├── .gitignore ├── LICENSE.md ├── README.md ├── TODO ├── ca.tex ├── changes-draft.html ├── countwords.sh ├── desc.txt ├── figures ├── .gitattributes ├── anyantidef.fig ├── anyantidef.pdf ├── anyantidef.pdf_t ├── arggraph.pdf ├── arggraph.xp ├── arggraph2.pdf ├── arggraph2.xp ├── ballisopen.fig ├── ballisopen.pdf ├── ballisopen.pdf_t ├── cauchy-pompeiu.fig ├── cauchy-pompeiu.pdf ├── cauchy-pompeiu.pdf_t ├── cayleydisc.fig ├── cayleydisc.pdf ├── cayleydisc.pdf_t ├── circlepath.pdf ├── circlepath.xp ├── compactbnd.fig ├── compactbnd.pdf ├── compactbnd.pdf_t ├── compactclosed.fig ├── compactclosed.pdf ├── compactclosed.pdf_t ├── conform.fig ├── conform.pdf ├── conform.pdf_t ├── conv1nsinxn.pdf ├── conv1nsinxn.xp ├── convexfunc.fig ├── convexfunc.pdf ├── convexfunc.pdf_t ├── cyclearoundcompact.fig ├── cyclearoundcompact.pdf ├── cyclearoundcompact.pdf_t ├── diffunderexample.pdf ├── diffunderexample.xp ├── disccauchy.fig ├── disccauchy.pdf ├── disccauchy.pdf_t ├── disconsubset.fig ├── disconsubset.pdf ├── disconsubset.pdf_t ├── dogandtree.fig ├── dogandtree.pdf ├── dogandtree.pdf_t ├── expplotlines.pdf ├── expplotlines.xp ├── figure-pdf15.sh ├── fillholes.fig ├── fillholes.pdf ├── fillholes.pdf_t ├── followbranch.fig ├── followbranch.pdf ├── followbranch.pdf_t ├── gammas.fig ├── gammas.pdf ├── gammas.pdf_t ├── harnacksgeneral.fig ├── harnacksgeneral.pdf ├── harnacksgeneral.pdf_t ├── homologousnothomotopic.fig ├── homologousnothomotopic.pdf ├── homologousnothomotopic.pdf_t ├── homotopy.fig ├── homotopy.pdf ├── homotopy.pdf_t ├── homotopyfixed.fig ├── homotopyfixed.pdf ├── homotopyfixed.pdf_t ├── imagexp.pdf ├── imagexp.xp ├── indexconstant.fig ├── indexconstant.pdf ├── indexconstant.pdf_t ├── intervalcon.fig ├── intervalcon.pdf ├── intervalcon.pdf_t ├── inversefuncRn.fig ├── inversefuncRn.pdf ├── inversefuncRn.pdf_t ├── inversefuncRn2.fig ├── inversefuncRn2.pdf ├── inversefuncRn2.pdf_t ├── inversion.fig ├── inversion.pdf ├── inversion.pdf_t ├── kkprime.fig ├── kkprime.pdf ├── kkprime.pdf_t ├── largestr.fig ├── largestr.pdf ├── largestr.pdf_t ├── lebesguedelta.fig ├── lebesguedelta.pdf ├── lebesguedelta.pdf_t ├── license.pdf ├── license2.pdf ├── localjordan.fig ├── localjordan.pdf ├── localjordan.pdf_t ├── logrealgraph.pdf ├── logrealgraph.xp ├── meanvalue.fig ├── meanvalue.pdf ├── meanvalue.pdf_t ├── ms-triang.fig ├── ms-triang.pdf ├── ms-triang.pdf_t ├── msboundary.fig ├── msboundary.pdf ├── msboundary.pdf_t ├── mscontfuncpt.fig ├── mscontfuncpt.pdf ├── mscontfuncpt.pdf_t ├── msopenset.fig ├── msopenset.pdf ├── msopenset.pdf_t ├── poisson-graph.pdf ├── poisson-graph.xp ├── poisson-kernel.pdf ├── polarcoords.fig ├── polarcoords.pdf ├── polarcoords.pdf_t ├── proofofdirichlet.fig ├── proofofdirichlet.pdf ├── proofofdirichlet.pdf_t ├── radiusconvcomplex.fig ├── radiusconvcomplex.pdf ├── radiusconvcomplex.pdf_t ├── radosthm.fig ├── radosthm.pdf ├── radosthm.pdf_t ├── rayjordan.fig ├── rayjordan.pdf ├── rayjordan.pdf_t ├── realexp.pdf ├── realexp.xp ├── residues.fig ├── residues.pdf ├── residues.pdf_t ├── rhalfcircle.fig ├── rhalfcircle.pdf ├── rhalfcircle.pdf_t ├── riemannmap.pdf ├── riemannmap.xp ├── riemannsphere.pdf ├── riemannsphere.xp ├── roots.fig ├── roots.pdf ├── roots.pdf_t ├── run-all-xp.sh ├── run-xp-topdf.sh ├── rungepolepush.fig ├── rungepolepush.pdf ├── rungepolepush.pdf_t ├── schwarzreflection.fig ├── schwarzreflection.pdf ├── schwarzreflection.pdf_t ├── seqcompactiscompact.fig ├── seqcompactiscompact.pdf ├── seqcompactiscompact.pdf_t ├── sequence-convergence-metric.pdf ├── sequence-convergence-metric.xp ├── spheremetric.fig ├── spheremetric.pdf ├── spheremetric.pdf_t ├── splittriang.fig ├── splittriang.pdf ├── splittriang.pdf_t ├── squarepath.pdf ├── squarepath.xp ├── starshaped.fig ├── starshaped.pdf ├── starshaped.pdf_t ├── svder.pdf ├── svder.xp ├── threediscs.fig ├── threediscs.pdf ├── threediscs.pdf_t ├── triang.fig ├── triang.pdf ├── triang.pdf_t ├── triangantidef.fig ├── triangantidef.pdf ├── triangantidef.pdf_t ├── trianggoursat.fig ├── trianggoursat.pdf ├── trianggoursat.pdf_t ├── twoannuli.fig ├── twoannuli.pdf ├── twoannuli.pdf_t ├── uniformconv.pdf ├── uniformconv.xp ├── varphiplot.pdf ├── varphiplot.xp ├── weiergeneral.fig ├── weiergeneral.pdf ├── weiergeneral.pdf_t ├── xyxsqysq.pdf ├── xyxsqysq.xp ├── zsqplot.pdf └── zsqplot.xp ├── getstats.sh ├── notations.tex ├── publish.sh ├── resizepdftocrownquatro.sh ├── slides ├── .gitignore ├── 1.1-p1.tex ├── 1.1-p2.tex ├── 1.1-p3.tex ├── 1.2-p1.tex ├── 1.2-p2.tex ├── 1.3.tex ├── 1.4-p1.tex ├── 1.4-p2.tex ├── 2.1.1.tex ├── 2.1.2.tex ├── 2.2.1.tex ├── 2.2.2.tex ├── 2.2.3.tex ├── 2.3-p1.tex ├── 2.3-p2.tex ├── 2.3-p3.tex ├── 2.4.1-2.4.2.tex ├── 2.4.3.tex ├── 2.4.4.tex ├── 3.1-p1.tex ├── 3.1-p2.tex ├── 3.1-p3.tex ├── 3.2.1.tex ├── 3.2.2.tex ├── 3.2.3.tex ├── 3.2.4.tex ├── 3.3.1.tex ├── 3.3.2.tex ├── 3.3.3.tex ├── 3.3.4.tex ├── 3.4.1.tex ├── 3.4.2.tex ├── 3.5.1.tex ├── 3.5.2.tex ├── 4.1.1.tex ├── 4.1.2.tex ├── 4.2.tex ├── 4.3-p1.tex ├── 4.3-p2.tex ├── 4.4-p1.tex ├── 4.4-p2.tex ├── 5.1.tex ├── 5.2.1.tex ├── 5.2.2.tex ├── 5.2.3.tex ├── 5.2.4.tex ├── 5.3-p1.tex ├── 5.3-p2.tex ├── 5.4.1.tex ├── 5.4.2.tex ├── 5.4.3.tex ├── 5.5.tex ├── 5.6.tex ├── 6.1.1-6.1.2.tex ├── 6.1.3.tex ├── 6.2.tex ├── 6.3.1.tex ├── 6.3.3.tex ├── 7.1.1.tex ├── 7.1.2.tex ├── 7.2.1.tex ├── 7.2.2.tex ├── 7.2.3-7.2.4.tex ├── README.md ├── ccaslides.sty ├── essential-sing.png ├── intro.tex ├── make-zip.sh └── vogtwidebar.sty └── vogtwidebar.sty /.gitignore: -------------------------------------------------------------------------------- 1 | ca.aux 2 | ca.idx 3 | ca.ilg 4 | ca.ind 5 | ca.log 6 | ca.out 7 | ca.pdf 8 | ca.toc 9 | ca.ist 10 | ca.glg 11 | ca.glo 12 | ca.gls 13 | ca.rubbercache 14 | ca.fdb_latexmk 15 | ca.fls 16 | ca.synctex.gz 17 | -------------------------------------------------------------------------------- /LICENSE.md: -------------------------------------------------------------------------------- 1 | # Dual license CC-BY-NC-SA and CC-BY-SA 2 | 3 | This work is dual licensed under the 4 | 5 | *Creative Commons Attribution-Non-commercial-Share Alike 4.0 International License* 6 | 7 | and the 8 | 9 | *Creative Commons Attribution-Share Alike 4.0 International License*. 