionalForm]]], ") provided that the value of u(x,y) gets close to the value ", Cell[BoxData[ FormBox[ StyleBox[\(u\_0\), FontSize->14], TraditionalForm]]], " as (x,y) gets close to (", Cell[BoxData[ FormBox[ StyleBox[\(x\_0\), FontSize->14], TraditionalForm]]], ",", Cell[BoxData[ FormBox[ StyleBox[\(y\_0\), FontSize->14], TraditionalForm]]], "). We write\n\t\t", Cell[BoxData[ RowBox[{ \(lim\_\(\((x, y)\) \[Rule] \((x\_0, y\_0)\)\)u \((x, y)\)\), "=", StyleBox[\(u\_0\), FontSize->14]}]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Theorem 2.1, Page 55.", FontWeight->"Bold"], " Let ", Cell[BoxData[ \(f \((z)\) = u \((x, y)\) + \ i\ v \((x, y)\)\)]], " be a complex function that is defined in some neighborhood of ", Cell[BoxData[ SubscriptBox[ StyleBox["z", FontSize->14], "0"]]], ", except perhaps at ", Cell[BoxData[ RowBox[{ SubscriptBox[ StyleBox["z", FontSize->14], "0"], "=", RowBox[{ SubscriptBox[ StyleBox["x", FontSize->14], "0"], "+", RowBox[{"i", " ", SubscriptBox[ StyleBox["y", FontSize->14], "0"]}]}]}]]], ". Then \n\t", Cell[BoxData[ RowBox[{\(lim\_\(z \[Rule] z\_0\)f \((z)\)\), "=", RowBox[{ SubscriptBox[ StyleBox["w", FontSize->14], "0"], "="}]}]]], Cell[BoxData[ \(u\_0 + i\ v\_0\)]], " \nif and only if\n\t", Cell[BoxData[ RowBox[{ \(lim\_\(\((x, y)\) \[Rule] \((x\_0, y\_0)\)\)u \((x, y)\)\), "=", StyleBox[\(u\_0\), FontSize->14]}]]], " and ", Cell[BoxData[ RowBox[{ \(lim\_\(\((x, y)\) \[Rule] \((x\_0, y\_0)\)\)v \((x, y)\)\), "=", SubscriptBox[ StyleBox["v", FontSize->14], "0"]}]]], ".\n\n", StyleBox["Theorem 2.2, Page 56.", FontWeight->"Bold"], " Let ", Cell[BoxData[ \(lim\_\(z \[Rule] z\_0\)f \((z)\) = A\)]], " and ", Cell[BoxData[ \(lim\_\(z \[Rule] z\_0\)g \((z)\) = B\)]], ". Then\n\t", Cell[BoxData[ \(lim\_\(z \[Rule] z\_0\)f \((z)\) + g \((z)\) = A + B\)]], ", \n\t", Cell[BoxData[ \(lim\_\(z \[Rule] z\_0\)f \((z)\) - g \((z)\) = A - B\)]], ", \n\t", Cell[BoxData[ \(lim\_\(z \[Rule] z\_0\)f \((z)\) g \((z)\) = A\ B\)]], ", \n\t", Cell[BoxData[ RowBox[{ RowBox[{\(lim\_\(z \[Rule] z\_0\)\), FractionBox[ StyleBox[\(f \((z)\)\), FontSize->12], StyleBox[\(g \((z)\)\), FontSize->12]]}], "=", FractionBox[ StyleBox["A", FontSize->12], StyleBox["B", FontSize->12]]}]]], ", where ", Cell[BoxData[ \(B \[NotEqual] 0\)]], ".\n\n", StyleBox["Theorem 2.3, Page 56.", FontWeight->"Bold"], " Let ", Cell[BoxData[ \(f \((z)\) = u \((x, y)\) + \ i\ v \((x, y)\)\)]], " be a defined in some neighborhood of ", Cell[BoxData[ SubscriptBox[ StyleBox["z", FontSize->14], "0"]]], ". Then ", Cell[BoxData[ \(f \((z)\)\)]], " is continuous at ", Cell[BoxData[ RowBox[{ SubscriptBox[ StyleBox["z", FontSize->14], "0"], "=", RowBox[{ SubscriptBox[ StyleBox["x", FontSize->14], "0"], "+", RowBox[{"i", " ", SubscriptBox[ StyleBox["y", FontSize->14], "0"]}]}]}]]], " if and only if ", Cell[BoxData[ \(u \((x, y)\)\)]], " and ", Cell[BoxData[ \(v \((x, y)\)\)]], " are continuous at ", Cell[BoxData[ RowBox[{"(", RowBox[{ SubscriptBox[ StyleBox["x", FontSize->14], "0"], ",", SubscriptBox[ StyleBox["y", FontSize->14], "0"]}], ")"}]]], ".\n\n", StyleBox["Theorem 2.4, Page 56.", FontWeight->"Bold"], " Suppose that ", Cell[BoxData[ \(f \((z)\)\)]], " and ", Cell[BoxData[ \(g \((z)\)\)]], " are continous at the point ", Cell[BoxData[ SubscriptBox[ StyleBox["z", FontSize->14], "0"]]], ". Then the following functions are continuous at ", Cell[BoxData[ SubscriptBox[ StyleBox["z", FontSize->14], "0"]]], ":\nTheir sum ", Cell[BoxData[ \(f \((z)\) + g \((z)\)\)]], ", \nTheir difference ", Cell[BoxData[ \(f \((z)\) - g \((z)\)\)]], ", \nTheir product ", Cell[BoxData[ \(f \((z)\) g \((z)\)\)]], ", \nTheir quotient ", Cell[BoxData[ FractionBox[ StyleBox[\(f \((z)\)\), FontSize->12], StyleBox[\(g \((z)\)\), FontSize->12]]]], ", provided that ", Cell[BoxData[ RowBox[{ RowBox[{"g", RowBox[{"(", SubscriptBox[ StyleBox["z", FontSize->14], "0"], ")"}]}], "\[NotEqual]", "0"}]]], ".