(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 67586, 1703]*) (*NotebookOutlinePosition[ 91945, 2588]*) (* CellTagsIndexPosition[ 91837, 2581]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["COMPLEX ANALYSIS: ", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["Mathematica 4.1", FontSize->18, FontSlant->"Italic", FontColor->RGBColor[1, 0, 1]], StyleBox[" Notebooks\n(c) John H. Mathews, and\nRussell W. Howell, 2002", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["\n", FontColor->RGBColor[1, 0, 0]], StyleBox["Complimentary software to accompany our textbook:", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]] }], "Text", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[TextData[{ StyleBox["COMPLEX ANALYSIS: for Mathematics and Engineering, \n4th Edition, \ 2001, ISBN: ", FontSize->18, FontColor->RGBColor[0, 0, 1]], StyleBox["0-7637-1425-9", FontFamily->"Times New Roman", FontSize->18, FontColor->RGBColor[0, 0, 0.996109]], StyleBox["\n", FontSize->18, FontColor->RGBColor[0, 1, 1]], StyleBox[ButtonBox["Jones & Bartlett Publishers, Inc.", ButtonData:>{ URL[ "http://www.jbpub.com/"], None}, ButtonStyle->"Hyperlink"], FontSize->18, FontWeight->"Bold"], StyleBox[" ", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], StyleBox[" \n", FontSize->18, FontColor->RGBColor[0, 1, 0]], StyleBox["40 Tall Pine Drive, Sudbury, MA 01776 \nTele. (800) 832-0034, \ FAX: (508) 443-8000 \nE-mail: mkt@jbpub.com \nInternet: ", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox[ButtonBox["http://www.jbpub.com/", ButtonData:>{ URL[ "http://www.jbpub.com/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], StyleBox[" ", FontSize->18, FontColor->RGBColor[1, 0, 1]], StyleBox["\n", FontSize->14], StyleBox["This free software is compliments of the authors.\n", FontSize->14, FontColor->RGBColor[1, 0, 1]], ButtonBox["John H. Mathews", ButtonData:>{ URL[ "http://www.ecs.fullerton.edu/~mathews/"], None}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->14, FontColor->RGBColor[1, 0, 1]], StyleBox["mathews@fullerton.edu \n", FontSize->16, FontColor->RGBColor[1, 0, 1]], ButtonBox["Russell W. Howell", ButtonData:>{ URL[ "http://www.westmont.edu/~howell"], None}, ButtonStyle->"Hyperlink"], StyleBox[", ", FontSize->16, FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontSize->14, FontColor->RGBColor[1, 0, 1]], StyleBox["howell@westmont.edu ", FontSize->16, FontColor->RGBColor[1, 0, 1]] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["CHAPTER 9", FontSize->18, FontColor->RGBColor[0.500008, 0.250004, 0.250004]], StyleBox[" ", FontSize->18], StyleBox["CONFORMAL MAPPING", FontSize->18, FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True, CellTags->"CHAPTER 9"], Cell[BoxData[{ \(\(S = \(T = \(w = \(W = \(x = \(y = \(z = \(Z = 0\)\)\)\)\)\)\);\)\ \), "\n", \(\(Clear[cos, f, F, g, graphs, Iden, S, set, sin, T, tan, w, W, w1, w2, w3, wdot, wdots, wplane, z, z1, z2, z3, zdot, zdots, zplane, Z, Zplane];\)\ \), "\[IndentingNewLine]", \(\(Needs["\"];\)\ \), "\[IndentingNewLine]", \(\(Needs["\"];\)\ \)}], "Input", InitializationCell->True] }, Closed]], Cell[TextData[{ StyleBox["Section 9", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox[".1", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox["\t", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[ButtonBox["Basic Properties of Conformal Mappings", ButtonData:>{"ca0901.nb", None}, ButtonStyle->"Hyperlink"], FontSize->14, FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Section 9.2", FontColor->RGBColor[0.500008, 0.250004, 0.250004]], " ", StyleBox["Bilinear Transformations", FontColor->RGBColor[0.500008, 0, 0.996109]], Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@0001f80@0006@000000000000001D0000E0000 00000000001?0`00C`<00215CDH00040R1d001@00002000000000000002l000080<005P200100@00 l000000000000000000004H0003H3000bP`004M4BD<1002000<00393]ST00000/P`000402@000eT6 00010284000000T0000V1Pl0201OATeBBE@Y80<0000N00D0000;0P0000050000308O01l01@800?L0 00<00@00001gMgL0EEEE0<`0b`3^k^h0IP1U0?g;0028R8P006Gm0?ooo`14A4@08R8R000000000000 00000000003k:O00M`400000e03b4P00b8/G002ni@110;`0lQ800:Cc4P00/?l04P140;OcM`0hc_<0 Mooo0?oob02;5`00^o530038G@0H01P0o18006H00000b8/05`2600:D0@3P_AX000400000b01M6000 C00000260P2D0@0004000Bn01X0100100000000000000000>00 _AX0063c4P00K000200A000P000000000i20004000001810000n00JG`00@cYL06Q_K@1UL640IfEL06MYIP1c04d0HGAX06E]H@1dJF<0 HD9Y06L^I`1YIP00g?L0M`PT05l000000000 00@00030>P3XMd<0>UaX06m]I@1`HFL0IEaW06UVL`1LCF40M6QU06eQM01YHf40@VUW02iWJ@1V08H0 0Y@100000000RDX0mWM800DD001V000000P00000QP02U040>aM500260P2D0I@00@8001810000@aL0 A@2600:D0@30ma800080000000000000n1YO0007002:0A00mA8009XAk@1ge`@00GD7002:0@0N0`00 00000000J0035P000001003Hm@0B06@0BZ]g09810P00G?D04P0>05_WM`30RaL00000000000005000 8040000800000040000008Se4P00/bP0l7L0000D000800@001P10000IP000000IP0000260P2D0ML0 105e00000000208000000000001f0P00053e018000000000/?lB0014]`3cMlP0cO=g0?ooo`3o1HH0 @`2600:D0@040000=08000<0000e00D000070@<0000R1000@`lP0<`0000P02000000020080000000 :0000200000P00000@08000000001000a0h00<@>00000@0000000000001gMgL0EEEE0Fn002l044001;b01;cY03o/000A00B07Oc]`3ccSP0oomg08090Qg0000G`00000000@003[00013MnP0J5`j06E]K`1WHG00 IeaU07=VJ@1QCE`0IFQd07AQK@1QHfT0IfU206UW;P2606H00I@20000001:R@00B7Of000D1@0006H0 00P008H00001U080AALk00:6002D0I@0008100014P0G@`00QP15006D0P0Bml000080000000000000 Ga[h0007000@0HX001;e0>dAVP04egL01gD1006:00000ah0000006P000005P<00@000?GH001T0180 Mj]:0081TP3eG0003P0B07OWF`0GRl000000000000005000004P000800010000000001;eR00X/`00 07O`000D000400P00AP006H000000000001V00:6003G0I@0M@440000000220000000000000000WH0 mE0000004P0000004_n`0;M40038Mo<0Mo?=0?ooo`261Ol0QP13006D0P092@T00000000000000000 0000000000000000000000T000071`L71`L71`L71`L71`L71`L71`L71`L71`L000T0000000000000 0000000000000000000000001`002@0410@410@410@410@410@410@410@410@4100700T0000410@4 10@410@410@410@410@410@410@400L000T010@410@400000P4410@41040004410@410@01`002@04 10@410@02000000:100000D0000010@4100700T0000410@4100820011P0000D500030`0410@400L0 00T010@410@400P820001P051@070P0300@410@01`002@0410@4104020P8208000D01`L72`0210@4 100700T0000410@40@0820P8208000L71`L200@410@400L000T010@4104000P820P820071`L71`80 0@@410@01`002@0410@400D5000000000000000000000@@4100700T000040@0000002`X21`080P05 1@0000L000@400L000T010@000L71`L71`L000P820801@0000L000@01`002@0400<300L71`L70006 00P820P000030004100700T000000`<300L71`0000H600P820P00`<010@400L000T010800`<300L0 00D01PH600P800<30`<010@01`002@0410X0000000051@061PH600030`<30`<0100700T0000410@2 0000000000H61PH:00<30`0000@400L000T010@410@:00<30P000000000000000@@410@01`002@04 10@4100500830`<30`<300H6004410@4100700T0000410@1000700<30`<30`001PH010@410@400L0 