10 | 11 | To view a copy of these licenses, visit http://creativecommons.org/licenses/by-nc-sa/4.0/ or http://creativecommons.org/licenses/by-sa/4.0/ or send a letter to 12 | 13 | Creative Commons 14 | PO Box 1866, Mountain View, CA 94042, USA. 15 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | Guide to Cultivating Complex Analysis: Working the Complex Field 2 | ---------------------------------------------------------------- 3 | 4 | One-semester graduate level mathematics textbook on basic complex analysis in 5 | one variable. Written for the graduate complex analysis course (5283) at 6 | Oklahoma State University. There is a bit more than a one-semester's worth of 7 | material in there to allow some choices. If the metric space appendix is also 8 | covered first, then it is quite a bit more than one-semester's worth. 9 | 10 | See https://www.jirka.org/ca/ 11 | 12 | * ca.tex is the LaTeX file 13 | * notations.tex is the list of notations 14 | * Figures are in figures/ 15 | * Some beamer slides are in slides/ 16 | * publish.sh compiles the file to pdf (probably overdoing it, but that doesn't hurt) 17 | * getstats.sh counts the numbers of sections, exercises, figures, etc. 18 | * resizepdftocrownquatro.sh resizes a pdf to Crown Quatro size (which is what a printed version uses) 19 | -------------------------------------------------------------------------------- /TODO: -------------------------------------------------------------------------------- 1 | More minor things to do/think about in the next version: 2 | 3 | * Mark important exercises, once that are used, or once that are important 4 | for the exposition, rather than just being exercises to try. 5 | 6 | * Perhaps add proposition about Laurent series convergence in 4.4 (that is, 7 | it converges on compact subsets of the annulus) we only say this in text 8 | right now. 9 | 10 | * Change Proposition 6.1.4 to be about a set of functions rather than a 11 | sequence? Note that all the other propositions/theorems are stated for 12 | sequences. Except in the Montel section where we switch to sets of 13 | functions. Perhaps we should move the whole 6.1 section also to sets where 14 | possible. On the other hand, we really only ever care for sequences. 15 | Does it make much difference? 16 | 17 | Things definitely for the next big version 18 | 19 | * Riemann surfaces 20 | 21 | * Blaschke products 22 | 23 | * Schottky's theorem (equivalence of annuli) should go somewhere, perhaps as exercise, but need the machinery 24 | 25 | * Inhomogeneous dbar solution (See Hormander) using Mittag-Leffler. 26 | 27 | * Hopf lemma (in a disc) 28 | 29 | Possibilities 30 | 31 | * Proof of our version of Green's in an appendix 32 | -------------------------------------------------------------------------------- /changes-draft.html: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/changes-draft.html -------------------------------------------------------------------------------- /countwords.sh: -------------------------------------------------------------------------------- 1 | #!/bin/sh 2 | detex ca.tex | grep -o -E '\w+' | tr '[A-Z]' '[a-z]' | sort | uniq -c | sort -nr 3 | -------------------------------------------------------------------------------- /desc.txt: -------------------------------------------------------------------------------- 1 | An introductory course in complex analysis for incoming graduate students. Created to teach Math 5283 at Oklahoma State University. The book has somewhat more material than could fit in a one-semester course, allowing some choices. There are also appendices on metric spaces and some basic analysis background to make for a longer and more complete course for those that have only had an introduction to basic analysis on the real line. 2 | 3 | Table of contents: 4 | 5 | Introduction 6 | 1. The Complex Plane 7 | 2. Holomorphic and Analytic Functions 8 | 3. Line Integrals and Rudimentary Cauchy Theorems 9 | 4. The Logarithm and Cauchy 10 | 5. Counting Zeros and Singularities 11 | 6. Montel and Riemann 12 | 7. Harmonic Functions 13 | 8. Weierstrass Factorization 14 | 9. Rational Approximation 15 | 10. Analytic Continuation 16 | Appendix A. Metric Spaces 17 | Appendix B. Results From Basic Analysis 18 | Appendix C. Basic Notation and Terminology 19 | 20 | Jiří Lebl is a Professor of Mathematics at the Oklahoma State University. 21 | -------------------------------------------------------------------------------- /figures/.gitattributes: -------------------------------------------------------------------------------- 1 | *.pdf binary 2 | -------------------------------------------------------------------------------- /figures/anyantidef.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #dddddd 11 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1800 630 18 18 1800 630 1818 630 12 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 -405 720 18 18 -405 720 -387 720 13 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1665 360 18 18 1665 360 1683 360 14 | 1 3 2 1 0 32 50 -1 -1 3.000 1 0.0000 1800 630 343 343 1800 630 2115 765 15 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 16 | 1800 630 1665 360 17 | 3 3 2 1 0 32 52 -1 20 1.000 0 0 0 6 18 | -765 405 675 1080 1305 225 3195 630 1575 1575 -540 1575 19 | -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 20 | 3 2 0 1 0 7 50 -1 -1 0.000 0 0 0 10 21 | -405 720 -90 900 -225 1125 360 1170 675 1485 1079 1426 22 | 1073 1021 1763 1258 1295 589 1800 630 23 | 0.000 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 24 | -0.500 0.000 25 | 4 0 0 50 -1 0 12 0.0000 2 180 315 1845 720 $q$\001 26 | 4 2 0 50 -1 0 12 0.0000 2 180 315 -450 720 $p$\001 27 | 4 1 0 50 -1 0 12 0.0000 2 150 345 2790 720 $U$\001 28 | 4 2 0 50 -1 0 12 0.0000 2 180 855 495 1485 $\\gamma$\001 29 | 4 0 0 50 -1 0 12 0.0000 2 150 300 1733 432 $z$\001 30 | -------------------------------------------------------------------------------- /figures/anyantidef.