\nTheir composition ", Cell[BoxData[ \(f \((g \((z)\))\)\)]], " provided that ", Cell[BoxData[ \(f \((z)\)\)]], " is continuous in a neighborhood of the point ", Cell[BoxData[ RowBox[{"g", RowBox[{"(", SubscriptBox[ StyleBox["z", FontSize->14], "0"], ")"}]}]]], "." }], "Text"], Cell[TextData[{ StyleBox["Example 2.11, Page 53.", FontWeight->"Bold"], " The function u(x,y) = ", StyleBox[" ", FontSize->16], Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(x\^3\), FontSize->14], StyleBox[ RowBox[{ SuperscriptBox[ StyleBox["x", FontSize->15], "2"], " ", "+", " ", \(y\^2\)}], FontSize->14]], TraditionalForm]], FontSize->16], " has the limit 0\nas (x,y) approaches (0,0)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[lim, r, \[Theta], u, u0, u1, u2, U, U0, x, y]; \n u[x_, y_]\ = \ x\^3\/\(x\^2 + y\^2\); \nPrint["\", u[x, y]]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(u0\ = \ u[x, y]; \nu1\ = \ Limit[u0, x \[Rule] 0]; \n u2\ = \ Limit[u1, y \[Rule] 0]; \n Print[\*"\"\< \!\(lim\+\(x \[Rule] 0\)\) \>\"", u[x, y], "\< = \>", u1]; \nPrint[ \*"\"\<\!\(lim\+\(y \[Rule] 0\)\) \!\(lim\+\(x \[Rule] 0\)\) \>\"", u[x, y], "\< = \>", u2]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(u0\ = \ u[x, y]; \nu1\ = \ Limit[u0, y \[Rule] 0]; \n u2\ = \ Limit[u1, x \[Rule] 0]; \n Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \>\"", u[x, y], "\< = \>", u1]; \nPrint[ \*"\"\<\!\(lim\+\(x \[Rule] 0\)\) \!\(lim\+\(y \[Rule] 0\)\) \>\"", u[x, y], "\< = \>", u2]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(U0\ = \ u[r\ Cos[\[Theta]], r\ Sin[\[Theta]]]; \n U[r_, \[Theta]_]\ = \ Simplify[U0]; \n lim\ = \ Limit[U[r, \[Theta]], r \[Rule] 0]; \n Print["\", U0]; \n Print["\", U[r, \[Theta]]]; \n Print[\*"\"\<\!\(lim\+\(r \[Rule] 0\)\) \>\"", U[r, \[Theta]], "\< = \>", lim]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(lim\ = \ Limit[U[r, \[Theta]], r \[Rule] 0]; \n Print[\*"\"\<\!\(lim\+\(r \[Rule] 0\)\) \>\"", U[r, \[Theta]], "\< = \>", lim]; \)], "Input", AspectRatioFixed->True], Cell["\<\ So, along all lines through the origin, the limit is 0. A more rigorous proof is needed for other approaches to the origin.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.12, Page 54.", FontWeight->"Bold"], " The function u(x,y) = ", StyleBox[" ", FontSize->14], Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(x\ y\), FontSize->14], StyleBox[ RowBox[{ FormBox[\(x\^2\), "TraditionalForm"], " ", "+", " ", FormBox[\(y\^2\), "TraditionalForm"]}], FontSize->14]], TraditionalForm]], FontSize->16], "\ndoes ", StyleBox["NOT", FontColor->RGBColor[1, 0, 1]], " have a limit as (x,y) approaches (0,0)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[lim, r, \[Theta], u, u0, u1, u2, U, U0, x, y]; \n u[x_, y_]\ = \(\ x\ y\)\/\(x\^2 + y\^2\); \n Print["\", u[x, y]]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(u0\ = \ u[x, y]; \nu1\ = \ Limit[u0, x \[Rule] 0]; \n u2\ = \ Limit[u1, y \[Rule] 0]; \n Print[\*"\"\< \!\(lim\+\(x \[Rule] 0\)\) \>\"", u[x, y], "\< = \>", u1]; \nPrint[ \*"\"\<\!\(lim\+\(y \[Rule] 0\)\) \!\(lim\+\(x \[Rule] 0\)\) \>\"", u[x, y], "\< = \>", u2]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(u0\ = \ u[x, y]; \nu1\ = \ Limit[u0, y \[Rule] 0]; \n u2\ = \ Limit[u1, x \[Rule] 0]; \n Print[\*"\"\< \!\(lim\+\(y \[Rule] 0\)\) \>\"", u[x, y], "\< = \>", u1]; \nPrint[ \*"\"\<\!\(lim\+\(x \[Rule] 0\)\) \!