00T010@410001`L00P<30`<020P01P0410@410@01`002@0410@400L:00000`<30P0020P000@410@4 100700T0000410@0000110400`80108000P010@410@400L000T010@410@410@4100300@410@10004 10@410@01`002@0410@410@410@4100410@410@40P@410@4100700T0000410@410@410@410@410@4 10@410@410@400L000T010@410@410@410@410@410@410@410@410@01`002@0410@410@410@410@4 10@410@410@410@410002@T000000000000000000000000000000000000000T9100002L1ool30000 00000140000<0000200000/0000@00005@0001D000090000400001D0000E00008@0000P0000:0000 40000000000000002@000100000O00007`000340000@10000@00000300400000MgMg05EEE@3<0dQ0>MgU0010P0000800000000001P000400=7>XP0?6000001200000002000006MgQP02U040 4P40002El000018001l000000000WP8000010000002XHel00?PJ05l0d0085P004`00000V00000000 000005ce4P000S<0kGL00000000895l008H209@14P010000UO00000000000800m18007iHi`1g22@0 G`000000000010000000000000000000hUSW07L8901O0000000000040000`3X0j7M303YLJ01_KFD0 L65W06ELI`1YIW<0G4eQ07AXI@1]HG@0JF=Q049YI`0^IfT0IP2600:D0@00000008U:0?IgB0055000 IP000008000008H00Y@103/GA@00QP80U06D0042000B0@000400343P000040000000000000MgMg05EEE@3;0<`0k^k^06D0IP00bod0R8R80?eU003oool0 A4A4028R8P0000000000000000000000l2Wk0001M`3D000001;b01N;b03U_P00_011000BlP0Blj@0 ok0004@04P1glkL0llhh0?ooM`38ool001N;04?a^`1Mb000600H000Bo00006H0RlP008H05`01U080 6[gP00010038000001QM0000C002QP00006D0010000HGLP0000000000000000000010;kT000401X0 0001000000000000h000000J_@0Blf0006`0014020008000000008_8001P01L0041c04P2e01`C`00 06iU000300000080lm`004004P00@^h00004004B003h000005lJ05`j@`1]KfP0HG1U05aUI`1VJFL0 C@1c06QdH@1QKFD0HfUd06U2H@1W;VL006IY01;cg0221`0007OW00000000000002000=`000004_<0 4_@l069?003d@0060l0@P000000000000804F<008Igi`01U080004B0?2E000B000001l000000002WP00 0@000000001OHjP06_P00=00G`005PP0000C000V00000000000001;eG00c0P0007O]0000001O90P0 0XH00181U00000400?2E00000020000001;d0>MHOP0T27L0001O0000000010000000000000000000 ieSR02@8M`0005l000000004000j`000@gOX06QL>P1UKFl0If5`06MLI@1cIVT0HDeL06EXM01dHFd0 HF=Y06MY@P1YIbh0QP1V006D0P000000BXT004QgmP0050D0001V0008002600000I@204DG>`02QP00 U06D00020@000A805d<008H0A@01U0804_O0000200000000000005lJn0001`00406:000Bm@3]4IX0 1=Mg00Me0@01RP0000000010<00;`0000@0000 000007MgM`1EEED0c03;0>k^kP1V06D0oL/008R8R000IOd0oooo04A4A00R8R80nbW`07L100000=@0 lQ800Bn01X010000>00_AX0063c4P00K000200A000P000G0600Ld000=@2B000Cg00IFh0000300020000 0=cc0180@03^@P001000000B0@000?P06Ul004dQ0>MgU0010P0000800000603AcZ803aP00000@P00Ha40igN601810000UO00000B000O0000WP80 00010:QSG`00n1X0G`3@00PF000C000002H005ce4P000S<0kGL000PTG`2D0A80UO000000P03d4P00 OUSW07L8901O000000@00>9Hi`00`3X0j7M303YLJ01_KFD0L65W06ELI`1YIW<0G4eQ07AXI@1]HG@0 JF=Q049YI`0^IfT0IP260029BP3fMdP01A@00008000008H0>aM509@1U000@aL0A@260<3g4P3h6Ul0 00L008X1403e4P00VQ7]07OG1001M@L008X101h3000006P00aH0003Hm@0B06@0BZ]g09810P00G?D0 4P0>05_WM`30RaL001@002010028mA800;"], "Graphics", ImageSize->{21, 21}], StyleBox[" ", FontColor->RGBColor[0.500008, 0, 0.996109]] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18, CellTags->"Section 9.2"], Cell[TextData[{ "\tThe prerequisite for this section is ", ButtonBox["Section 2.6", ButtonData:>{"ca0206.nb", None}, ButtonStyle->"Hyperlink"], " ", ButtonBox["The Reciprocal Transformation w = 1/z.", ButtonData:>{"ca0206.nb", None}, ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "\tAnother important class of elementary mappings was studied by ", ButtonBox["August Ferdinand M\[ODoubleDot]bius", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mobius.\ html"], None}, ButtonStyle->"Hyperlink"], " (1790-1868). These mappings are conveniently expressed as the quotient \ of two linear expressions and are commonly known as linear fractional or \ bilinear transformations. They arise naturally in mapping problems involving \ the function arctan(z). In this section we will show how they are used to \ map a disk one-to-one and onto a half-plane.\n\n\tLet a, b, c, and d \ denote four complex constants with the restriction that ", Cell[BoxData[ \(a\ d \[NotEqual] b\ c\)]], ". Then the function\n\n\t", Cell[BoxData[ \(w = \(S \((z)\) = \(a\ z\ + \ b\)\/\(c\ z\ + \ d\)\)\)]], "\n\nis called a ", StyleBox["bilinear transformation", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " or ", ButtonBox["M\[ODoubleDot]bius Transformation", ButtonData:>{ URL[ "http://mathworld.wolfram.com/MoebiusTransformation.html"], None}, ButtonStyle->"Hyperlink"], " ", " or ", StyleBox["linear fractional transformation", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ". If the expression for S(z) is multiplied through by the quantity ", Cell[BoxData[ \(c\ z\ + \ d\)]], ", then the resulting expression has the bilinear form ", Cell[BoxData[ \(c\ w\ z\ - \ a\ z\ + \ d\ w\ - \ b\ = \ 0\)]], ". We can collect terms involving z and write ", Cell[BoxData[ \(z \((c\ w\ - a\ )\)\ = \ \(-d\)\ w\ + \ b\)]], ". For values of ", Cell[BoxData[ \(w \[NotEqual] a\/c\)]], " the inverse transformation is given by\n\n\t", Cell[BoxData[ \(z = \(\(S\^\(-1\)\) \((w)\) = \(\(-d\)\ w + b\)\/\(c\ w - a\)\)\)]], "." }], "Text"], Cell[TextData[{ "\tWe can extend ", Cell[BoxData[ \(S\ \ and\ \ S\^\(-1\)\)]], " to mappings in the extended complex plane. The value ", Cell[BoxData[ \(S \((\[Infinity])\)\)]], " should be chosen to equal the limit of ", Cell[BoxData[ \(S \((\[Infinity])\)\)]], " as ", Cell[BoxData[ \(z \[Rule] \[Infinity]\)]], ". Therefore we define\n\n\t", Cell[BoxData[ \(S \((\[Infinity])\)\ = \ \(lim\+\(z\ \[Rule] \ \[Infinity]\)\ \ S \ \((z)\)\ = \ \(lim\+\(z\ \[Rule] \ \[Infinity]\)\ \ \(a\ \ + \ b\/z\)\/\(c\ \ \ + \ d\/z\)\ = \ \ a\/c\)\)\)]], ", \n\nand the inverse is ", Cell[BoxData[ \(\(S\^\(-1\)\) \((a\/c)\)\ = \ \[Infinity]\)]], ". Similarly, the value ", Cell[BoxData[ \(\(S\^\(-1\)\) \((\[Infinity])\)\)]], " is obtained by a limit\n\n\t", Cell[BoxData[ \(\(S\^\(-1\)\) \((\[Infinity])\)\ \[Equal] \ lim\+\(w \[Rule] \ \[Infinity]\)\ \ \(S\^\(-1\)\) \((w)\)\ \[Equal] \ \ lim\+\(w\ \[Rule] \ \[Infinity]\)\ \ \(\(-d\)\ + b\/w\)\/\(c\ - a\/w\)\ \ \[Equal] \ \ \(-\ d\)\/c\)]], ",\n\nand the inverse is ", Cell[BoxData[ \(S \((\(-\ d\)\/c)\)\ = \ \[Infinity]\)]], ". With these extensions we conclude that the transformation ", Cell[BoxData[ \(w = S \((z)\)\)]], " is a one-to-one mapping of the extended complex z-plane onto the \ extended complex w-plane. " }], "Text"], Cell[TextData[{ StyleBox["\n", FontWeight->"Bold"], StyleBox["Example 9.3, Page 364.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Show that ", Cell[BoxData[ \(w = \(S \((z)\) = \(\[ImaginaryI]\ \((1\ - \ z)\)\)\/\(1\ + \ z\)\)\ \)], AspectRatioFixed->True], " maps the unit disk ", Cell[BoxData[ \(\(\(|\)\(z\)\(|\)\(\(<\)\(1\)\)\)\)]], " one-to-one and onto the upper half plane ", Cell[BoxData[ \(Im[w] > 0\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 9.3.