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/anyantidef.pdf -------------------------------------------------------------------------------- /figures/anyantidef.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{anyantidef.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{4144sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 | \gdef\SetFigFont#1#2#3#4#5{% 8 | \reset@font\fontsize{#1}{#2pt}% 9 | \fontfamily{#3}\fontseries{#4}\fontshape{#5}% 10 | \selectfont}% 11 | \fi\endgroup% 12 | \begin{picture}(4059,1485)(-850,-848) 13 | \put(1846,119){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$q$}% 14 | }}}} 15 | \put(-449,119){\makebox(0,0)[rb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$p$}% 16 | }}}} 17 | \put(2791,119){\makebox(0,0)[b]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$U$}% 18 | }}}} 19 | \put(496,-646){\makebox(0,0)[rb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\gamma$}% 20 | }}}} 21 | \put(1734,407){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$z$}% 22 | }}}} 23 | \end{picture}% 24 | -------------------------------------------------------------------------------- /figures/arggraph.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/arggraph.pdf -------------------------------------------------------------------------------- /figures/arggraph.xp: -------------------------------------------------------------------------------- 1 | /* -*-ePiX-*- */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | P F(double x, double y) 6 | { 7 | return P(x, y, 2*Atan(y/(x+sqrt(x*x+y*y)))); 8 | } 9 | 10 | P FF(double r, double th) 11 | { 12 | double x = r*Cos(th); 13 | double y = r*Sin(th); 14 | return F(x,y); 15 | } 16 | 17 | domain RR(P(sqrt(0.000001),-M_PI+0.00001), P(sqrt(2), M_PI-0.00001), mesh(5,40), mesh(10,80)); 18 | 19 | int main() 20 | { 21 | picture(P(-1.5,-M_PI*0.7), P(1.5,M_PI*0.7), "2.5x2.5in"); 22 | 23 | begin(); 24 | //border(); 25 | 26 | camera.at(P(-8,-8,35)); 27 | 28 | arrow(P(-1.9,0,0), P(1.9,0,0)); 29 | label(P(1.9,0), P(2,4), "$\\operatorname{Re} z$", r); 30 | 31 | arrow(P(0,0,0), P(0,2.1,0)); 32 | label(P(0,2.2), P( 2,8), "$\\operatorname{Im} z$", br); 33 | 34 | plain (); 35 | fill(White()); 36 | surface(FF, RR); // polar 37 | 38 | // that bit of the y axis that's above the graph 39 | line(P(0,-1.9,0), P(0,0,0)); 40 | 41 | tikz_format(); 42 | end(); 43 | } 44 | -------------------------------------------------------------------------------- /figures/arggraph2.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/arggraph2.pdf -------------------------------------------------------------------------------- /figures/arggraph2.xp: -------------------------------------------------------------------------------- 1 | /* -*-ePiX-*- */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | double addz = 0.0; 6 | double zscale = 0.3; 7 | 8 | P F(double x, double y) 9 | { 10 | return P(x, y, zscale*Atan(y/(x+sqrt(x*x+y*y)))+ addz); 11 | } 12 | 13 | P FF(double r, double th) 14 | { 15 | double x = r*Cos(th); 16 | double y = r*Sin(th); 17 | return F(x,y); 18 | } 19 | 20 | domain RR(P(sqrt(0.000001),-M_PI+0.00001), P(sqrt(2), M_PI-0.00001), mesh(5,40), mesh(10,80)); 21 | 22 | int main() 23 | { 24 | picture(P(-1.5,-M_PI*0.7), P(1.5,M_PI*0.7), "2.5x2.5in"); 25 | 26 | begin(); 27 | //border(); 28 | 29 | camera.at(P(8,8,10)); 30 | 31 | arrow(P(-1.9,0,0), P(1.9,0,0)); 32 | label(P(1.9,0), P(0,-2), "$\\operatorname{Re} z$", br); 33 | 34 | arrow(P(0,-1.9,0), P(0,1.9,0)); 35 | label(P(0,1.9), P(0,-2), "$\\operatorname{Im} z$", bl); 36 | 37 | plain (); 38 | fill(White()); 39 | surface(FF, RR); // polar 40 | fill(false); 41 | 42 | pen(2); 43 | plot(FF, RR.resize2(0,M_PI).slice1(1)); 44 | dot(P(1,0,0)); 45 | plain(); 46 | 47 | addz=zscale*M_PI; 48 | fill(White()); 49 | surface(FF, RR); // polar 50 | fill(false); 51 | 52 | pen(2); 53 | /* When moving camera, the 0.75 must be adjusted*/ 54 | plot(FF, RR.resize2(-M_PI*0.75,0).slice1(1)); 55 | dot(P(1,0,zscale*M_PI)); 56 | plain(); 57 | 58 | 59 | //addz=2*M_PI; 60 | //surface(FF, RR); // polar 61 | 62 | arrow(P(0,0,0), P(0,0,1.7)); 63 | 64 | tikz_format(); 65 | end(); 66 | } 67 | -------------------------------------------------------------------------------- /figures/ballisopen.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.6a 2 | Landscape 3 | Center 4 | 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$B(x,\\delta)$\001 31 | 4 0 0 50 -1 0 12 0.0000 2 195 720 992 436 $\\alpha$\001 32 | 4 0 0 50 -1 0 12 0.0000 2 195 315 968 644 $y$\001 33 | -------------------------------------------------------------------------------- /figures/ballisopen.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/ballisopen.pdf -------------------------------------------------------------------------------- /figures/ballisopen.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{ballisopen.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{4144sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 | \gdef\SetFigFont#1#2#3#4#5{% 8 | \reset@font\fontsize{#1}{#2pt}% 9 | \fontfamily{#3}\fontseries{#4}\fontshape{#5}% 10 | \selectfont}% 11 | \fi\endgroup% 12 | \begin{picture}(1500,1498)(241,-855) 13 | \put(901,-286){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$x$}% 14 | }}}} 15 | \put(454,-156){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\delta$}% 16 | }}}} 17 | \put(1285,287){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$z$}% 18 | }}}} 19 | \put(690,-615){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$B(x,\delta)$}% 20 | }}}} 21 | \put(993,403){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\alpha$}% 22 | }}}} 23 | \put(969,195){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$y$}% 24 | }}}} 25 | \end{picture}% 26 | 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-0.500 -1.000 -0.500 -0.500 28 | 4 0 0 50 -1 0 12 0.0000 2 150 345 630 2205 $U$\001 29 | 4 0 0 50 -1 0 12 0.0000 2 165 1080 2786 2430 $\\Delta_r(z)$\001 30 | 4 0 0 50 -1 0 12 0.0000 2 150 300 1895 2398 $z$\001 31 | 4 0 0 50 -1 0 12 0.0000 2 150 885 3864 1840 $\\Gamma$\001 32 | -------------------------------------------------------------------------------- /figures/cauchy-pompeiu.