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If ", Cell[BoxData[ \(f' \((z\_0)\) \[NotEqual] 0\)]], ", then we can express f in the form \n\n\t", Cell[BoxData[ \(f \((z)\) = f \((z\_0)\) + f' \((z\_0)\) \((z - z\_0)\) + \[Eta] \((z)\) \((z - z\_0)\)\)]], ", where ", Cell[BoxData[ \(\[Eta] \((z)\) \[Rule] \(0\ \ as\ \ z \[Rule] z\_0\)\)]], ". \n\nIf z is near ", Cell[BoxData[ \(z\_0\)]], ", then the transformation ", Cell[BoxData[ \(w = f \((z)\)\)]], " has the ", StyleBox["linear approximation", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " \n\n\t", Cell[BoxData[ \(S \((z)\) = A + B \((z - z\_0)\)\)]], ", where ", Cell[BoxData[ \(A = \(f \((z\_0)\)\ \ and\ \ B = f' \((z\_0)\)\)\)]], ". \n\nSince ", Cell[BoxData[ \(\[Eta] \((z)\) \[Rule] 0\)]], " when ", Cell[BoxData[ \(z \[Rule] z\_0\)]], ", it is reasonable that for points near ", Cell[BoxData[ \(z\_0\)]], " the transformation ", Cell[BoxData[ \(w = f \((z)\)\)]], " has an effect much like the linear mapping ", Cell[BoxData[ \(w = S \((z)\)\)]], ". The effect of the linear mapping S is a rotation of the plane through \ the angle ", Cell[BoxData[ \(\[Alpha]\ = \ arg[f' \((z\_0)\)]\)]], ", followed by a magnification by the factor ", Cell[BoxData[ \(\(\(|\)\(f' \((z\_0)\)\)\(|\)\)\)]], ", followed by a rigid translation by the vector ", Cell[BoxData[ \(A + Bz\_0\)]], ". Consequently, the mapping ", Cell[BoxData[ \(w = S \((z)\)\)]], " preserves angles at the point ", Cell[BoxData[ \(z\_0\)]], ". We now show that the mapping ", Cell[BoxData[ \(w = f \((z)\)\)]], " also preserves angles at ", Cell[BoxData[ \(z\_0\)]], ".\n\n\tA mapping ", Cell[BoxData[ \(w = f \((z)\)\)]], " is said to be angle preserving, or ", ButtonBox["Conformal", ButtonData:>{ URL[ "http://mathworld.wolfram.com/ConformalTransformation.html"], None}, ButtonStyle->"Hyperlink"], " ", StyleBox[" at ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(z\_0\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", if it preserves angles between oriented curves in magnitude as well as \ in orientation. Also, we will learn that if f is a conformal mapping then \ orthogonal curves are mapped onto orthogonal curves. 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" }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[f, Iden, wplane, z, zplane];\)\ \), "\n", \(\(Iden[z_] = z;\)\ \), "\n", \(\(f[z_] = Cos[z];\)\ \), "\[IndentingNewLine]", \(\(zdot = Graphics[{Magenta, PointSize[0.03], Point[{Re[Z], Im[Z]}]}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(wdot = Graphics[{Magenta, PointSize[0.03], Point[{Re[f[Z]], Im[f[Z]]}]}, DisplayFunction \[Rule] Identity];\)\ \), "\n", \(\(zplane = CartesianMap[ Iden, {0.6, 1.4, 0.1}, {\(-1.4\), \(-0.6\), 0.1}, \[IndentingNewLine]PlotRange \[Rule] {{0, 1.5}, {\(-1.5\), 0.0}}, AspectRatio \[Rule] 1, \[IndentingNewLine]Ticks \[Rule] {Range[0, 2.0, 0.5], Range[\(-2.0\), 0.0, 0.5]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Blue, Cyan}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(wplane = CartesianMap[ f, {0.6, 1.4, 0.1}, {\(-1.4\), \(-0.6\), 0.1}, \[IndentingNewLine]PlotRange \[Rule] {{0, 1.88}, {0.0, 1.88}}, AspectRatio \[Rule] 1, \[IndentingNewLine]Ticks \[Rule] {Range[0, 2, 0.5], Range[0, 2, 0.5]}, \[IndentingNewLine]AxesLabel \[Rule] {"\", \ "\"}, PlotStyle \[Rule] {Blue, Cyan}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(Show[zplane, zdot, PlotRange \[Rule] {{0, 1.5}, {\(-1.5\), 0.0}}, AspectRatio \[Rule] 1, \[IndentingNewLine]Ticks \[Rule] {Range[0, 2.0, 0.5], Range[\(-2.0\), 0.0, 0.5]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\ \[IndentingNewLine]", \(\(Print["\", Z];\)\ \), "\n", \(\(Show[wplane, wdot, PlotRange \[Rule] {{0, 1.88}, {0.0, 1.88}}, AspectRatio \[Rule] 1, \[IndentingNewLine]Ticks \[Rule] {Range[0, 2, 0.5], Range[0, 2, 