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell[TextData[{ "Enter the function ", Cell[BoxData[ \(S \((z)\) = \ \(\[ImaginaryI]\ \((1\ - \ z)\)\)\/\(1\ + \ z\)\)], AspectRatioFixed->True], "." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[z];\)\ \), "\[IndentingNewLine]", \(\(Clear[eqn, Iden, solset, S, w, z];\)\ \), "\n", \(\(Iden[z_]\ = \ z;\)\ \), "\n", \(\(S[z_]\ = \ \(\[ImaginaryI]\ \((1 - z)\)\)\/\(1 + z\);\)\ \), "\n", \(\(Print["\", S[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ To show S(z) is one-to-one, find the inverse function.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\[IndentingNewLine]\(eqn = w \[Equal] S[z];\)\ \), "\[IndentingNewLine]", \(\(solset\ = \ Solve[eqn, z];\)\ \), "\[IndentingNewLine]", \(\(S1[w_] = solset\_\(\(\[LeftDoubleBracket]\)\(1, 1, \ 2\)\(\[RightDoubleBracket]\)\);\)\ \), "\[IndentingNewLine]", \(\(Print[eqn];\)\ \), "\n", \(\(Print[solset];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\\"", S1[w]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ Check out the inverse function.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Print["\", S[z]];\)\ \), "\n", \(\(Print[\*"\"\\"", S1[w]];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(S\^\(-1\)\)[S[z]] = \>\"", S1[S[z]], "\< = \>", Together[S1[S[z]]]];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\\"", S[S1[w]], "\< = \>", Together[S[S1[w]]]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "\nConsider the unit circle C: |z| = 1. Since ", Cell[BoxData[ \(z = \(\(S\^\(-1\)\) \((w)\) = \(\[ImaginaryI]\ - \ w\)\/\(\ \[ImaginaryI]\ + \ w\)\)\)]], ", the image of these points will satisfy ", Cell[BoxData[ \(\(\(\ \)\(\(\(|\)\(w + \[ImaginaryI]\)\(|\)\) = \(\(|\)\(w + \ \[ImaginaryI]\)\(|\)\)\)\)\)]], ". ", "\n", "Square both sides of the above equation and obtain ", Cell[BoxData[ \(\(\(|\)\(u + \[ImaginaryI]\ v + \[ImaginaryI]\)\( | \^2\)\) = \(\(|\)\ \(\(-u\) - \[ImaginaryI]\ v + \[ImaginaryI]\)\( | \^2\)\)\)]], " which can be simplified." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(eqn = u\^2 + \((v + 1)\)\^2 == u\^2 + \((\(-v\) + 1)\)\^2;\)\ \), "\[IndentingNewLine]", \(\(Print[eqn];\)\ \), "\[IndentingNewLine]", \(\(Print[Simplify[eqn]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell["\<\ The circle C divides the z-plane into two portions, and its image, v = 0, \ is the u-axis which divides the w-plane into two portions. The image of a \ point interior to C will determine which half plane is the image of the \ unit disk. For example\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(Print["\", S[0]];\)\(\ \)\(\[IndentingNewLine]\) \)\)\)], "Input"], Cell[TextData[{ "\nTherefore, the image of the unit disk ", Cell[BoxData[ \(\(\(|\)\(z\)\(|\)\(\(<\)\(1\)\)\)\)]], " is the upper half plane ", Cell[BoxData[ \(Im[w] > 0\)]], ". \n\nTo visualize that S(z) is onto we will consider a graph. The \ graph is not conclusive, but it appears to confirm our suspicion." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[f, Iden, z];\)\ \), "\n", \(\(Iden[z_]\ = \ z;\)\ \), "\n", \(\(S[ z_]\ = \(\(\ \)\(\[ImaginaryI]\ \((1 - z)\)\)\)\/\(1 + z\);\)\ \), \ "\n", \(\(PolarMap[ Iden, {0.000005, 0.999995, 0.11111}, {0, 2 \[Pi], \[Pi]\/12}, PlotRange \[Rule] {{\(-1.05\), 1.05}, {\(-1.05\), 1.05}}, \[IndentingNewLine]Ticks \[Rule] {Range[\(-1\), 1, 1], Range[1, 1, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Pink, Magenta}];\)\ \ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\n", \(\(PolarMap[ S\ \ \ , {0.000005, 0.999995, 0.11111}, {0, 2 \[Pi], \[Pi]\/12}, Ticks \[Rule] {Range[\(-3\), 3, 1], Range[0, 3, 1]}, AspectRatio \[Rule] 1\/2, PlotRange \[Rule] {{\(-3\), 3}, {0, 3}}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Pink, Magenta}];\)\ \ \), "\[IndentingNewLine]", \(\(Print["\ 0,\>"];\)\ \), "\n", \(\(Print["\", S[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nWe see that the transformation ", Cell[BoxData[ \(w = \(S \((z)\) = \(\[ImaginaryI]\ \((1\ - \ z)\)\)\/\(1\ + \ z\)\)\ \)], AspectRatioFixed->True], " maps the unit disk ", Cell[BoxData[ \(\(\(|\)\(z\)\(|\)\(\(<\)\(1\)\)\)\)]], " one-to-one and onto the upper half plane ", Cell[BoxData[ \(Im[w] > 0\)]], ". " }], "Text"] }, Closed]], Cell[TextData[{ StyleBox["\nTheorem 9.3 (The Implicit Formula), Page 365.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " There exists a unique bilinear transformation that maps three distinct \ points ", Cell[BoxData[ RowBox[{\(z\_1\), ",", \(z\_2\), ",", RowBox[{ StyleBox["and", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], " ", \(z\_3\)}]}]]], " onto three distinct points ", Cell[BoxData[ RowBox[{\(w\_1\), ",", \(w\_2\), ",", RowBox[{ StyleBox["and", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], " ", \(w\_3\)}]}]]], ", respectively. An implicit formula for the mapping is given by the \ equation \n\n\t", Cell[BoxData[ \(\(\((z\ - \ z\_1)\)\ \((z\_2\ - \ z\_3)\)\)\/\(\(\ \)\(\((z\ - \ z\ \_3)\) \((z\_2\ - \ z\_1)\)\)\) = \(\((w\ - \ w\_1)\)\ \((w\_2\ - \ \ w\_3)\)\)\/\(\((w\ - \ w\_3)\) \((w\_2\ - \ w\_1)\)\(\ \)\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["Proof of Theorem 9.3, see text Page 365.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 9.4, Page 366.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Construct the bilinear transformation w = S(z) that maps the points \ ", Cell[BoxData[ \(z\_1 = \(-\[ImaginaryI]\), \ z\_2 = 1, \ z\_3 = \[ImaginaryI]\)]], " onto the points ", Cell[BoxData[ \(w\_1 = \(-1\), \ \ w\_2 = 0, \ \ w\_3 = 1\)]], ", respectively." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 9.4.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell["\<\ Enter the three points and their images and solve for w = S(z). \ \ \>", "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[w, z];\)\ \), "\[IndentingNewLine]", \(\(Clear[formula, solset, S];\)\ \), "\n", \(\(formula\ = \ \(\((z - z\_1)\)\ \((z\_2 - z\_3)\)\)\/\(\(\ \)\(\((z - \ z\_3)\) \((z\_2 - z\_1)\)\)\) \[Equal] \(\((w - w\_1)\)\ \((w\_2 - \ w\_3)\)\)\/\(\(\((w - w\_3)\) \((w\_2 - w\_1)\)\)\(\ \)\);\)\ \), "\n", \(\(Print[formula];\)\ \), "\n", \(\(z\_1\ = \ \(-\[ImaginaryI]\);\)\ \ \), "\[IndentingNewLine]", \(\(z\_2\ = \ 1;\)\ \ \), "\[IndentingNewLine]", \(\(z\_3\ = \ \[ImaginaryI];\)\ \ \ \), "\n", \(\(w\_1\ = \ \(-1\);\)\ \ \ \), "\[IndentingNewLine]", \(\(w\_2\ = \ 0;\)\ \ \), "\[IndentingNewLine]", \(\(w\_3\ = \ 1;\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(z\_1\) = \>\"", z\_1, \*"\"\<, \!