\(lim\+\(y \[Rule] 0\)\) \>\"", u[x, y], "\< = \>", u2]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(U0\ = \ u[r\ Cos[\[Theta]], r\ Sin[\[Theta]]]; \n U[r_, \[Theta]_]\ = \ Simplify[U0]; \n lim\ = \ Limit[U[r, \[Theta]], r \[Rule] 0]; \n Print["\", U0]; \n Print["\", U[r, \[Theta]]]; \n Print[\*"\"\<\!\(lim\+\(r \[Rule] 0\)\) \>\"", U[r, \[Theta]], "\< = \>", lim]; \)], "Input"], Cell[TextData[{ "Since this value is dependent on the angle \[Theta] of approach to 0.\n\ Therefore, u(x,y) = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(x\ y\), FontSize->14], StyleBox[ RowBox[{ FormBox[\(x\^2\), "TraditionalForm"], " ", "+", " ", FormBox[\(y\^2\), "TraditionalForm"]}], FontSize->14]], TraditionalForm]], FontSize->16], " does ", StyleBox["NOT", FontColor->RGBColor[1, 0, 1]], " have a limit as (x,y) approaches (0,0)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example on page 56.", FontWeight->"Bold"], " Find the limit of f(z) = ", Cell[BoxData[ FormBox[ StyleBox[\(z\^2\), FontSize->14], TraditionalForm]]], " - 2z + 1 as z \[Rule] 1 + i." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, lim, z]; \nf[z_]\ = \ z\^2\ - \ 2 z\ + \ 1; \n lim\ = \ Limit[f[z], z \[Rule] 1 + \[ImaginaryI]]; \n Print["\", f[z]]; \n Print[\*"\"\<\!\(lim\+\(z \[Rule] 1 + \[ImaginaryI]\)\) \>\"", f[z], "\< = \>", lim]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.13, Page 57.", FontWeight->"Bold"], " Show that the polynomial function given by\n w = P(z) = ", Cell[BoxData[ FormBox[ StyleBox[\(a\_0\), FontSize->14], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ StyleBox[\(a\_1\), FontSize->14], TraditionalForm]]], "z + ", Cell[BoxData[ FormBox[ StyleBox[\(a\_2\), FontSize->14], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[\(z\^2\), FontSize->14], TraditionalForm]]], " + ... + ", Cell[BoxData[ FormBox[ StyleBox[\(a\_n\), FontSize->14], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[\(z\^n\), FontSize->14], TraditionalForm]]], "\nis continuous at each point ", Cell[BoxData[ FormBox[ StyleBox[\(z\_0\), FontSize->14], TraditionalForm]]], " in the complex plane." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["For illustration, ", FontColor->RGBColor[1, 0, 1]], "we use n = 5." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[a, lim, P, z, z0]; \nP[z_]\ = \ Sum[a[n] z\^n, \ {n, 0, 5}]; \n lim\ = \ Limit[P[z], \ z \[Rule] z0]; \nPrint["\< P[z] = \>", P[z]]; \nPrint["\< \>"]; \nPrint[\*"\"\< P[\!\(z\_0\)] = \>\"", P[z0]]; \n Print[\*"\"\<\!\(lim\+\(z \[Rule] z\_0\)\) P[z] = \>\"", lim]; \n Print[\*"\"\< Is P[\!\(z\_0\)] = \!\(lim\+\(z \[Rule] z\_0\)\) P[z] ? \>\ \"", lim\ \ == \ \ P[z0]]; \)], "Input", AspectRatioFixed->True], Cell["Therefore P(z) is continuous.", "Text"], Cell[TextData[{ "\n", StyleBox["Example 2.14, Page 58.", FontWeight->"Bold"], " \nFind the limit of f(z) = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[ RowBox[{ FormBox[\(z\^2\), "TraditionalForm"], " ", "-", " ", \(2 i\)}], FontSize->14], StyleBox[ RowBox[{ FormBox[\(z\^2\), "TraditionalForm"], " ", "-", " ", \(2 z\), " ", "+", " ", "2"}], FontSize->14]], TraditionalForm]]], " as z \[Rule] 1 + i." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, g, lim, z]; \n f[z_]\ = \ \(z\^2\ - \ 2 \[ImaginaryI]\)\/\(z\^2\ - \ 2 z\ + \ 2\); \nPrint["\", f[z]]; \)], "Input", AspectRatioFixed->True], Cell["Try to evaluate f(1+i).", "Text"], Cell[BoxData[ \(\(Print["\", f[1 + \[ImaginaryI]]]; \)\)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(lim\ = \ Limit[f[z], z \[Rule] 1 + \[ImaginaryI]]; \n Print[\*"\"\<\!