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/cauchy-pompeiu.pdf -------------------------------------------------------------------------------- /figures/cauchy-pompeiu.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{cauchy-pompeiu.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{4144sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 | \gdef\SetFigFont#1#2#3#4#5{% 8 | 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version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #d9d9d9 11 | 0 33 #eaeaea 12 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1575 1305 23 23 1575 1305 1598 1305 13 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1575 2295 23 23 1575 2295 1598 2295 14 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 810 1440 23 23 810 1440 833 1440 15 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 16 | 0 1800 3150 1800 17 | 2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2 18 | 0 0 1.00 60.00 120.00 19 | 0 0 1.00 60.00 120.00 20 | 1575 1305 810 1440 21 | 2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2 22 | 0 0 1.00 60.00 120.00 23 | 0 0 1.00 60.00 120.00 24 | 810 1440 1575 2295 25 | 2 2 0 0 0 32 52 -1 20 0.000 0 0 -1 0 0 5 26 | 45 1035 3105 1035 3105 1800 45 1800 45 1035 27 | 4 0 0 50 -1 0 12 0.0000 2 150 270 1665 1350 $i$\001 28 | 4 0 0 50 -1 0 12 0.0000 2 150 330 1665 2385 $-i$\001 29 | 4 2 0 50 -1 0 12 0.0000 2 150 300 720 1440 $z$\001 30 | 4 1 0 50 -1 0 12 0.0000 2 150 510 2565 1530 $\\bH$\001 31 | -------------------------------------------------------------------------------- /figures/cayleydisc.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/cayleydisc.pdf -------------------------------------------------------------------------------- /figures/cayleydisc.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{cayleydisc.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{4144sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 | \gdef\SetFigFont#1#2#3#4#5{% 8 | \reset@font\fontsize{#1}{#2pt}% 9 | \fontfamily{#3}\fontseries{#4}\fontshape{#5}% 10 | \selectfont}% 11 | \fi\endgroup% 12 | \begin{picture}(3174,1411)(-11,-1606) 13 | 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-------------------------------------------------------------------------------- 1 | /* slightly modified sample from epix samples gallery */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | int main() 6 | { 7 | double slope; 8 | double c; 9 | double x; 10 | 11 | picture(P(-1.3,-1.3), P(1.3,1.3), "1.5x1.5in"); 12 | 13 | begin(); 14 | //border(); 15 | 16 | //line(P(-0.3,0),P(1.3,0)); 17 | //line(P(0,-0.3),P(0,1.3)); 18 | //h_axis(3); 19 | //v_axis(3); 20 | // 21 | 22 | arrow_inset(0.9); 23 | arrow(P(0,0),P(0.4,0)); 24 | arrow(P(0,0),P(0,0.4)); 25 | arrow_inset(0); 26 | 27 | 28 | bold(); 29 | solid(); 30 | arc_arrow(P(0,0),1,0,2*M_PI/3); 31 | arc_arrow(P(0,0),1,2*M_PI/3,4*M_PI/3); 32 | arc_arrow(P(0,0),1,4*M_PI/3,2*M_PI); 33 | 34 | dot(P(1,0)); 35 | label(P(1.09,0), P(0,7), "$t=0$", r); 36 | label(P(1.09,0), P(0,-7), "$t=2\\pi$", r); 37 | 38 | dot(P(0,1)); 39 | label(P(0,1), P(0,2), "$t=\\frac{\\pi}{2}$", t); 40 | 41 | dot(P(-1,0)); 42 | label(P(-1,0), P(-5,0), "$t = \\pi$", l); 43 | 44 | tikz_format(); 45 | end(); 46 | } 47 | 48 | -------------------------------------------------------------------------------- /figures/compactbnd.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.6a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #41534c 11 | 0 33 #555555 12 | 0 34 #888888 13 | 0 35 #e9e9e9 14 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 990 945 18 18 990 945 1008 945 15 | 1 3 2 1 0 35 53 -1 20 1.000 1 0.0000 990 945 946 946 990 945 1936 945 16 | 1 3 2 1 0 7 47 -1 -1 1.000 1 0.0000 990 945 630 630 990 945 1620 945 17 | 1 3 2 1 0 7 48 -1 -1 1.000 1 0.0000 990 945 315 315 990 945 1305 945 18 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 19 | 0 0 1.00 60.00 120.00 20 | 990 945 1935 810 21 | 2 1 0 1 0 7 48 -1 -1 0.000 0 0 -1 1 0 2 22 | 0 0 1.00 60.00 120.00 23 | 990 945 1395 450 24 | 2 1 0 1 0 7 48 -1 -1 0.000 0 0 -1 1 0 2 25 | 0 0 1.00 60.00 120.00 26 | 990 945 945 630 27 | 3 1 0 1 0 34 51 -1 20 0.000 0 0 0 5 28 | 135 990 225 90 1125 450 1395 900 180 1665 29 | 1.000 1.000 1.000 1.000 1.000 30 | 3 2 0 1 0 7 50 -1 -1 0.000 0 1 0 3 31 | 0 0 1.00 60.00 120.00 32 | 2115 1575 1791 1179 1183 1064 33 | 0.000 -1.000 0.000 34 | 4 0 0 50 -1 0 12 0.0000 2 165 345 315 495 $K$\001 35 | 4 0 0 50 -1 0 12 0.0000 2 165 315 810 900 $1$\001 36 | 4 0 0 50 -1 0 12 0.0000 2 165 315 1305 765 $2$\001 37 | 4 0 0 50 -1 0 12 0.0000 2 195 720 675 1755 $B(p,3)$\001 38 | 4 0 0 50 -1 0 12 0.0000 2 195 720 765 1440 $B(p,2)$\001 39 | 4 0 0 50 -1 0 12 0.0000 2 165 315 1710 1035 $3$\001 40 | 4 0 0 50 -1 0 12 0.0000 2 195 720 1867 1746 $B(p,1)$\001 41 | 4 0 0 50 -1 0 12 0.0000 2 195 315 869 1099 $p$\001 42 | -------------------------------------------------------------------------------- /figures/compactbnd.pdf: -------------------------------------------------------------------------------- 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-------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.6a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #41534c 11 | 0 33 #555555 12 | 0 34 #888888 13 | 0 35 #dddddd 14 | 0 36 #bbbbbb 15 | 0 37 #cccccc 16 | 0 38 #eeeeee 17 | 0 39 #aaaaaa 18 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 990 945 18 18 990 945 1008 945 19 | 1 3 0 1 0 7 47 -1 -1 0.000 1 0.0000 990 945 405 405 990 945 1395 945 20 | 1 3 0 1 0 7 48 -1 -1 0.000 1 0.0000 990 945 270 270 990 945 1260 945 21 | 1 3 0 0 0 36 47 -1 -1 0.000 1 0.0000 990 945 811 811 990 945 1801 945 22 | 1 3 0 1 0 36 47 -1 -1 3.000 1 0.0000 990 945 811 811 990 945 1800 900 23 | 1 3 0 0 0 36 56 -1 20 0.000 1 0.0000 990 945 810 810 990 945 1800 945 24 | 1 3 0 0 0 35 54 -1 20 0.000 1 0.0000 990 945 270 270 990 945 1260 945 25 | 1 3 0 0 0 37 55 -1 20 0.000 1 0.0000 990 945 405 405 990 945 1395 945 26 | 1 3 0 0 0 38 53 -1 20 1.500 1 0.0000 990 945 186 186 990 945 1170 900 27 | 1 3 0 1 0 36 47 -1 -1 0.000 1 0.0000 990 945 186 186 990 945 1176 945 28 | 2 2 0 0 0 39 61 -1 20 0.000 0 0 -1 0 0 5 29 | -90 0 2970 