\(z\_2\) = \>\"", z\_2, \*"\"\<, \!\(z\_3\) = \>\"", z\_3];\)\ \), "\n", \(\(Print[\*"\"\< \!\(w\_1\) = \>\"", w\_1, \*"\"\<, \!\(w\_2\) = \>\"", w\_2, \*"\"\<, \!\(w\_3\) = \>\"", w\_3];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[formula];\)\ \), "\n", \(\(solset\ = \ Solve[formula, w];\)\ \), "\[IndentingNewLine]", \(\(Print[solset];\)\ \), "\n", \(\(S[z_]\ = \ solset\_\(\(\[LeftDoubleBracket]\)\(1, 1, \ 2\)\(\[RightDoubleBracket]\)\);\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\", S[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nCheck our work and by looking at the images of ", Cell[BoxData[ \(z\_1, \ z\_2, \ z\_3\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Print[\*"\"\< \!\(z\_1\) = \>\"", z\_1, \*"\"\<, \!\(z\_2\) = \>\"", z\_2, \*"\"\<, \!\(z\_3\) = \>\"", z\_3];\)\ \), "\n", \(\(Print[\*"\"\< \!\(w\_1\) = \>\"", w\_1, \*"\"\<, \!\(w\_2\) = \>\"", w\_2, \*"\"\<, \!\(w\_3\) = \>\"", w\_3];\)\ \), "\n", \(\(Print[\*"\"\\"", S[z\_1], \*"\"\<, S[\!\(z\_2\)] = \>\"", S[z\_2], \*"\"\<, S[\!\(z\_3\)] = \>\"", S[z\_3]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ To visualize the mapping w = S(z) we will consider a graph. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[f, Iden, wplane, z, zplane];\)\ \), "\n", \(\(Iden[z_] = z;\)\ \), "\n", \(\(S[ z_] = \(-\(\(\[ImaginaryI]\ \((\(-1\) + z)\)\)\/\(1 + z\)\)\);\)\ \), "\[IndentingNewLine]", \(\(set\ = \ \ {{0, \(-1\)}, {1, 0}, {0, 1}};\)\ \), "\[IndentingNewLine]", \(\(zdots = Graphics[{{Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(2\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(3\)\(\ \[RightDoubleBracket]\)\)]}}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(set\ = \ \ {{\(-1\), 0}, {0, 0}, {1, 0}};\)\ \), "\[IndentingNewLine]", \(\(wdots = Graphics[{{Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(2\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(3\)\(\ \[RightDoubleBracket]\)\)]}}, DisplayFunction \[Rule] Identity];\)\ \), "\n", \(\(zplane = CartesianMap[ Iden, {0, 1, 0.125}, {\(-1\), 1, 0.125}, \[IndentingNewLine]PlotRange \[Rule] {{\(-0.1\), 1.1}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 2, \[IndentingNewLine]Ticks \[Rule] {Range[0, 1, 1], Range[\(-1\), 1, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {DarkGreen, Green}, DisplayFunction \[Rule] Identity];\)\ \ \), "\n", \(\(wplane = CartesianMap[ S, {0, 1, 0.125}, {\(-1\), 1, 0.125}, \[IndentingNewLine]PlotRange \[Rule] {{\(-1.2\), 1.2}, {\(-0.2\), 1.0}}, AspectRatio \[Rule] 1\/2, \[IndentingNewLine]Ticks \[Rule] {Range[\(-1\), 1, 1], Range[\(-1\), 1, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {DarkGreen, Green}, DisplayFunction \[Rule] Identity];\)\ \ \), "\[IndentingNewLine]", \(\(Show[zplane, zdots, PlotRange \[Rule] {{\(-0.1\), 1.1}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 2, \[IndentingNewLine]Ticks \[Rule] {Range[0, 1, 1], Range[\(-1\), 1, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\ \[IndentingNewLine]", \(\(Print[\*"\"\\"", z\_1, \*"\"\<, \!\(z\_2\) = \>\"", z\_2, \*"\"\<, \!\(z\_3\) = \>\"", z\_3, "\<,\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\[IndentingNewLine]", \(\(Show[wplane, wdots, \[IndentingNewLine]PlotRange \[Rule] {{\(-1.2\), 1.2}, {\(-0.2\), 1.0}}, AspectRatio \[Rule] 1\/2, \[IndentingNewLine]Ticks \[Rule] {Range[\(-1\), 1, 1], Range[\(-1\), 1, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\ \[IndentingNewLine]", \(\(Print[\*"\"\\"", w\_1, \*"\"\<, \!\(w\_2\) = \>\"", w\_2, \*"\"\<, \!\(w\_3\) = \>\"", w\_3];\)\ \), "\[IndentingNewLine]", \(\(Print["\"\ , S[z]];\)\ \), "\[IndentingNewLine]", \(\(Print["\", z\_1, "\<] = \>", S[z\_1], \[IndentingNewLine]"\<, S[\>", z\_2, "\<] = \>", S[z\_2], \[IndentingNewLine]"\<, S[\>", z\_3, "\<] = \>", S[z\_3]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"] }, Closed]], Cell[TextData[{ "\n", StyleBox["Example 9.5, Page 366.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Find the bilinear transformation w = S(z) that maps the points ", Cell[BoxData[ \(z\_1 = \(-2\), \ \ z\_2 = \(-1\) - \[ImaginaryI], \ \ z\_3 = 0\)]], " onto the points ", Cell[BoxData[ \(w\_1 = \(-1\), \ \ w\_2 = 0, \ \ w\_3 = 1\)]], ", respectively." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 9.5.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell["\<\ Enter the three points and their images and solve for w = S(z). \ \ \>", "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[w, z];\)\ \), "\[IndentingNewLine]", \(\(Clear[formula, solset, S];\)\ \), "\n", \(\(formula\ = \ \(\((z - z\_1)\)\ \((z\_2 - z\_3)\)\)\/\(\(\ \)\(\((z - \ z\_3)\) \((z\_2 - z\_1)\)\)\) \[Equal] \(\((w - w\_1)\)\ \((w\_2 - \ w\_3)\)\)\/\(\(\((w - w\_3)\) \((w\_2 - w\_1)\)\)\(\ \)\);\)\ \), "\n", \(\(Print[formula];\)\ \), "\n", \(\(z\_1\ = \ \(-2\);\)\ \ \ \), "\[IndentingNewLine]", \(\(z\_2\ = \ \(-1\) - \[ImaginaryI];\)\ \ \), "\[IndentingNewLine]", \(\(z\_3\ = \ 0;\)\ \), "\n", \(\(w\_1\ = \ \(-1\);\)\ \ \ \), "\[IndentingNewLine]", \(\(w\_2\ = \ 0;\)\ \ \), "\[IndentingNewLine]", \(\(w\_3\ = \ 1;\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(z\_1\) = \>\"", z\_1, \*"\"\<, \!\(z\_2\) = \>\"", z\_2, \*"\"\<, \!\(z\_3\) = \>\"", z\_3];\)\ \), "\n", \(\(Print[\*"\"\< \!\(w\_1\) = \>\"", w\_1, \*"\"\<, \!\(w\_2\) = \>\"", w\_2, \*"\"\<, \!\(w\_3\) = \>\"", w\_3];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[formula];\)\ \), "\n", \(\(solset\ = \ Solve[formula, w];\)\ \), "\[IndentingNewLine]", \(\(Print[solset];\)\ \), "\n", \(\(S[z_]\ = \ solset\_\(\(\[LeftDoubleBracket]\)\(1, 1, \ 2\)\(\[RightDoubleBracket]\)\);\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\", S[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nThis is equivalent to the formula ", Cell[BoxData[ \(S \((z)\) = \(\((1 - \[ImaginaryI])\) z\ \ + \ \ 2\)\/\(\((1 + \ \[ImaginaryI])\) z\ \ + \ \ 2\)\)]], ", and can be verified by a computation." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Print[ S[z], "\< = \>", \(\((1 - \[ImaginaryI])\) z\ + \ 2\)\/\(\((1 + \ \[ImaginaryI])\) z\ + \ 2\)];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[S[z], "\< = \>", Simplify[\(\((1 - \[ImaginaryI])\) z\ + \ 2\)\/\(\((1 + \ \[ImaginaryI])\) z\ + \ 2\)]];\)\ \), "\n", \(\(Print["\", S[z]\ \[Equal] \ Simplify[\(\((1 - \[ImaginaryI])\) z\ + \ 2\)\/\(\((1 + \ \[ImaginaryI])\) z\ + \ 2\)]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nCheck our work and look at the images of ", Cell[BoxData[ \(z\_1, \ z\_2, \ z\_3\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Print[\*"\"\< \!\(z\_1\) = \>\"", z\_1, \*"\"\<, \!