\(lim\+\(z \[Rule] 1 + \[ImaginaryI]\)\) \>\"", f[z], "\< = \>", lim]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(g[z_]\ = \ Factor[f[z]]; \nPrint["\< f[z] = \>", f[z]]; \n Print["\", g[z]]; \n Print["\< g[1+\[ImaginaryI]] = \>", g[1 + \[ImaginaryI]]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 2", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Section 2.5 ", StyleBox["Branches of Functions", FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18, CellTags->"Section 2.5"], Cell[TextData[{ " In Section 2.3 we defined the principal square root function and \ investigated some of its properties. We now investigate the ", StyleBox["branches", FontColor->RGBColor[1, 0, 1]], " of the square root function." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "First load the ComplexMap package. Make sure this is done only ONCE in \ any ", StyleBox["Mathematica", FontSlant->"Italic"], " session" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Get["\"]\)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example 2.15, Page 60.", FontWeight->"Bold"], " Find the image of the plane cut along the \nnegative x-axis under the \ mappings \nw = ", Cell[BoxData[ FormBox[ StyleBox[\(f\_1\), FontSize->14], TraditionalForm]]], "(z) = ", Cell[BoxData[ FormBox[ RowBox[{" ", StyleBox[\(z\^\(1/2\)\), FontSize->14]}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[\(r\^\(1/2\)\), FontSize->14], TraditionalForm]]], " (cos(", Cell[BoxData[ FormBox[ StyleBox[\(\[Theta]\/2\), FontSize->14], TraditionalForm]]], ") + i sin(", Cell[BoxData[ FormBox[ StyleBox[\(\[Theta]\/2\), FontSize->14], TraditionalForm]]], ")), and\nw = ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontSize->14], "2"], TraditionalForm]]], "(z) = - ", Cell[BoxData[ FormBox[ StyleBox[\(z\^\(1/2\)\), FontSize->14], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[\(r\^\(1/2\)\), FontSize->14], TraditionalForm]]], " (cos(", Cell[BoxData[ FormBox[ StyleBox[\(\(\[Theta] + 4 \[Pi]\)\/2\), FontSize->14], TraditionalForm]]], ") + i sin(", Cell[BoxData[ FormBox[ StyleBox[\(\(\[Theta] + 4 \[Pi]\)\/2\), FontSize->14], TraditionalForm]]], "))." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, wplane, z, zplane]; \nIden[z_]\ = \ z; \n f[z_]\ = \ \@z; \n zplane\ = PolarMap[Iden, {0, 4, 0.5}, {\(-3.14\), \ 3.14, \ 3.14\/12}, PlotRange \[Rule] {{\(-4\), 4.4}, {\(-4\), 4.4}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-4\), 4, 2], Range[\(-4\), 4, 2]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n wplane\ = \ PolarMap[f\ , {0, 4, 0.5}, {\(-3.14\), \ 3.14, \ 3.14\/12}, PlotRange \[Rule] {{\(-0.2\), 2.2}, {\(-2.2\), 2.2}}, AspectRatio \[Rule] 2, Ticks \[Rule] {Range[0, 2, 1], Range[\(-2\), 2, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \n Print["\< The mapping w = \>", f[z]]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, wplane, z, zplane]; \nIden[z_]\ = \ z; \n f[z_]\ = \ \(-\@z\); \n zplane\ = \ PolarMap[Iden, {0, 4, 0.5}, {\(-3.14\), \ 3.14, \ 3.14\/12}, PlotRange \[Rule] {{\(-4\), 4.4}, {\(-4\), 4.4}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-4\), 4, 2], Range[\(-4\), 4, 2]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n wplane\ = \ PolarMap[f\ , {0, 4, 0.5}, {\(-3.14\), \ 3.14, \ 3.14\/12}, PlotRange \[Rule] {{\(-2.2\), 0.2}, {\(-2.2\), 2.2}}, AspectRatio \[Rule] 2, Ticks \[Rule] {Range[\(-2\), 0, 1], Range[\(-2\), 2, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \n Print["\< The mapping w = \>", f[z]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["\nExample 2.16, Page 61.", FontWeight->"Bold"], " Show that the function ", Cell[BoxData[ FormBox[ StyleBox[\(f\_1\), FontSize->14], TraditionalForm]]], "(z) is discontinuous along the negative real axis,\nwhere w = ", Cell[BoxData[ FormBox[ StyleBox[\(f\_1\), FontSize->14], TraditionalForm]]], "(z) = ", Cell[BoxData[ FormBox[ RowBox[{" ", StyleBox[\(z\^\(1/2\)\), FontSize->14]}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ StyleBox[\(r\^\(1/2\)\), FontSize->14], TraditionalForm]]], " (cos(", Cell[BoxData[ FormBox[ StyleBox[\(\[Theta]\/2\), FontSize->14], TraditionalForm]]], ") + i sin(", Cell[BoxData[ FormBox[ StyleBox[\(\[Theta]\/2\), FontSize->14], TraditionalForm]]], "))." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Clear[f, r, \[Theta]]; \n f[r_, \[Theta]_]\ = \ \(\@r\) \((Cos[\[Theta]\/2] + \[ImaginaryI]\ Sin[\[Theta]\/2])\); \n lim1\ = PowerExpand[Limit[f[r, \[Theta]], \[Theta] \[Rule] \[Pi]]]; \n lim2\ = PowerExpand[Limit[f[r, \[Theta]], \[Theta] \[Rule] \(-\[Pi]\)]]; \nPrint["\", f[r, \[Theta]]]\), \(Print[\*"\"\<\!\(lim\+\(\[Theta] \[Rule] \[Pi]\)\) f[z] = \>\"", lim1]; \nPrint[\*"\"\<\!\(lim\+\(\[Theta] \[Rule] \(-\[Pi]\)\)\) f[z] = \>\"", lim2]; \)}], "Input"], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 2", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Section 2.6 ", FontSize->18], StyleBox["The Reciprocal Transformation w = ", FontSize->18, FontColor->RGBColor[1, 0, 1]], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox["z", FontSize->14]], FontColor->RGBColor[1, 0, 1]], TraditionalForm]]] }], "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"Section 2.6"], Cell[TextData[{ " The ", StyleBox["reciprocal transformation", FontColor->RGBColor[1, 0, 1]], " is: ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", RowBox[{\(f(z)\), "=", FractionBox[ StyleBox["1", FontSize->14], StyleBox["z", FontSize->14]]}]}], TraditionalForm]]], ". It maps the \"extended complex z-plane\" one-to-one and onto the \ \"extended complex w-plane\".\n\n It is convenient to extend the system \ of complex numbers by joining to it an \"ideal\" point denoted by \ \[Infinity] and called the ", StyleBox["point at infinity", FontColor->RGBColor[1, 0, 1]], ". This new set is called the extended complex plane." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "First load the ComplexMap package. Make sure this is done only ONCE in \ any ", StyleBox["Mathematica", FontSlant->"Italic"], " session." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Get["\"]\)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Example 2.17, Page 67.", FontWeight->"Bold"], " Show that the image of the right half plane Re(z) > ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox["2", FontSize->14]], FontSize->14], TraditionalForm]]], "\nunder the reciprocal transformation w = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox["z", FontSize->14]], TraditionalForm]]], " is the disk |w - 1| < 1." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, ineq1, ineq2, ineq3, ineq4, wplane, z, zplane]; \n Print[ineq1\ = \ x\ > \ 1\/2]; \n Print[ineq2\ = \ ReplaceAll[ineq1 \ , {x \[Rule] u\/\(u\^2\ + \ v\^2\), y \[Rule] \(-v\)\/\(u\^2\ + \ v\^2\)}]]; \n Print[ineq3\ = \ Distribute[\ 2 \((u\^2\ + \ v\^2)\)\ ineq2, \ Greater]]; \n Print[ineq4\ = \ Distribute[\ 1\ - \ 2 u\ + \ ineq3, \ Greater]]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, wplane, z, zplane]; \nIden[z_]\ = \ z; \n f[z_]\ = \ 1\/z; \n zplane\ = \ CartesianMap[Iden, \ {0.5, 10, 0.5}, {\(-5\), 5, 0.25}, PlotRange \[Rule] {{\(-0.1\), 10}, {\(-5\), 5}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[0, 10, 2], Range[\(-5\), 5, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \ \n wplane\ = \ CartesianMap[f, \ {0.5, 10, 0.5}, {\(-5\), 5, 0.25}, PlotRange \[Rule] {{\(-0.1\), 2.1}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[0, 2, 1], Range[\(-2\), 2, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \ \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \n Print["\< The mapping w = \>", f[z]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.18, Page 67.", FontWeight->"Bold"], " Find the image of the portion of the right half plane\nRe(z) > ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox["2", FontSize->14]], TraditionalForm]]], " that lies inside the circle |z - ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox["2", FontSize->14]], TraditionalForm]]], "| < 1 under the transformation w = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox["2", FontSize->14]], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, ineq1, ineq2, ineq3, ineq4, ineq5, ineq6, ineq7, ineq8, ineq9, wplane, z, zplane]; \nPrint[ineq1\ = \ x\ > \ 1\/2]; \n Print[ineq2\ = ReplaceAll[ineq1 \ , {x \[Rule] u\/\(u\^2\ + \ v\^2\), y \[Rule] \(-v\)\/\(u\^2\ + \ v\^2\)}]]; \n Print[ineq3\ = \ Distribute[\ 2 \((u\^2\ + \ v\^2)\)\ ineq2, \ Greater]]; \n Print[ineq4\ = \ Distribute[\ 1\ - \ 2 u\ + \ ineq3, \ Greater]]; \)], "Input", AspectRatioFixed->True], Cell["Which is the disk |w - 1| < 1.", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Print[ineq1\ = \ x\^2\ - \ x\ + \ 1\/4\ + \ y\^2\ < \ 1]; \n Print[ineq2\ = \ ReplaceAll[ineq1 \ , {x \[Rule] u\/\(u\^2\ + \ v\^2\), y \[Rule] \(-v\)\/\(u\^2\ + \ v\^2\)}]]; \n Print[ineq3\ = \ MapAll[Together, ineq2]]; \n Print[ineq4\ = \ Distribute[\ 4 \((u\^2\ + \ v\^2)\)\ ineq3, \ Less]]; \nPrint[ineq5\ = \ MapAll[Expand, ineq4]]; \n Print[ineq6\ = \ Distribute[\ 4 u\ - u\^2\ - \ v\^2\ + \ ineq5, \ Less]]; \n Print[ineq7\ = \ Distribute[\ 1\/3\ ineq6, \ Less]]; \n Print[ineq8\ = \ MapAll[Expand, ineq7]]; \n Print[ineq9\ = \ Distribute[\ \((2\/3)\)\^2\ + ineq8, \ Less]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "Which is the exterior of the circle |w + ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["2", FontSize->14], StyleBox["3", FontSize->14]], TraditionalForm]]], "| = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["4", FontSize->14], StyleBox["3", FontSize->14]], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Print[ineq4]; \nPrint[ineq9]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "The image is the crescent shaped region in the w-plane which is the\n\ portion of the disk |w - 1| < 1 that lies outside the circle |w + ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["2", FontSize->14], StyleBox["3", FontSize->14]], TraditionalForm]]], "| = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["4", FontSize->14], StyleBox["3", FontSize->14]], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[f, Iden, wplane, W, z, zplane, Z]; \nIden[z_]\ = \ z; \n f[z_]\ = \ 1\/z; \nZ[z_]\ = \ Iden[z + 1\/2]; \n W[z_]\ = \ f[z + 1\/2]; \n zplane\ = \ PolarMap[Z\ , {0, 1, 0.1}, {\(-\[Pi]\)\/2, \[Pi]\/2, \[Pi]\/12}, PlotRange \[Rule] {{\(-0.1\), 2.1}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-6\), 4, 1], Range[\(-2\), 8, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n wplane\ = \ PolarMap[W\ , {0, 1, 0.1}, {\(-\[Pi]\)\/2, \[Pi]\/2, \[Pi]\/12}, PlotRange \[Rule] {{\(-0.1\), 2.1}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[0, 2, 1], Range[\(-1\), 1, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \n Print["\< The mapping w = \>", f[z]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 2.19, Page 69.", FontWeight->"Bold"], " Find the images of the vertical lines x = a and the\nhorizontal lines \ y = b under the mapping w = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox["z", FontSize->14]], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Clear[a, ac, al, b, bc, bl, circlea, circleb, hc, hl, horiz, x, y, u, v, vc, verti, vl, wplane, zplane]; \nPrint[eq1\ = \ x\ == \ a]; \n Print[eq2\ = \ ReplaceAll[eq1 \ , {x \[Rule] u\/\(u\^2\ + \ v\^2\), y \[Rule] \(-v\)\/\(u\^2\ + \ v\^2\)}]]; \n Print[eq3\ = \ Distribute[\ \((u\^2\ + \ v\^2)\)\/a\ \ eq2, \ Equal]]; \nPrint[eq4\ = \ Distribute[\ \(-\ u\)\/a\ + \ eq3, \ Equal]]; \n Print[eq5\ = \ Distribute[\ 1\/\((2 a)\)\^2\ + \ eq4, \ Equal]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "The image is the circle |w - ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox[\(2 a\), FontSize->14]], TraditionalForm]]], "| = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox[\(|2 a | \), FontSize->14]], TraditionalForm]]], ",\nwith center ", Cell[BoxData[ \(TraditionalForm\`w\_0\)], FontSize->14], " = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox[\(2 a\), FontSize->14]], TraditionalForm]]], " and radius ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], RowBox[{ StyleBox["|", FontSize->14], RowBox[{ StyleBox[\(2 a\), FontSize->14], "|"}]}]], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Print[eq1\ = \ y\ == \ b]; \n Print[eq2\ = \ ReplaceAll[eq1 \ , {x \[Rule] u\/\(u\^2\ + \ v\^2\), y \[Rule] \(-v\)\/\(u\^2\ + \ v\^2\)}]]; \n Print[eq3\ = \ Distribute[\ \((u\^2\ + \ v\^2)\)\/b\ \ eq2, \ Equal]]; \nPrint[eq4\ = \ Distribute[\ v\/b\ + \ eq3, \ Equal]]; \n Print[eq5\ = \ Distribute[\(\ 1\)\/\((2 b)\)\^2\ + \ eq4, \ Equal]]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ "The image is the circle |w +", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(\ i\), FontSize->14], StyleBox[\(2 b\), FontSize->14]], TraditionalForm]]], "| = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox[\(|2 b | \), FontSize->14]], TraditionalForm]]], ",\nwith center ", Cell[BoxData[ FormBox[ StyleBox[\(w\_0\), FontSize->14], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox[\(\ \(-i\)\), FontSize->14], StyleBox[\(2 b\), FontSize->14]], TraditionalForm]]], " and radius ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["1", FontSize->14], StyleBox[\(|2 b | \), FontSize->14]], TraditionalForm]]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Now make some horizontal and vertical lines:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(hl[b_]\ = \ Line[{{\(-2\), b}, {2, b}}]; \n horiz\ = \ {hl[\(-1\)], hl[\(-1\)\/2], hl[1\/2], hl[1]}; \n bl\ = \ Graphics[horiz]; \nvl[a_]\ = \ Line[{{a, \(-2\)}, {a, 2}}]; \n verti\ = \ {vl[\(-1\)], vl[\(-1\)\/2], vl[1\/2], vl[1]}; \n al\ = \ Graphics[verti]; \)], "Input", AspectRatioFixed->True], Cell["Now make some images of horizontal and vertical lines:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(hc[b_]\ = \ Circle[{0, \(-1\)\/\(2 b\)}, 1\/Abs[2 b]]; \n circleb\ = \ {hc[\(-1\)], hc[\(-1\)\/2], hc[1\/2], hc[1]}; \n bc\ = \ Graphics[circleb]; \n vc[a_]\ = \ Circle[{1\/\(2 a\), 0}, 1\/Abs[2 a]]; \n circlea\ = \ {vc[\(-1\)], vc[\(-1\)\/2], vc[1\/2], vc[1]}; \n ac\ = \ Graphics[circlea]; \)], "Input", AspectRatioFixed->True], Cell[BoxData[ \(zplane\ = \ Show[al, bl, PlotRange \[Rule] {{\(-2.2\), 2.2}, {\(-2.2\), 2.2}}, AspectRatio \[Rule] 1, Axes \[Rule] True, Ticks \[Rule] {Range[\(-4\), 4, 1], Range[\(-4\), 4, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n wplane\ = \ Show[ac, bc, PlotRange \[Rule] {{\(-2.2\), 2.2}, {\(-2.2\), 2.2}}, AspectRatio \[Rule] 1, Axes \[Rule] True, Ticks \[Rule] {Range[\(-4\), 4, 1], Range[\(-4\), 4, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] Identity]; \n Show[GraphicsArray[{zplane, wplane}], DisplayFunction \[Rule] $DisplayFunction]; \ \n Print[\*"\"\< The mapping w = \!