0 2970 1890 -90 1890 -90 0 30 | 3 2 0 1 0 38 53 -1 -1 3.000 0 1 0 3 31 | 0 0 1.00 60.00 120.00 32 | 1845 1395 1485 1350 1230 1154 33 | 0.000 -1.000 0.000 34 | 3 2 0 1 0 38 53 -1 -1 3.000 0 1 0 3 35 | 0 0 1.00 60.00 120.00 36 | 1845 855 1485 810 1212 955 37 | 0.000 -1.000 0.000 38 | 3 2 0 1 0 38 53 -1 -1 3.000 0 1 0 3 39 | 0 0 1.00 60.00 120.00 40 | 1845 360 1350 450 1070 915 41 | 0.000 -1.000 0.000 42 | 3 1 0 1 0 34 51 -1 20 0.000 0 0 0 5 43 | 0 945 90 45 990 405 1260 855 45 1620 44 | 1.000 1.000 1.000 1.000 1.000 45 | 4 0 0 50 -1 0 12 0.0000 2 180 720 630 1575 $C(x,1)$\001 46 | 4 0 0 50 -1 0 12 0.0000 2 180 1890 1890 450 $C(x,\\nicefrac{1}{4})$\001 47 | 4 0 0 50 -1 0 12 0.0000 2 180 1890 1890 1485 $C(x,\\nicefrac{1}{2})$\001 48 | 4 0 0 50 -1 0 12 0.0000 2 180 1890 1890 945 $C(x,\\nicefrac{1}{3})$\001 49 | 4 0 0 50 -1 0 12 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-------------------------------------------------------------------------------- /figures/conform.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Inches 5 | Letter 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #898989 11 | 0 33 #d8d8d8 12 | 5 1 0 1 0 7 50 -1 -1 0.000 0 1 1 0 1591.782 2519.061 1950 2325 1575 2112 1526 2117 13 | 0 0 1.00 60.00 120.00 14 | 6 3702 1895 5847 3005 15 | 5 1 0 1 0 7 50 -1 -1 0.000 0 1 1 0 4263.386 2481.697 4379 2090 3963 2205 3932 2243 16 | 0 0 1.00 60.00 120.00 17 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 4272 2514 13 13 4272 2514 4285 2514 18 | 2 1 2 1 0 7 50 -1 -1 1.000 0 0 -1 0 0 3 19 | 3743 2090 4273 2515 4447 1911 20 | 3 2 0 1 0 7 50 -1 -1 0.000 0 0 0 5 21 | 4678 2556 4409 2648 4273 2515 4136 2426 3802 2310 22 | 0.000 -0.500 -0.500 -0.500 0.000 23 | 3 2 0 1 0 7 50 -1 -1 0.000 0 0 0 5 24 | 3965 2834 4122 2813 4273 2515 4388 2186 4604 1914 25 | 0.000 -0.500 -0.500 -0.500 0.000 26 | 4 0 0 50 -1 0 12 0.0000 2 150 675 4021 2030 $\\theta$\001 27 | 4 2 0 50 -1 0 12 0.0000 2 180 495 4197 2622 $f(p)$\001 28 | 4 0 0 50 -1 0 12 0.0000 2 180 1140 4707 2605 $f\\circ\\alpha$\001 29 | 4 0 0 50 -1 0 12 0.0000 2 180 1275 4148 2960 $f\\circ\\gamma$\001 30 | -6 31 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1574 2550 13 13 1574 2550 1587 2550 32 | 2 1 2 1 0 7 50 -1 -1 1.000 0 0 -1 0 0 3 33 | 1500 1875 1575 2550 2250 2175 34 | 3 2 0 1 0 7 50 -1 -1 0.000 0 0 0 5 35 | 1500 3075 1575 2925 1575 2550 1500 2250 1350 2175 36 | 0.000 -0.500 -0.500 -0.500 0.000 37 | 3 2 0 1 0 7 50 -1 -1 0.000 0 0 0 5 38 | 825 2700 1125 2850 1575 2550 1875 2475 2250 2475 39 | 0.000 -0.500 -0.500 -0.500 0.000 40 | 3 0 0 1 0 7 50 -1 -1 3.000 0 1 0 3 41 | 0 0 1.00 60.00 120.00 42 | 2533 2404 2977 2291 3448 2439 43 | 0.000 1.000 0.000 44 | 4 0 0 50 -1 0 12 0.0000 2 150 675 1798 2122 $\\theta$\001 45 | 4 0 0 50 -1 0 12 0.0000 2 180 720 1599 3028 $\\alpha$\001 46 | 4 0 0 50 -1 0 12 0.0000 2 180 315 1623 2696 $p$\001 47 | 4 0 0 50 -1 0 12 0.0000 2 180 855 875 2658 $\\gamma$\001 48 | 4 1 0 50 -1 0 12 0.0000 2 150 270 2980 2251 $f$\001 49 | -------------------------------------------------------------------------------- /figures/conform.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/conform.pdf -------------------------------------------------------------------------------- /figures/conform.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{conform.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{3947sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 | \gdef\SetFigFont#1#2#3#4#5{% 8 | \reset@font\fontsize{#1}{#2pt}% 9 | \fontfamily{#3}\fontseries{#4}\fontshape{#5}% 10 | \selectfont}% 11 | \fi\endgroup% 12 | \begin{picture}(3909,1234)(814,-2258) 13 | \put(4022,-1191){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\theta$}% 14 | }}}} 15 | \put(4198,-1783){\makebox(0,0)[rb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$f(p)$}% 16 | }}}} 17 | \put(4708,-1766){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$f\circ\alpha$}% 18 | }}}} 19 | \put(4149,-2121){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$f\circ\gamma$}% 20 | }}}} 21 | \put(1799,-1283){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\theta$}% 22 | }}}} 23 | \put(1600,-2189){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\alpha$}% 24 | }}}} 25 | \put(1624,-1857){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$p$}% 26 | }}}} 27 | \put(876,-1819){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\gamma$}% 28 | }}}} 29 | \put(2981,-1412){\makebox(0,0)[b]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$f$}% 30 | }}}} 31 | \end{picture}% 32 | -------------------------------------------------------------------------------- /figures/conv1nsinxn.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/conv1nsinxn.pdf -------------------------------------------------------------------------------- /figures/conv1nsinxn.xp: -------------------------------------------------------------------------------- 1 | #include "epix.h" 2 | #include 3 | #include 4 | using namespace ePiX; 5 | 6 | int n = 1; 7 | 8 | double f(double t) 9 | { 10 | return (1.0/n)*sin(t*n); 11 | } 12 | 13 | int main() 14 | { 15 | int i; 16 | picture(P(-0.1,-1), P(7,1), "5x1.6in"); 17 | 18 | begin(); 19 | 20 | //border(); 21 | h_axis(P(0,0),P(7,0),7); 22 | v_axis(P(0,-1),P(0,1),2); 23 | 24 | plain(); 25 | 26 | for (i=10; i >= 1; i--) { 27 | n=i; 28 | pen(Black(1.0 - 0.1*(i-1))); 29 | plot(f, -0.1, 7.0, 40*i); 30 | } 31 | 32 | //n=10; 33 | //pen(Black(0.4)); 34 | //plot(f, 0.0, 5.0, 250); 35 | 36 | //n=20; 37 | //pen(Black(0.2)); 38 | //plot(f, 0.0, 5.0, 500); 39 | 40 | tikz_format(); 41 | end(); 42 | } 43 | 44 | -------------------------------------------------------------------------------- /figures/convexfunc.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #d9d9d9 11 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 12 | 495 2025 2250 1710 13 | 2 1 2 1 0 7 50 -1 -1 3.000 0 0 -1 0 0 2 14 | 990 1935 990 2295 15 | 2 1 2 1 0 7 50 -1 -1 3.000 0 0 -1 0 0 2 16 | 1980 1755 1980 2295 17 | 3 2 0 1 0 7 50 -1 -1 0.000 0 0 0 5 18 | 540 1665 990 1935 1530 1980 1980 1755 2250 1485 19 | 0.000 -0.500 -0.500 -0.500 0.000 20 | 4 1 0 50 -1 0 12 0.0000 2 180 705 1485 2160 $y=f(x)$\001 21 | 4 1 0 50 -1 0 12 0.0000 2 180 750 1440 1710 $y=g(x)$\001 22 | 4 1 0 50 -1 0 12 0.0000 2 150 825 1980 2430 $x=\\beta$\001 23 | 4 1 0 50 -1 0 12 0.0000 2 180 930 990 2430 $x=\\alpha$\001 24 | -------------------------------------------------------------------------------- /figures/convexfunc.