\(z\_2\) = \>\"", z\_2, \*"\"\<, \!\(z\_3\) = \>\"", z\_3];\)\ \), "\n", \(\(Print[\*"\"\< \!\(w\_1\) = \>\"", w\_1, \*"\"\<, \!\(w\_2\) = \>\"", w\_2, \*"\"\<, \!\(w\_3\) = \>\"", w\_3];\)\ \), "\n", \(\(Print[\*"\"\\"", S[z\_1], \*"\"\<, S[\!\(z\_2\)] = \>\"", S[z\_2], \*"\"\<, S[\!\(z\_3\)] = \>\"", S[z\_3]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ To visualize the mapping w = S(z) we will consider a graph. \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[f, Iden, wplane, z, zplane];\)\ \), "\n", \(\(Iden[z_] = z;\)\ \), "\n", \(\(S[ z_] = \(-\(\(\[ImaginaryI]\ \((\((1 + \[ImaginaryI])\) + z)\)\)\/\(\((1 - \[ImaginaryI])\) + z\)\)\);\)\ \), "\[IndentingNewLine]", \(\(set\ = \ \ {{\(-2\), 0}, {\(-1\), \(-1\)}, {0, 0}};\)\ \), "\[IndentingNewLine]", \(\(zdots = Graphics[{{Magenta, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)]}, {Magenta, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(2\)\(\ \[RightDoubleBracket]\)\)]}, {Magenta, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(3\)\(\ \[RightDoubleBracket]\)\)]}}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(set\ = \ \ {{\(-1\), 0}, {0, 0}, {1, 0}};\)\ \), "\[IndentingNewLine]", \(\(wdots = Graphics[{{Magenta, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)]}, {Magenta, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(2\)\(\ \[RightDoubleBracket]\)\)]}, {Magenta, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(3\)\(\ \[RightDoubleBracket]\)\)]}}, DisplayFunction \[Rule] Identity];\)\ \), "\n", \(\(zplane = CartesianMap[ Iden, {\(-2\), 0, 0.125}, {\(-1\), 1, 0.125}, \[IndentingNewLine]PlotRange \[Rule] {{\(-2.1\), 0.1}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 1, \[IndentingNewLine]Ticks \[Rule] {Range[\(-2\), 0, 1], Range[\(-1\), 1, 1]}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Blue, Cyan}, DisplayFunction \[Rule] Identity];\)\ \), "\n", \(\(wplane = CartesianMap[ S, {\(-2\), 0, 0.125}, {\(-1\), 1, 0.125}, \[IndentingNewLine]PlotRange \[Rule] {{\(-1.2\), 1.2}, {\(-0.4\), 2.0}}, AspectRatio \[Rule] 1, \[IndentingNewLine]Ticks \[Rule] {Range[\(-1\), 1, 1], Range[\(-1\), 2, 1]}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Blue, Cyan}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(Show[zplane, zdots, PlotRange \[Rule] {{\(-2.1\), 0.1}, {\(-1.1\), 1.1}}, \[IndentingNewLine]AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-2\), 0, 1], Range[\(-1\), 1, 1]}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \), \ "\[IndentingNewLine]", \(\(Print[\*"\"\\"", z\_1, \*"\"\<, \!\(z\_2\) = \>\"", z\_2, \*"\"\<, \!\(z\_3\) = \>\"", z\_3, "\<,\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\[IndentingNewLine]", \(\(Show[wplane, wdots, PlotRange \[Rule] {{\(-1.2\), 1.2}, {\(-0.4\), 2.0}}, \[IndentingNewLine]AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-1\), 1, 1], Range[\(-1\), 2, 1]}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \), \ "\[IndentingNewLine]", \(\(Print[\*"\"\\"", w\_1, \*"\"\<, \!\(w\_2\) = \>\"", w\_2, \*"\"\<, \!\(w\_3\) = \>\"", w\_3, "\<,\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\"\ , S[z]];\)\ \), "\[IndentingNewLine]", \(\(Print["\", z\_1, "\<] = \>", S[z\_1], \[IndentingNewLine]"\<, S[\>", z\_2, "\<] = \>", S[z\_2], \[IndentingNewLine]"\<, S[\>", z\_3, "\<] = \>", S[z\_3]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"] }, Closed]], Cell[TextData[{ "\n", StyleBox["Example 9.6, Page 367.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Show that the mapping ", Cell[BoxData[ \(w = \(S \((z)\) = \(\((1 - \[ImaginaryI])\) z\ \ + \ \ 2\)\/\(\((1 + \ \[ImaginaryI])\) z\ \ + \ \ 2\)\)\)], AspectRatioFixed->True], " maps the disk ", Cell[BoxData[ \(\(\(|\)\(z + 1\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\)]], " one-to-one and onto the upper half plane ", Cell[BoxData[ \(Im[w] > 0\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 9.6.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell[TextData[{ "Enter the function ", Cell[BoxData[ \(S \((z)\) = \(\((1 - \[ImaginaryI])\) z\ \ + \ \ 2\)\/\(\((1 + \ \[ImaginaryI])\) z\ \ + \ \ 2\)\)], AspectRatioFixed->True], ". " }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[S, w, W, z, Z];\)\ \), "\[IndentingNewLine]", \(\(Clear[d, g, solset, w1, w2, w3, z1, z2, z3];\)\ \), "\n", \(\(S[ z_]\ = \ \(\((1 - \[ImaginaryI])\) z\ + \ 2\)\/\(\((1 + \ \[ImaginaryI])\) z\ + \ 2\);\)\ \), "\n", \(\(Print["\", S[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ To show S(z) is one-to-one, find the inverse function.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(solset\ = \ Solve[w \[Equal] S[z], z];\)\ \), "\n", \(\(Print[solset];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\\"", solset\_\(\(\[LeftDoubleBracket]\)\(1, 1, \ 2\)\(\[RightDoubleBracket]\)\)];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ To show S(z) is onto, use the method of oriented points on the boundary \ curve.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(z\_1\ = \ \(-2\);\)\ \ \), "\[IndentingNewLine]", \ \(\(z\_2\ = \ \(-1\) - \[ImaginaryI];\)\ \), "\[IndentingNewLine]", \(\(z\_3\ = \ 0;\)\ \ \), "\n", \(\(w\_1\ = \ \(-1\);\)\ \ \), "\[IndentingNewLine]", \(\(w\_2\ = \ 0;\)\ \ \), "\[IndentingNewLine]", \(\(w\_3\ = \ 1;\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(z\_1\) = \>\"", z\_1, \*"\"\<, \!\(z\_2\) = \>\"", z\_2, \*"\"\<, \!\(z\_3\) = \>\"", z\_3];\)\ \), "\n", \(\(Print[\*"\"\< \!\(w\_1\) = \>\"", w\_1, \*"\"\<, \!\(w\_2\) = \>\"", w\_2, \*"\"\<, \!\(w\_3\) = \>\"", w\_3];\)\ \), "\n", \(\(Print[\*"\"\\"", S[z\_1], \*"\"\<, S[\!\(z\_2\)] = \>\"", S[z\_2], \*"\"\<, S[\!\(z\_3\)] = \>\"", S[z\_3]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nTherefore, the image of the disk ", Cell[BoxData[ \(\(\(|\)\(z + 1\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\)]], " is the upper half plane ", Cell[BoxData[ \(Im[w] > 0\)]], ". In order to graph the image of the disk ", Cell[BoxData[ \(\(\(|\)\(z + 1\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\)]], " under w = S(z) we use the change of variable and find the image of ", Cell[BoxData[ \(\(\(|\)\(z\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\)]], " under ", Cell[BoxData[ \(w = \(g \((z)\) = S \((z - 1)\)\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\n\(g[z_]\ = \ S[z - 1];\)\ \), "\n", \(\(Print["\< S(z) = \>", S[z]];\)\ \), "\n", \(\(Print["\", g[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ To plot the graph we use the shifted disk and the functions d[z] = z-1 and \ g[z] = S[z-1].