\(1\/z\)\>\""]; \)], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["GoTo ", FontSize->16, FontColor->RGBColor[0, 1, 0]], ButtonBox["Chapter 2", ButtonData:>"CHAPTER", ButtonStyle->"Hyperlink"] }], "Text"] }, Closed]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 640}, {0, 452}}, AutoGeneratedPackage->None, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{502, 253}, WindowMargins->{{0, Automatic}, {Automatic, 5}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. Make modifications to any definition using \ commands in the Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], PageWidth->WindowWidth, ScriptMinSize->9], Cell[StyleData[All, "Presentation"], PageWidth->WindowWidth, ScriptMinSize->12, FontSize->16], Cell[StyleData[All, "Condensed"], PageWidth->WindowWidth, CellBracketOptions->{"Margins"->{1, 1}, "Widths"->{0, 5}}, ScriptMinSize->8, FontSize->11], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ScriptMinSize->5, FontSize->10, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the \ Notebook level.\ \>", "Text"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, CellFrameLabelMargins->6, StyleMenuListing->None] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellMargins->{{12, Inherited}, {20, 40}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subtitle", 0}, {"Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->36, FontWeight->"Bold"], Cell[StyleData["Title", "Presentation"], CellMargins->{{24, 10}, {20, 40}}, LineSpacing->{1, 0}, FontSize->44], Cell[StyleData["Title", "Condensed"], CellMargins->{{8, 10}, {4, 8}}, FontSize->20], Cell[StyleData["Title", "Printout"], CellMargins->{{2, 10}, {15, 30}}, FontSize->24] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{12, Inherited}, {10, 15}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}, { "Subsubtitle", 0}}, FontFamily->"Helvetica", FontSize->24], Cell[StyleData["Subtitle", "Presentation"], CellMargins->{{24, 10}, {15, 20}}, LineSpacing->{1, 0}, FontSize->36], Cell[StyleData["Subtitle", "Condensed"], CellMargins->{{8, 10}, {4, 4}}, FontSize->14], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{2, 10}, {10, 15}}, FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubtitle"], CellMargins->{{12, Inherited}, {10, 20}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, CounterIncrements->"Subsubtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontFamily->"Helvetica", FontSize->14, FontSlant->"Italic"], Cell[StyleData["Subsubtitle", "Presentation"], CellMargins->{{24, 10}, {10, 20}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Subsubtitle", "Condensed"], CellMargins->{{8, 10}, {8, 12}}, FontSize->12], Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {8, 10}}, 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"], 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"], 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"], 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"], 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.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{45, 10}, {5, 7}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], 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", CellHorizontalScrolling->True, 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", FontColor->RGBColor[0, 0, 1]], 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", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], 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"], ImageSize->{175, 175}, ImageMargins->{{38, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Printout"], ImageSize->{250, 250}, ImageMargins->{{30, Inherited}, {Inherited, 0}}, FontSize->9] }, 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, ScriptLevel->0, SingleLetterItalics->True, StyleMenuListing->None, 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[ 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