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/convexfunc.pdf -------------------------------------------------------------------------------- /figures/convexfunc.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{convexfunc.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{4144sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 | \gdef\SetFigFont#1#2#3#4#5{% 8 | \reset@font\fontsize{#1}{#2pt}% 9 | \fontfamily{#3}\fontseries{#4}\fontshape{#5}% 10 | \selectfont}% 11 | \fi\endgroup% 12 | \begin{picture}(1779,1026)(484,-1660) 13 | \put(1486,-1321){\makebox(0,0)[b]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$y=f(x)$}% 14 | }}}} 15 | \put(1441,-871){\makebox(0,0)[b]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$y=g(x)$}% 16 | }}}} 17 | \put(1981,-1591){\makebox(0,0)[b]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$x=\beta$}% 18 | }}}} 19 | \put(991,-1591){\makebox(0,0)[b]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$x=\alpha$}% 20 | }}}} 21 | \end{picture}% 22 | -------------------------------------------------------------------------------- /figures/cyclearoundcompact.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #dddddd 11 | 0 33 #afafaf 12 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 495 810 18 18 495 810 513 810 13 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 4392 855 18 18 4392 855 4410 855 14 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1245 115 18 18 1245 115 1263 115 15 | 1 3 0 2 0 32 52 -1 20 0.000 1 0.0000 4410 855 836 836 4410 855 5246 869 16 | 1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 4410 855 757 757 4410 855 5167 867 17 | 2 1 0 1 0 32 50 -1 20 0.000 0 0 -1 1 0 2 18 | 1 1 1.00 60.00 120.00 19 | 266 712 257 737 20 | 2 1 0 1 0 32 50 -1 20 0.000 0 0 -1 1 0 2 21 | 1 1 1.00 60.00 120.00 22 | 1059 585 1055 573 23 | 2 1 0 1 0 33 50 -1 20 3.000 0 0 -1 1 0 2 24 | 1 1 1.00 60.00 120.00 25 | 5083 508 5087 515 26 | 2 2 0 0 0 33 54 -1 20 0.000 0 0 -1 0 0 5 27 | 3038 -40 5744 -40 5744 1764 3038 1764 3038 -40 28 | 3 3 0 1 0 32 50 -1 -1 0.000 0 0 0 7 29 | 940 395 490 305 310 575 182 1133 625 1160 1075 980 30 | 1120 755 31 | -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 32 | 3 3 2 1 0 32 53 -1 20 1.000 0 0 0 8 33 | -90 495 180 180 630 135 1485 630 1485 1215 630 1395 34 | -360 1125 -270 630 35 | -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 36 | 3 3 0 2 0 33 52 -1 20 0.000 0 0 0 5 37 | 810 405 492 454 318 954 900 945 945 585 38 | -0.500 -0.500 -0.500 -0.500 -0.500 39 | 3 3 2 1 0 7 51 -1 20 1.000 0 0 0 9 40 | 3909 731 4095 450 4455 450 4681 307 4860 855 4730 1078 41 | 4230 1080 3960 990 4140 900 42 | -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 43 | -0.500 44 | 3 2 0 1 0 33 50 -1 -1 3.000 0 1 0 3 45 | 0 0 1.00 60.00 120.00 46 | 1459 901 2402 773 3877 1113 47 | 0.000 -0.500 0.000 48 | 3 2 0 1 0 33 50 -1 -1 3.000 0 1 0 3 49 | 0 0 1.00 60.00 120.00 50 | 1301 110 2507 290 4365 846 51 | 0.000 -0.500 0.000 52 | 4 1 0 50 -1 0 12 0.0000 2 150 345 675 585 $K$\001 53 | 4 0 0 50 -1 0 12 0.0000 2 150 315 540 855 $0$\001 54 | 4 0 0 50 -1 0 12 0.0000 2 150 345 -180 990 $U$\001 55 | 4 0 0 50 -1 0 12 0.0000 2 150 315 4455 945 $0$\001 56 | 4 1 0 50 -1 0 12 0.0000 2 150 525 3398 209 ``$K$''\001 57 | 4 1 0 50 -1 0 12 0.0000 2 180 3210 4435 1303 $\\psi\\bigl(\\varphi(U\\setminus K)\\bigr)$\001 58 | 4 1 0 50 -1 0 12 0.0000 2 180 1545 2365 597 $\\psi \\circ \\varphi$\001 59 | 4 2 0 50 -1 0 12 0.0000 2 180 660 1188 166 $\\infty$\001 60 | -------------------------------------------------------------------------------- /figures/cyclearoundcompact.pdf: -------------------------------------------------------------------------------- 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-------------------------------------------------------------------------------- /figures/diffunderexample.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/diffunderexample.pdf -------------------------------------------------------------------------------- /figures/diffunderexample.xp: -------------------------------------------------------------------------------- 1 | /* -*-ePiX-*- */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | P F(double x, double y) 6 | { 7 | if (x < 0.00001) 8 | return P(x, y, 0.0); 9 | else if (x > 0.99999) 10 | return P(x, y, y); 11 | else 12 | return P(x, y, (exp(y*log(x))-1)/log(x)); 13 | } 14 | 15 | domain R1(P(0,0), P(1.0/10.0,1), mesh(2,20), mesh(30,20)); 16 | domain R2(P(1.0/10.0,0), P(1,1), mesh(18,20), mesh(18,20)); 17 | 18 | int main() 19 | { 20 | picture(P(-0.1,-0.1), P(1.7,1.7), "2.5x2.5in"); 21 | 22 | begin(); 23 | 24 | camera.at(P(-3,-6,3)); 25 | 26 | arrow(P(-0.1,0,0), P(1.3,0,0)); 27 | label(P(1.3,0), P(0,-4), "$x$", b); 28 | 29 | arrow(P(0,-0.1,0), P(0,1.3,0)); 30 | label(P(0,1.3), P( -4,0), "$y$", l); 31 | 32 | plain (); 33 | fill(White()); 34 | surface(F, R1); // rectangular 35 | surface(F, R2); // rectangular 36 | 37 | arrow(P(0, 0, 0), P(0,0,1.2)); 38 | label(P(0,0,1.2), P(-4,0), "$z$", l); 39 | 40 | tikz_format(); 41 | end(); 42 | } 43 | -------------------------------------------------------------------------------- /figures/disccauchy.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #41534c 11 | 0 33 #555555 12 | 0 34 #888888 13 | 0 35 #e9e9e9 14 | 5 1 1 1 0 7 50 -1 -1 4.000 0 1 0 0 1494.182 939.044 2692 1019 1189 -222 905 1985 15 | 5 1 1 1 0 7 50 -1 -1 4.000 0 1 0 0 1502.477 921.072 373 295 2627 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| 2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 3 42 | 0 0 1.00 30.00 45.00 43 | 2295 508 2324 552 2415 517 44 | 2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 3 45 | 0 0 1.00 30.00 45.00 46 | 2444 513 2500 494 2452 391 47 | 2 1 1 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 3 48 | 908 1982 2030 544 2689 1000 49 | 2 1 1 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 3 50 | 2272 -122 2145 923 373 301 51 | 4 2 0 50 -1 0 12 0.0000 2 180 795 2459 1461 $\\gamma$\001 52 | 4 1 0 50 -1 0 12 0.0000 2 180 855 1739 438 $\\alpha$\001 53 | 4 0 0 50 -1 0 12 0.0000 2 165 510 460 1111 $c_2$\001 54 | 4 0 0 50 -1 0 12 0.0000 2 165 510 661 1587 $c_1$\001 55 | 4 1 0 50 -1 0 12 0.0000 2 150 300 2071 849 $z$\001 56 | -------------------------------------------------------------------------------- /figures/disccauchy.