\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[S, z];\)\ \), "\[IndentingNewLine]", \(\(d[z_]\ = \ z - 1;\)\ \), "\n", \(\(S[ z_]\ = \ \(\((1 - \[ImaginaryI])\) z\ + \ 2\)\/\(\((1 + \ \[ImaginaryI])\) z\ + \ 2\);\)\ \), "\[IndentingNewLine]", \(\(g[z_]\ = \ S[z - 1];\)\ \), "\[IndentingNewLine]", \(\(set\ = \ \ {{\(-2\), 0}, {\(-1\), \(-1\)}, {0, 0}};\)\ \), "\[IndentingNewLine]", \(\(zdots = Graphics[{{Blue, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)]}, {Blue, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(2\)\(\ \[RightDoubleBracket]\)\)]}, {Blue, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(3\)\(\ \[RightDoubleBracket]\)\)]}}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(set\ = \ \ {{\(-1\), 0}, {0, 0}, {1, 0}};\)\ \), "\[IndentingNewLine]", \(\(wdots = Graphics[{{Blue, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)]}, {Blue, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(2\)\(\ \[RightDoubleBracket]\)\)]}, {Blue, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(3\)\(\ \[RightDoubleBracket]\)\)]}}, DisplayFunction \[Rule] Identity];\)\ \), "\n", \(\(zplane = PolarMap[ d\ \ \ , {0.00005, 0.99995, 0.1111}, {0, 2 \[Pi], \[Pi]\/12}, \[IndentingNewLine]PlotRange \[Rule] \ {{\(-2.05\), 0.05}, {\(-1.05\), 1.05}}, \[IndentingNewLine]Ticks \[Rule] {Range[\(-2\), 1, 1], Range[1, 1, 1]}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Pink}, DisplayFunction \[Rule] Identity];\)\ \ \), "\n", \(\(wplane = PolarMap[ g\ \ \ , {0.00005, 0.99995, 0.1111}, {0, 2 \[Pi], \[Pi]\/12}, \[IndentingNewLine]Ticks \[Rule] \ {Range[\(-3\), 3, 1], Range[0, 3, 1]}, \[IndentingNewLine]AspectRatio \[Rule] 1\/2, PlotRange \[Rule] {{\(-3\), 3}, {0, 3}}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Pink}, DisplayFunction \[Rule] Identity];\)\ \ \), "\[IndentingNewLine]", \(\(Show[zplane, zdots, PlotRange \[Rule] {{\(-2.05\), 0.05}, {\(-1.05\), 1.05}}, \[IndentingNewLine]Ticks \[Rule] {Range[\(-2\), 1, 1], Range[1, 1, 1]}, \[IndentingNewLine]AspectRatio \[Rule] 1, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\ \[IndentingNewLine]", \(\(Print["\"];\)\ \), "\[IndentingNewLine]", \(\(Show[wplane, wdots, PlotRange \[Rule] {{\(-3\), 3}, {0, 3}}, AspectRatio \[Rule] 1\/2, \[IndentingNewLine]Ticks \[Rule] {Range[\(-3\), 3, 1], Range[0, 3, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\ \[IndentingNewLine]", \(\(Print["\ 0,\>"];\)\ \), "\n", \(\(Print["\", S[z]];\)\ \), "\[IndentingNewLine]", \(\(Print["\", z\_1, "\<] = \>", S[z\_1], \[IndentingNewLine]"\<, S[\>", z\_2, "\<] = \>", S[z\_2], \[IndentingNewLine]"\<, S[\>", z\_3, "\<] = \>", S[z\_3]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nWe see that the transformation ", Cell[BoxData[ \(w = \(S \((z)\) = \(\((1 - \[ImaginaryI])\) z\ \ + \ \ 2\)\/\(\((1 + \ \[ImaginaryI])\) z\ \ + \ \ 2\)\)\)], AspectRatioFixed->True], " maps the disk ", Cell[BoxData[ \(\(\(|\)\(z + 1\)\(|\)\(\(<\)\(\ \)\(1\)\)\)\)]], " one-to-one and onto the upper half plane ", Cell[BoxData[ \(Im[w] > 0\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[TextData[{ StyleBox["\nCorollary 9.1 (The Implicit Formula with a point at Infinity), \ Page 367.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " In equation (9-18) the point at infinity can be introduced as one of the \ prescribed points in either the z plane or the w plane." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["Proof of Corollary 9.1, see text Page 367.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Example 9.7, Page 368.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Find the bilinear transformation w = S(z) that maps the \ crescent-shaped region that lies inside the disk ", Cell[BoxData[ \(\(\(|\)\(z - 2\)\(|\)\(\(<\)\(\ \)\(2\)\)\)\)]], " and outside the circle ", Cell[BoxData[ \(\(\(|\)\(z - 1\)\(|\)\) = \ 1\)]], " onto a horizontal strip." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 9.7.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell[TextData[{ "Enter three points and their images and solve for w = S(z). For \ convenience we choose ", Cell[BoxData[ \(z\_1 = 4, \ z\_2 = 2 + 2 \[ImaginaryI], \ z\_3 = 0\)], AspectRatioFixed->True], " which are mapped onto the points ", Cell[BoxData[ \(w\_1 = 0, \ w\_2 = 1, \ w\_3 = \[Infinity]\)], AspectRatioFixed->True], ", respectively. In this case we remove the terms involving ", Cell[BoxData[ \(w\_3\ = \ \[Infinity]\)], AspectRatioFixed->True], ", in the implicit formula, because this implies that ", Cell[BoxData[ \(\(w\_2\ - \ w\_3\)\/\(w\ - \ w\_3\) = \(\(w\_2\ - \ \ \[Infinity]\)\/\(w\ - \ \[Infinity]\) = 1\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[S, T, w, W, z, Z];\)\ \), "\[IndentingNewLine]", \(\(Clear[formula, Iden, solset];\)\ \), "\n", \(\(formula\ = \ \(\((z - z\_1)\) \((z\_2 - z\_3)\)\)\/\(\((z - z\_3)\) \ \((z\_2 - z\_1)\)\)\ \[Equal] \ \(w - w\_1\)\/\(w\_2 - w\_1\);\)\ \), "\n", \(\(Print[formula];\)\ \), "\n", \(\(z\_1\ = \ 4;\)\ \ \ \), "\[IndentingNewLine]", \(\(z\_2\ = \ 2 + 2 \[ImaginaryI];\)\ \ \), "\[IndentingNewLine]", \(\(z\_3\ = \ 0;\)\ \ \), "\n", \(\(w\_1\ = \ 0;\)\ \ \), "\[IndentingNewLine]", \(\(w\_2\ = \ 1;\)\ \ \), "\[IndentingNewLine]", \(\(w\_3\ = \ \[Infinity];\)\ \ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(z\_1\) = \>\"", z\_1, \*"\"\<, \!\(z\_2\) = \>\"", z\_2, \*"\"\<, \!\(z\_3\) = \>\"", z\_3];\)\ \), "\n", \(\(Print[\*"\"\< \!\(w\_1\) = \>\"", w\_1, \*"\"\<, \!\(w\_2\) = \>\"", w\_2, \*"\"\<, \!\(w\_3\) = \>\"", w\_3];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[formula];\)\ \), "\n", \(\(solset\ = \ Solve[formula, w];\)\ \), "\[IndentingNewLine]", \(\(Print[solset];\)\ \), "\n", \(\(S[z_]\ = \ solset\_\(\(\[LeftDoubleBracket]\)\(1, 1, \ 2\)\(\[RightDoubleBracket]\)\);\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\[IndentingNewLine]", \(\(Print["\", S[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nCheck our work and look at the images of ", Cell[BoxData[ \(z\_1, \ z\_2, \ z\_3\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Print[\*"\"\< \!\(z\_1\) = \>\"", z\_1, \*"\"\<, \!\(z\_2\) = \>\"", z\_2, \*"\"\<, \!\(z\_3\) = \>\"", z\_3];\)\ \), "\n", \(\(Print[\*"\"\< \!\(w\_1\) = \>\"", w\_1, \*"\"\<, \!\(w\_2\) = \>\"", w\_2, \*"\"\<, \!\(w\_3\) = \>\"", w\_3];\)\ \), "\n", \(\(Print[\*"\"\\"", S[z\_1], \*"\"\<, S[\!\(z\_2\)] = \>\"", S[z\_2], \*"\"\<, S[\!