0.5]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\n", \(\(Print["\", \ f[Z], "\<,\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\", f[z], "\<,\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\", Z, "\<) = \>", f[Z], "\< = \>", N[f[Z]]];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\", N[Arg[\(f'\)[Z]]]];\)\ \), "\n", \(\(Print["\< Scale = \>", N[Abs[\(f'\)[Z]]]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ StyleBox["\n(c)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Consider the mapping w = cos(z) near ", Cell[BoxData[ \(z = \[Pi] + \[ImaginaryI]\)], AspectRatioFixed->True], "." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Z\ = \ \[Pi] + \[ImaginaryI];\)\ \ \ \ \), "\n", \(\(Print["\< f(z) = \>", f[z]];\)\ \), "\n", \(\(Print["\< f'(z) = \>", \(f'\)[z]];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print["\< At z = \>", Z];\)\ \), "\[IndentingNewLine]", \(\(Print["\< f(\>", Z, "\<) = \>", f[Z]];\)\ \), "\[IndentingNewLine]", \(\(Print["\< f(\>", Z, "\<) = \>", N[f[Z]]];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print["\< f'(\>", Z, "\<) = \>", \(f'\)[Z], "\< = \>", N[\(f'\)[Z]]];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print["\", Arg[\(f'\)[Z]]];\)\ \), "\[IndentingNewLine]", \(\(Print["\", N[Arg[\(f'\)[Z]]]];\)\ \), "\n", \(\(Print["\< Scale = |\>", \(f'\)[Z], "\<| = \>", Abs[\(f'\)[Z]]];\)\ \), "\[IndentingNewLine]", \(\(Print["\< Scale = |\>", \(f'\)[Z], "\<| = \>", N[Abs[\(f'\)[Z]]]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nThe image of a small rectangle about ", Cell[BoxData[ \(z = \[Pi] + \[ImaginaryI]\)], AspectRatioFixed->True], " is a curvilinear region about w = - cosh(1). " }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[f, Iden, wplane, z, zplane];\)\ \), "\n", \(\(Iden[z_] = z;\)\ \), "\n", \(\(f[z_] = Cos[z];\)\ \), "\[IndentingNewLine]", \(\(zdot = Graphics[{Red, PointSize[0.03], Point[{Re[Z], Im[Z]}]}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(wdot = Graphics[{Red, PointSize[0.03], Point[{Re[f[Z]], Im[f[Z]]}]}, DisplayFunction \[Rule] Identity];\)\ \), "\n", \(\(zplane = CartesianMap[ Iden, {\[Pi] - 0.5, \[Pi] + 0.5, 0.125}, {0.5, 1.5, 0.125}, \[IndentingNewLine]PlotRange \[Rule] {{0, 4}, {0, 2}}, AspectRatio \[Rule] 1\/2, \[IndentingNewLine]Ticks \[Rule] {Range[0, 4, 1], Range[0, 2, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {DarkGreen, Green}, DisplayFunction \[Rule] Identity];\)\ \ \), "\[IndentingNewLine]", \(\(wplane = CartesianMap[ f, {\[Pi] - 0.5, \[Pi] + 0.5, 0.125}, {0.5, 1.5, 0.125}, \[IndentingNewLine]PlotRange \[Rule] {{\(-3.2\), 0}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 1, \[IndentingNewLine]Ticks \[Rule] {Range[\(-3\), 0, 1], Range[\(-1\), 1, 0.5]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {DarkGreen, Green}, DisplayFunction \[Rule] Identity];\)\ \ \), "\[IndentingNewLine]", \(\(Show[zplane, zdot, PlotRange \[Rule] {{0, 4}, {0, 2}}, AspectRatio \[Rule] 1\/2, \[IndentingNewLine]Ticks \[Rule] {Range[0, 4, 1], Range[0, 2, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\ \[IndentingNewLine]", \(\(Print["\", Z];\)\ \), "\n", \(\(Show[wplane, wdot, PlotRange \[Rule] {{\(-3.2\), 0}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 1, \[IndentingNewLine]Ticks \[Rule] {Range[\(-3\), 0, 1], Range[\(-1\), 1, 0.5]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\n", \(\(Print["\", \ f[Z], "\<,\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\", f[z], "\<,\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\", Z, "\<) = \>", f[Z], "\< = \>", N[f[Z]]];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\", Arg[\(f'\)[Z]]];\)\ \), "\n", \(\(Print["\< Scale = \>", Abs[\(f'\)[Z]]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"] }, Closed]], Cell[TextData[{ "\n\tLet f be a nonconstant analytic function. If ", Cell[BoxData[ \(f' \((z\_0)\) = 0\)]], ", then ", Cell[BoxData[ \(z\_0\)]], " is called a ", StyleBox["critical point", FontSlant->"Italic"], " of f, and the mapping w=f(z) is not conformal at ", Cell[BoxData[ \(z\_0\)]], ". The next result shows what happens at a critical point." }], "Text"], Cell[TextData[{ StyleBox["\nTheorem 9.2 (", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ButtonBox["Nonconformal Map", ButtonData:>{ URL[ "http://mathworld.wolfram.com/NonconformalMap.html"], None}, ButtonStyle->"Hyperlink"], StyleBox["), Page 358.