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/disccauchy.pdf 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100.00 7 | Single 8 | -2 9 | 1200 2 10 | 1 2 0 1 0 7 50 -1 -1 0.000 1 0.0000 10237 3330 923 360 11160 3690 9315 2970 11 | 1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 10215 2700 127 127 10215 2700 10305 2790 12 | 1 4 0 1 0 0 50 -1 20 0.000 1 0.0000 8295 3740 23 23 8272 3740 8318 3740 13 | 1 4 0 1 0 0 50 -1 20 0.000 1 0.0000 10229 3306 23 23 10206 3306 10252 3306 14 | 1 2 0 1 0 7 52 -1 -1 0.000 1 0.0000 8212 3847 2992 743 5220 3847 11204 3847 15 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 3 16 | 10080 3465 10215 3240 10395 3465 17 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 18 | 10215 3240 10215 2835 19 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 20 | 10125 2925 10350 2925 21 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 3 22 | 9360 3645 9405 3555 9450 3645 23 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 3 24 | 9540 3645 9585 3555 9630 3645 25 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 26 | 9666 3423 9721 3352 27 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 28 | 9343 3374 10125 2925 29 | 2 1 0 1 0 0 50 -1 20 0.000 0 0 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//border(); 19 | 20 | plain(); 21 | 22 | //pen(Black()); 23 | //bold(); 24 | solid(); 25 | 26 | pen(Black(darkband)); 27 | fill(Black(darkband)); 28 | rect(P(-0.4,-1.1),P(0.0,1.1)); 29 | 30 | pen(Black(lightband)); 31 | fill(Black(lightband)); 32 | rect(P(-1.1,0.0),P(1.1,3*M_PI*(0.06666666666))); 33 | 34 | 35 | nofill(); 36 | pen(Black()); 37 | 38 | for (x = -1.0; x <= 1.0001; x += 0.2) { 39 | line(P(x,-1.1),P(x,1.1)); 40 | } 41 | 42 | clip_box(P(1.1+1,-1.1),P(1.1+1+2.2,1.1)); 43 | pen(Black(darkband)); 44 | fill(Black(darkband)); 45 | circle(P(1.1+1+1.1,0),exp(0.0)); 46 | pen(Black(0.0)); 47 | fill(Black(0.0)); 48 | circle(P(1.1+1+1.1,0),exp(-0.4)); 49 | 50 | pen(Black(lightband)); 51 | fill(Black(lightband)); 52 | triangle(P(1.1+1+1.1,0.0), 53 | P(1.1+1+1.1+2.0,0.0), 54 | P(1.1+1+1.1+2.0*cos(3*M_PI*(0.06666666666)),2.0*sin(3*M_PI*(0.06666666666)))); 55 | 56 | nofill(); 57 | pen(Black()); 58 | clip_box(); 59 | 60 | 61 | 62 | 63 | clip_box(P(1.1+1,-1.1),P(1.1+1+2.2,1.1)); 64 | for (x = -3.2; x <= 1.0001; x += 0.2) { 65 | circle(P(1.1+1+1.1,0),exp(x)); 66 | } 67 | clip_box(); 68 | 69 | line_style("- - - - "); 70 | 71 | for (y = -M_PI*0.33333333; y <= M_PI*0.33333333+0.1; y += M_PI*0.066666666) { 72 | line(P(-1.1,y),P(1.1,y)); 73 | } 74 | 75 | clip_box(P(1.1+1,-1.1),P(1.1+1+2.2,1.1)); 76 | for (y = -M_PI; y <= M_PI-0.01; y += M_PI*0.066666666666) { 77 | line(//P(1.1+1+1.1,0), 78 | P(1.1+1+1.1+0.043*cos(y),0.043*sin(y)), 79 | P(1.1+1+1.1+1.6*cos(y),1.6*sin(y))); 80 | } 81 | clip_box(); 82 | 83 | solid(); 84 | plain(); 85 | 86 | arrow(P(1.2,0),P(2.0,0)); 87 | 88 | fill(Black()); 89 | circle(P(1.1+1+1.1,0),exp(-3.4)); 90 | 91 | fill(White()); 92 | circle(P(1.1+1+1.1,0),exp(-4.6)); 93 | 94 | tikz_format(); 95 | end(); 96 | } 97 | 98 | -------------------------------------------------------------------------------- /figures/figure-pdf15.sh: -------------------------------------------------------------------------------- 1 | #!/bin/sh 2 | cp -f $1 $1.bak 3 | gs 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-------------------------------------------------------------------------------- 1 | /* -*-ePiX-*- */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | P F(double x, double y) 6 | { 7 | return P(x, y, exp(x)*sin(y)/10.0); 8 | } 9 | 10 | domain R1(P(-4, -6), P(0,6), mesh(10,20), mesh(10,20)); 11 | domain R2(P(0, -6), P(4,6), mesh(10,20), mesh(20,60)); 12 | 13 | int main() 14 | { 15 | picture(P(-7.1,-7.6), P(7.4,4.7), "2.5x1.9in"); 16 | 17 | begin(); 18 | //border(); 19 | 20 | camera.at(P(17,-10,15)); 21 | 22 | arrow(P(0,0,0), P(5,0,0)); 23 | label(P(5,0), P(0,-2), "$x$", b); 24 | 25 | arrow(P(0,0,0), P(0,7.5,0)); 26 | label(P(0,7.5), P( 0,2), "$y$", t); 27 | 28 | plain (); 29 | fill(White()); 30 | surface(F, R1); // rectangular 31 | surface(F, R2); // rectangular 32 | 33 | arrow(P(0, 0, 0), P(0,0,5.0)); 34 | //label(P(0,0,5.0), P(0,4), "$z$", t); 35 | 36 | bbold(); 37 | //plot(F,R1.slice2(0)); 38 | //plot(F,R2.slice2(0)); 39 | line(P(-4,0,0),P(4,0,0)); 40 | 41 | tikz_format(); 42 | end(); 43 | } 44 | -------------------------------------------------------------------------------- /figures/indexconstant.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #d9d9d9 11 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 12 | 1 1 1.00 60.00 120.00 13 | 4454 733 4430 743 14 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 15 | 1 1 1.00 60.00 120.00 16 | 2813 1368 2748 1379 17 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 18 | 1 1 1.00 60.00 120.00 19 | 3112 1734 3064 1734 20 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 21 | 1 1 1.00 60.00 120.00 22 | 1544 728 1650 693 23 | 3 3 0 1 0 7 50 -1 -1 0.000 0 0 0 8 24 | 450 1665 2025 630 4815 1845 5310 495 2700 1395 1260 1350 25 | 2070 765 3645 1665 26 | -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 27 | 4 2 0 50 -1 0 12 0.0000 2 180 855 1170 855 $\\Gamma$\001 28 | 4 1 0 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| camera.at(P(8,8,10)); 27 | 28 | arrow(P(-1.9,0,0), P(1.9,0,0)); 29 | label(P(1.9,0), P(0,-2), "$\\operatorname{Re} z$", br); 30 | 31 | arrow(P(0,-1.9), P(0,1.9,0)); 32 | label(P(0,1.9), P(0,-2), "$\\operatorname{Im} z$", bl); 33 | 34 | plain (); 35 | fill(White()); 36 | surface(FF, RR); // polar 37 | fill(false); 38 | 39 | pen(2); 40 | plot(FF, RR.resize2(-M_PI,M_PI).slice1(1)); 41 | dot(P(1,0,0)); 42 | plain(); 43 | 44 | // that bit of the axes that's above the graph 45 | line(P(0,-1,0), P(0,1,0)); 46 | line(P(-1,0,0), P(1,0,0)); 47 | 48 | arrow(P(0,0,0), P(0,0,1.7)); 49 | 50 | tikz_format(); 51 | end(); 52 | } 53 | -------------------------------------------------------------------------------- /figures/meanvalue.