\(z\_3\)] = \>\"", S[z\_3]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nAnd we need to look at the images of three more points", "; ", " ", Cell[BoxData[ \(z\_4 = 1 - \[ImaginaryI], \ z\_5 = 2, \ z\_6 = 1 + \[ImaginaryI]\)], AspectRatioFixed->True], " and ", Cell[BoxData[ \(w\_4 = \(-2\) + \[ImaginaryI], \ w\_5 = \[ImaginaryI], \ w\_6 = 2 + \[ImaginaryI]\)], AspectRatioFixed->True], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]z\_4\ = \ \ 1 - \[ImaginaryI]; \ \ \ \ \ z\_5\ = \ 2; \ \ \ \ z\_6\ = \ 1 + \[ImaginaryI];\ \), "\n", \(w\_4\ = \ \(-2\) + \[ImaginaryI]; \ \ \ w\_5\ = \ \[ImaginaryI]; \ \ \ \ w\_6\ = \ 2 + \[ImaginaryI];\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\< \!\(z\_4\) = \>\"", z\_4, \*"\"\<, \!\(z\_5\) = \>\"", z\_5, \*"\"\<, \!\(z\_6\) = \>\"", z\_6];\)\ \), "\n", \(\(Print[\*"\"\< \!\(w\_4\) = \>\"", w\_4, \*"\"\<, \!\(w\_5\) = \>\"", w\_5, \*"\"\<, \!\(w\_6\) = \>\"", w\_6];\)\ \), "\n", \(\(Print[\*"\"\\"", S[z\_4], \*"\"\<, S[\!\(z\_5\)] = \>\"", S[z\_5], \*"\"\<, S[\!\(z\_6\)] = \>\"", S[z\_6]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ To illustrate the mapping, consider the inverse image of the horizontal \ strip. We will need to use the inverse function.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Iden[w_]\ = \ w;\)\ \), "\n", \(\(solset\ = \ Solve[formula, z];\)\ \), "\n", \(\(T[w_]\ = \ solset\_\(\(\[LeftDoubleBracket]\)\(1, 1, \ 2\)\(\[RightDoubleBracket]\)\);\)\ \), "\n", \(\(Print["\", S[z]];\)\ \), "\n", \(\(Print["\", T[w]];\)\ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print["\", T[S[z]], "\< = \>", Simplify[T[S[z]]]];\)\ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print["\", S[T[w]], "\< = \>", Simplify[S[T[w]]]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ Now plot the graphs using T[w] then the identity map.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(set\ = \ \ {{4, 0}, {2, 2}, {1, \(-1\)}, {2, 0}, {1, 1}};\)\ \), "\[IndentingNewLine]", \(\(zdots = Graphics[{{Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(2\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(3\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(4\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(5\)\(\ \[RightDoubleBracket]\)\)]}}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(set\ = \ \ {{0, 0}, {1, 0}, {\(-2\), 1}, {0, 1}, {2, 1}};\)\ \), "\[IndentingNewLine]", \(\(wdots = Graphics[{{Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(2\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(3\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(4\)\(\ \[RightDoubleBracket]\)\)]}, {Red, PointSize[0.04], Point[set\_\(\(\[LeftDoubleBracket]\)\(5\)\(\ \[RightDoubleBracket]\)\)]}}, DisplayFunction \[Rule] Identity];\)\ \), "\[IndentingNewLine]", \(\(zplane = CartesianMap[ T\ \ \ , {\(-9\), 9, 0.3}, {0.005, 0.995, 0.09}, \[IndentingNewLine]PlotRange \[Rule] {{\(-0.05\), 4.05}, {\(-2.05\), 2.05}}, \[IndentingNewLine]Ticks \[Rule] {Range[0, 4, 1], Range[2, 2, 1]}, \[IndentingNewLine]AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {DarkGreen, Green}, DisplayFunction \[Rule] Identity];\)\ \ \), "\n", \(\(wplane = CartesianMap[ Iden, {\(-9\), 9, 0.9}, {0.005, 0.995, 0.09}, \[IndentingNewLine]PlotRange \[Rule] {{\(-9\), 9}, {0, 1}}, AspectRatio \[Rule] 1\/3, \[IndentingNewLine]Ticks \[Rule] {Range[\(-9\), 9, 3], Range[0, 1, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {DarkGreen, Green}, DisplayFunction \[Rule] Identity];\)\ \ \), "\[IndentingNewLine]", \(\(Show[zplane, zdots, PlotRange \[Rule] {{\(-0.05\), 4.05}, {\(-2.05\), 2.05}}, AspectRatio \[Rule] 1, \[IndentingNewLine]Ticks \[Rule] {Range[0, 4, 1], Range[2, 2, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\ \[IndentingNewLine]", \(\(Print["\"];\)\ \), \ "\[IndentingNewLine]", \(\(Print["\"];\)\ \ \), "\[IndentingNewLine]", \(\(Show[wplane, wdots, PlotRange \[Rule] {{\(-9\), 9}, {0, 1}}, AspectRatio \[Rule] 1\/3, \[IndentingNewLine]Ticks \[Rule] {Range[\(-9\), 9, 3], Range[0, 1, 1]}, AxesLabel \[Rule] {"\", "\"}, DisplayFunction \[Rule] $DisplayFunction];\)\ \ \), "\ \[IndentingNewLine]", \(\(Print["\"];\)\ \), "\n\ ", \(\(Print["\", S[z]];\)\ \), "\[IndentingNewLine]", \(\(Print["\", z\_1, "\<] = \>", S[z\_1], \[IndentingNewLine]"\<, S[\>", z\_2, "\<] = \>", S[z\_2], \[IndentingNewLine]"\<, S[\>", z\_3, "\<] = \>", S[z\_3]];\)\ \), "\[IndentingNewLine]", \(\(Print["\", z\_4, "\<] = \>", S[z\_4], \[IndentingNewLine]"\<, S[\>", z\_5, "\<] = \>", S[z\_5], \[IndentingNewLine]"\<, S[\>", z\_6, "\<] = \>", S[z\_6]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nWe have constructed the transformation ", Cell[BoxData[ \(w = \(S[ z] = \(-\(\(\[ImaginaryI]\ \((\(-4\)\ + \ z)\)\)\/z\)\)\)\)]], " and see that the image of the crescent-shaped region that lies inside \ the disk ", Cell[BoxData[ \(\(\(|\)\(z - 2\)\(|\)\(\(<\)\(\ \)\(2\)\)\)\)]], " and outside the circle ", Cell[BoxData[ \(\(\(|\)\(z - 1\)\(|\)\) = \ 1\)]], " is the horizontal strip ", Cell[BoxData[ \(0 < v < 1\)]], ". " }], "Text"] }, Closed]], 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 bilinear transformations.", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.996109, 0, 0.996109]], "\n\n", StyleBox["1.", FontWeight->"Bold"], " Avital, Shmuel and Shlomo Libeskind ''An Algebraic and Geometric Approach \ to Two Step Iteration of Bilinear Functions (in The Teaching of Math.),'' Am. \ Math. M., Vol. 91, No. 1. (Jan., 1984), pp. 53-56, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["2.", FontWeight->"Bold"], " Beardon, Alan F., ''Curvature, Circles, and Conformal Maps (in Notes),'' \ Am. Math. M., Vol. 94, No. 1. (Jan., 1987), pp. 48-53, ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["3.", FontWeight->"Bold"], " Booker, T. Hoy, (1989), ''Bilinear Basics,'' Math. Mag., V. 62, No. 4, \ pp. 262-267.\n\n", StyleBox["4.", FontWeight->"Bold"], " Boyd, James N., (1985), ''A Property of Inversion in Polar Coordinates, \ The Math. Teach., V. 78, No. 1, pp. 60-61.\n\n", StyleBox["5.", FontWeight->"Bold"], " Brickman, Louis, ''The Symmetry Principle for Mobius Transformations (in \ Notes),'' Am. Math. M., Vol. 100, No. 8. (Oct., 1993), pp. 781-782., ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["6.", FontWeight->"Bold"], " Budden, F. J. (1969), ''Transformation Geometry in the Plane by Complex \ Number Methods,'' The Math. Gazette, V. 53, No. 383, pp. 19-31.\n\n", StyleBox["7.", FontWeight->"Bold"], " Cohen, Martin P., (1983), ''Inversion in a Circle: A Different Kind of \ Transformation,'' The Math. Teach., V. 86, No. 8, pp. 620-623.