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " Let f(z) be analytic at ", Cell[BoxData[ \(z\_0\)]], ". If ", Cell[BoxData[ \(f' \((z\_0)\) = 0, f'' \((z\_0)\) = 0, \ ... , \(f\^\((k - 1)\)\) \((z\_0)\) = 0\)]], " ", Cell[BoxData[ SuperscriptBox["", TagBox["", Derivative], MultilineFunction->None]]], "and \n", Cell[BoxData[ \(\(f\^\((k)\)\) \((z\_0)\) \[NotEqual] 0\)]], ", then the mapping w = f(z) magnifies angles at the vertex ", Cell[BoxData[ \(z\_0\)]], " by the factor k." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["Proof of Theorem 9.2, see text Page 358.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 9.2, Page 358.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Graph the transformation ", Cell[BoxData[ \(w = \(f \((z)\) = z\^2\)\)], AspectRatioFixed->True], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 9.2.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[{ "Enter the function ", Cell[BoxData[ \(f \((z)\) = z\^2\)], AspectRatioFixed->True], " and graph the mapping." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[f, Iden, z];\)\ \), "\n", \(\(Iden[z_]\ = \ z;\)\ \), "\n", \(\(f[z_]\ = \ z\^2;\)\ \), "\n", \(\(CartesianMap[ Iden, {0, 1, 0.1}, {0, 1, 0.1}, \[IndentingNewLine]Ticks \[Rule] {Range[0, 1, 0.5], Range[0, 1, 0.5]}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Pink}];\)\ \ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\n", \(\(CartesianMap[\ f\ \ , {0, 1, 0.1}, {0, 1, 0.1}, \[IndentingNewLine]Ticks \[Rule] {Range[\(-1\), 1, 1], Range[0, 2, 1]}, \[IndentingNewLine]AspectRatio \[Rule] 1, PlotRange \[Rule] {{\(-1\), 1}, {0, 2}}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Pink}];\)\ \ \), "\[IndentingNewLine]", \(\(Print["\", \ u \[Equal] a\^2 - v\^2\/\(4 a\^2\), "\< and \>", u \[Equal] \(-b\^2\) + v\^2\/\(4\ b\^2\)];\)\ \), "\n", \(\(Print["\", f[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[TextData[{ "\n\tAnother property of a conformal mapping ", Cell[BoxData[ \(w = f \((z)\)\)]], " is obtained by considering the modulus of ", Cell[BoxData[ \(f' \((z\_0)\)\)]], ". If ", Cell[BoxData[ \(z\_1\)]], " is near ", Cell[BoxData[ \(z\_0\)]], ", then in the equation ", Cell[BoxData[ \(f \((z\_1)\) = f \((z\_0)\) + f' \((z\_0)\) \((z\_1 - z\_0)\) + \[Eta] \((z\_1)\) \((z\_1 - z\_0)\)\)]], " we can neglect the term ", Cell[BoxData[ \(\[Eta] \((z\_1)\) \((z\_1 - z\_0)\)\)]], ", and obtain the approximation \n\n\t", Cell[BoxData[ RowBox[{\(w\_1 - w\_0\), "=", RowBox[{\(f \((z\_1)\) - f \((z\_0)\)\), StyleBox["\[TildeTilde]", FontSize->14], " ", \(f' \((z\_0)\) \((z\_1 - z\_0)\)\)}]}]]], ". \n\nUsing this relation, we see that the distance ", Cell[BoxData[ \(\(\(|\)\(w\_1 - w\_0\)\(|\)\)\)]], " between the images of the points ", Cell[BoxData[ \(z\_1\)]], " and ", Cell[BoxData[ \(z\_0\)]], " is given approximately by ", Cell[BoxData[ \(\(\(|\)\(f' \((z\_0)\) || z\_1 - z\_0\)\(|\)\)\)]], ". Therefore we say that the transformation ", Cell[BoxData[ \(w = f \((z)\)\)]], " changes small distances near ", Cell[BoxData[ \(z\_0\)]], " by the ", StyleBox["scale factor", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ \(\(\(|\)\(f' \((z\_0)\)\)\(|\)\)\)]], ". \n" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Library Research Experience for Undergraduates", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.996109]]], "Text"], Cell[TextData[{ StyleBox["Project I. Write a report on conformal mapping.", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.996109, 0, 0.996109]], "\n\n", StyleBox["1.", FontWeight->"Bold"], " Bruch, John C. and Roger C. Wood (1972), ''Teaching Complex Variables \ with an Interactive Computer System,'' IEEE Trans. on Ed., E-15, No. 1, pp. \ 73-80.\n\n", StyleBox["2.", FontWeight->"Bold"], " Bruch, John C., (1975), ''The Use of Interactive Computer Graphics in the \ Conformal Mapping Area,'' Computers and Graphics, V. 1, pp. 361-374.\n\n", StyleBox["3.", FontWeight->"Bold"], " D'Angelo, John P.,''Mapping Theorems in Complex Analysis (in Progress \ Reports),'' Am. Math. M., Vol. 91, No. 7. (Aug. - Sep., 1984), pp. 413-414, \ ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["4.", FontWeight->"Bold"], " Frederick, Carl and Eric L. Schwartz, (1990), ''Conformal image \ warping,'' IEEE Computer Graphics and Applications, V. 10, pp. 54-61.\n\n", StyleBox["5.", FontWeight->"Bold"], " Kasner, Edward and John De Cicco, ''Geometry of Scale Curves in Conformal \ Maps,'' American J. of Math., Vol. 67, No. 1. (Jan., 1945), pp. 157-166, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["6.", FontWeight->"Bold"], " Miser, Hugh J. (1942),'' Regions and their 'Patterns' in Conformal \ Mapping,'' Math. Mag., Vol. 16 pp. 333-337.\n\n", StyleBox["7.", FontWeight->"Bold"], " Moppert, C. F., ''Deduction of Cardano's Formula by Conformal Mapping (in \ Math. Notes),''Am. Math. M., Vol. 59, No. 5. (May, 1952), pp. 310-314, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["8.", FontWeight->"Bold"], " Novinger, W. P., ''An Elementary Approach to the Problem of Extending \ Conformal Maps to the Boundary (in Classroom Notes),'' Am. Math. M., Vol. 82, \ No. 3. (Mar., 1975), pp. 279-282, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["9.", FontWeight->"Bold"], " Richardson, S., ''An Identity Arising in a Problem of Conformal Mapping \ (in Classroom Notes),'' SIAM Review, Vol. 31, No. 3. (Sep., 1989), pp. \ 484-485, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["10.", FontWeight->"Bold"], " Walsh, J. L., ''On the Circles of Curvature of the Images of Circles \ under a Conformal Map,'' Am. Math. M., Vol. 46, No. 8. (Oct., 1939), pp. \ 472-485, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["11.", FontWeight->"Bold"], " Warschawski, S. E., ''On Conformal Mapping of Infinite Strips,'' Trans. \ of the Am. Math. Soc., Vol. 51, No. 2. (Mar., 1942), pp. 280-335, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["12.", FontWeight->"Bold"], " Williams, Richard K., ''A Note on Conformality (in Classroom Notes),'' \ Am. Math. M., Vol. 80, No. 3. (Mar., 1973), pp. 299-300, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " " }], "Text"], Cell[TextData[{ StyleBox["Project II. Write a report on numerical conformal mapping.", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.996109, 0, 0.996109]], "\n\n", StyleBox["1.", FontWeight->"Bold"], " Chakravarthy, Sukumar and Dale Anderson, ''Numerical Conformal Mapping,'' \ Math. of Computation, Vol. 33, No. 147. (Jul., 1979), pp. 953-969, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["2.", FontWeight->"Bold"], " Hayes, John K., David K. Kahaner and Richard G. Kellner, ''An Improved \ Method for Numerical Conformal Mapping,'' Math. of Computation, Vol. 26, No. \ 118. (Apr., 1972), pp. 327-334+s1-s28, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["3.", FontWeight->"Bold"], " Mastin, C. Wayne, (1987), ''Numerical conformal mapping,'' Comp. Meth. in \ App. Mech. and Eng., V. 63, pp. 209-211.\n\n", StyleBox["4.", FontWeight->"Bold"], " Wegmann, Rudolf, ''On Fornberg's Numerical Method for Conformal Mapping \ SIAM J. on Numerical Analysis, Vol. 23, No. 6. 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