\n\n", StyleBox["8.", FontWeight->"Bold"], " Eljoseph, Nathan, ''On the Iteration of Linear Fractional \ Transformations,'' Am. Math. M., Vol. 75, No. 4. (Apr., 1968), pp. 362-366, \ ", StyleBox[ButtonBox["Jstor.", ButtonData:>{ URL[ "http://www.jstor.org/"], None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], " \n\n", StyleBox["9.", FontWeight->"Bold"], " Amir-Moez, (1967), ''Conformal Linear Transformations,'' Math. Mag., Vol. \ 40, pp. 268-270." }], "Text"] }, Closed]], Cell[TextData[{ StyleBox["Section 9.2", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0.250004, 0.250004]], StyleBox[" ", FontSize->16, FontWeight->"Bold"], StyleBox["Exercises for Bilinear Transformations\n", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0.996109, 0.500008, 0.996109]], StyleBox["See textbook page 370.", FontSize->16, FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]] }], "Text"] }, Open ]], Cell[TextData[{ StyleBox["Section 9", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox[".3", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox["\t", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[ButtonBox["Mappings Involving Elementary Functions", ButtonData:>{"ca0903.nb", None}, ButtonStyle->"Hyperlink"], FontSize->14, FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["Section 9", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox[".4", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.250004]], StyleBox["\t", FontFamily->"New Century Schlbk", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[ButtonBox["Mappings by Trigonometric Functions", ButtonData:>{"ca0904.nb", None}, ButtonStyle->"Hyperlink"], FontSize->14, FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["GO TO THE", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[0, 0.996109, 0]], StyleBox[" ", FontSize->18, FontWeight->"Bold"], StyleBox[ButtonBox["Table of Contents", ButtonData:>{"content.nb", None}, ButtonStyle->"Hyperlink"], FontWeight->"Bold"], "\n" }], "Text"] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, AutoGeneratedPackage->None, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{756, 598}, WindowMargins->{{31, Automatic}, {Automatic, 6}}, 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[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontSize->16, FontColor->RGBColor[1, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{ "CHAPTER 9"->{ Cell[4362, 144, 318, 12, 35, "Text", Evaluatable->False, CellTags->"CHAPTER 9"]}, "Section 9.2"->{ Cell[5796, 193, 10697, 142, 41, "Text", Evaluatable->False, CellTags->"Section 9.2"]} } *) (*CellTagsIndex CellTagsIndex->{ {"CHAPTER 9", 91592, 2569}, {"Section 9.2", 91709, 2573} } *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1727, 52, 615, 20, 109, "Text", Evaluatable->False], Cell[2345, 74, 1980, 65, 248, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[4362, 144, 318, 12, 35, "Text", Evaluatable->False, CellTags->"CHAPTER 9"], Cell[4683, 158, 480, 8, 110, "Input", InitializationCell->True] }, Closed]], Cell[5178, 169, 593, 20, 32, "Text"], Cell[CellGroupData[{ Cell[5796, 193, 10697, 142, 41, "Text", Evaluatable->False, CellTags->"Section 9.2"], Cell[16496, 337, 295, 9, 37, "Text"], Cell[16794, 348, 1949, 49, 298, "Text"], Cell[18746, 399, 1419, 38, 260, "Text"], Cell[20168, 439, 547, 20, 56, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[20740, 463, 103, 2, 33, "Text"], Cell[20846, 467, 180, 6, 37, "Text"], Cell[21029, 475, 371, 7, 165, "Input"], Cell[21403, 484, 129, 5, 52, "Text", Evaluatable->False], Cell[21535, 491, 596, 13, 211, "Input"], Cell[22134, 506, 104, 5, 52, "Text", Evaluatable->False], Cell[22241, 513, 525, 9, 173, "Input"], Cell[22769, 524, 675, 18, 75, "Text", Evaluatable->False], Cell[23447, 544, 284, 6, 110, "Input"], Cell[23734, 552, 326, 8, 71, "Text", Evaluatable->False], Cell[24063, 562, 138, 3, 70, "Input"], Cell[24204, 567, 396, 11, 90, "Text", Evaluatable->False], Cell[24603, 580, 1234, 24, 325, "Input"], Cell[25840, 606, 381, 13, 56, "Text"] }, Closed]], Cell[26236, 622, 1424, 38, 112, "Text", Evaluatable->False], Cell[27663, 662, 178, 4, 33, "Text", Evaluatable->False], Cell[27844, 668, 469, 15, 71, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[28338, 687, 103, 2, 33, "Text"], Cell[28444, 691, 92, 3, 33, "Text"], Cell[28539, 696, 1712, 32, 486, "Input"], Cell[30254, 730, 189, 7, 52, "Text", Evaluatable->False], Cell[30446, 739, 604, 12, 110, "Input"], Cell[31053, 753, 135, 5, 52, "Text", Evaluatable->False], Cell[31191, 760, 3826, 76, 677, "Input"] }, Closed]], Cell[35032, 839, 463, 14, 68, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[35520, 857, 103, 2, 33, "Text"], Cell[35626, 861, 92, 3, 33, "Text"], Cell[35721, 866, 1707, 32, 486, "Input"], Cell[37431, 900, 291, 8, 57, "Text", Evaluatable->False], Cell[37725, 910, 614, 13, 182, "Input"], Cell[38342, 925, 183, 7, 52, "Text", Evaluatable->False], Cell[38528, 934, 594, 12, 110, "Input"], Cell[39125, 948, 135, 5, 52, "Text", Evaluatable->False], Cell[39263, 955, 4043, 80, 667, "Input"] }, Closed]], Cell[43321, 1038, 569, 19, 54, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[43915, 1061, 103, 2, 33, "Text"], Cell[44021, 1065, 217, 7, 38, "Text"], Cell[44241, 1074, 410, 9, 147, "Input"], Cell[44654, 1085, 129, 5, 52, "Text", Evaluatable->False], Cell[44786, 1092, 412, 9, 131, "Input"], Cell[45201, 1103, 155, 6, 52, "Text", Evaluatable->False], Cell[45359, 1111, 954, 19, 230, "Input"], Cell[46316, 1132, 618, 19, 71, "Text", Evaluatable->False], Cell[46937, 1153, 242, 5, 110, "Input"], Cell[47182, 1160, 166, 6, 52, "Text", Evaluatable->False], Cell[47351, 1168, 3611, 70, 702, "Input"], Cell[50965, 1240, 472, 15, 57, "Text", Evaluatable->False] }, Closed]], Cell[51452, 1258, 361, 9, 68, "Text", Evaluatable->False], Cell[51816, 1269, 180, 4, 33, "Text", Evaluatable->False], Cell[51999, 1275, 491, 15, 71, "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[52515, 1294, 103, 2, 33, "Text"], Cell[52621, 1298, 771, 21, 58, "Text", Evaluatable->False], Cell[53395, 1321, 1660, 32, 486, "Input"], Cell[55058, 1355, 183, 7, 52, "Text", Evaluatable->False], Cell[55244, 1364, 606, 12, 110, "Input"], Cell[55853, 1378, 442, 15, 52, "Text", Evaluatable->False], Cell[56298, 1395, 867, 16, 150, "Input"], Cell[57168, 1413, 196, 6, 52, "Text", Evaluatable->False], Cell[57367, 1421, 684, 15, 230, "Input"], Cell[58054, 1438, 126, 5, 52, "Text", Evaluatable->False], Cell[58183, 1445, 4154, 81, 681, "Input"], Cell[62340, 1528, 515, 16, 74, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[62892, 1549, 166, 3, 34, "Text"], Cell[63061, 1554, 2485, 68, 414, "Text"] }, Closed]], Cell[65561, 1625, 477, 16, 57, "Text"] }, Open ]], Cell[66053, 1644, 594, 20, 35, "Text"], Cell[66650, 1666, 590, 20, 35, "Text"], Cell[67243, 1688, 339, 13, 57, "Text"] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)