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #dddddd 11 | 0 33 #c7c7c7 12 | 1 3 0 1 0 0 50 -1 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-------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.6a 2 | Landscape 3 | Center 4 | Inches 5 | Letter 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #e6e6e6 11 | 0 33 #d5d5d5 12 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 3225 1425 26 26 3225 1425 3246 1440 13 | 1 3 2 1 0 33 51 -1 20 1.000 1 0.0000 3200 1439 586 586 3200 1439 3786 1431 14 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 15 | 0 0 1.00 60.00 120.00 16 | 3225 1425 2792 1041 17 | 3 3 2 1 0 32 52 -1 20 1.000 0 0 0 9 18 | 450 1950 1025 1520 300 825 1562 899 3524 739 4647 1480 19 | 3364 2106 1851 1746 1150 2057 20 | -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 21 | -1.000 22 | 4 0 0 50 -1 0 12 0.0000 2 165 315 3243 1571 $x$\001 23 | 4 0 0 50 -1 0 12 0.0000 2 165 345 1057 1132 $V$\001 24 | 4 0 0 50 -1 0 12 0.0000 2 180 1080 2971 1843 $B(x,\\delta)$\001 25 | 4 0 0 50 -1 0 12 0.0000 2 165 675 3023 1201 $\\delta$\001 26 | 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}}}} 15 | \put(1058,-293){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$V$}% 16 | }}}} 17 | \put(2972,-1004){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$B(x,\delta)$}% 18 | }}}} 19 | \put(3024,-362){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\delta$}% 20 | }}}} 21 | \end{picture}% 22 | -------------------------------------------------------------------------------- /figures/poisson-graph.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/poisson-graph.pdf -------------------------------------------------------------------------------- /figures/poisson-graph.xp: -------------------------------------------------------------------------------- 1 | /* -*-ePiX-*- */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | P F(double x, double y) 6 | { 7 | double f = (1/(2*M_PI))*(-(y*y+x*x-1)/(y*y+x*x-2*x+1)); 8 | 9 | return P(x, y, f); 10 | } 11 | 12 | P FF(double r, double th) 13 | { 14 | double x = r*Cos(th); 15 | double y = r*Sin(th); 16 | return F(x,y); 17 | } 18 | 19 | //domain RR(P(sqrt(0.000001),0), P(0.9999999, 2*M_PI), mesh(15,40), mesh(15*5,40*5)); // mesh(10,120), mesh(10,240)); 20 | domain RR(P(sqrt(0.000001),0.25*M_PI), P(0.9999999, (1.75)*M_PI), mesh(15,30), mesh(15*2,30*2)); // mesh(10,120), mesh(10,240)); 21 | domain RR2(P(sqrt(0.000001),0), P(0.9999999, 0.25*M_PI), mesh(15,5), mesh(15*3,5*3)); // mesh(10,120), mesh(10,240)); 22 | domain RR3(P(sqrt(0.000001),-0.25*M_PI), P(0.9999999, 0), mesh(15,5), mesh(15*3,5*3)); // mesh(10,120), mesh(10,240)); 23 | 24 | int main() 25 | { 26 | picture(P(-1.08,-0.83), P(1.25,1.55), "2.5x2in"); 27 | 28 | begin(); 29 | //border(); 30 | 31 | camera.at(P(2,-8,8)); 32 | 33 | arrow(P(-1.1,0,0), P(1.25,0,0)); 34 | //label(P(2.1,0), P(2,4), "$\\operatorname{Re} z$", r); 35 | 36 | arrow(P(0,-1.1,0), P(0,1.25,0)); 37 | //label(P(0,2.2), P( 2,8), "$\\operatorname{Im} z$", br); 38 | 39 | plain (); 40 | fill(White()); 41 | clip_box(P(-1.1,-1.1,0),P(1.1,1.1, 2)); 42 | surface(FF, RR); // polar 43 | surface(FF, RR2); // polar 44 | surface(FF, RR3); // polar 45 | 46 | arrow(P(0,0,(1/(2*M_PI))), P(0,0,1.5)); 47 | 48 | tikz_format(); 49 | end(); 50 | } 51 | -------------------------------------------------------------------------------- /figures/poisson-kernel.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/poisson-kernel.pdf -------------------------------------------------------------------------------- /figures/polarcoords.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Inches 5 | Letter 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 5 1 0 1 0 7 50 -1 -1 0.000 0 1 0 0 1405.662 1581.390 1875 1575 1810 1343 1776 1293 11 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 12 | 0 0 1.00 90.00 150.00 13 | 1200 1575 2700 1575 14 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 15 | 0 0 1.00 90.00 150.00 16 | 1425 1800 1425 675 17 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 18 | 5 1 1.00 60.00 60.00 19 | 1425 1575 2175 975 20 | 4 0 0 50 -1 0 12 0.0000 2 150 675 1919 1425 $\\theta$\001 21 | 4 0 0 50 -1 0 12 0.0000 2 150 270 1623 1187 $r$\001 22 | 4 0 0 50 -1 0 12 0.0000 2 165 1155 1932 909 $re^{i\\theta}$\001 23 | -------------------------------------------------------------------------------- /figures/polarcoords.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/polarcoords.pdf 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\put(1933,-70){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$re^{i\theta}$}% 18 | }}}} 19 | \end{picture}% 20 | -------------------------------------------------------------------------------- /figures/proofofdirichlet.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #d9d9d9 11 | 0 33 #eaeaea 12 | 5 1 0 1 0 7 50 -1 -1 0.000 0 0 1 1 885.887 2249.822 1631 1232 1875 1467 2041 1743 13 | 0 0 1.00 60.00 120.00 14 | 0 0 1.00 60.00 120.00 15 | 5 1 0 1 0 33 51 -1 20 0.000 0 0 0 0 584.237 2520.709 495 945 1782 1493 2158 2402 16 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1588 1305 23 23 1588 1305 1611 1305 17 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1430 2034 23 23 1430 2034 1453 2034 18 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1982 1779 23 23 1982 1779 2005 1779 19 | 2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2 20 | 0 0 1.00 60.00 120.00 21 | 0 0 1.00 60.00 120.00 22 | 1975 1779 1434 2031 23 | 2 3 0 0 0 33 52 -1 20 0.000 0 0 -1 0 0 7 24 | 497 945 455 1835 674 2392 2159 2400 2089 2269 628 998 25 | 497 945 26 | 4 0 0 50 -1 0 12 0.0000 2 180 1785 1891 1409 $\\text{angle}<\\delta$\001 27 | 4 2 0 50 -1 0 12 0.0000 2 165 1710 1548 1414 $z_0=e^{i\\theta_0}$\001 28 | 4 2 0 50 -1 0 12 0.0000 2 150 405 1982 2315 $\\D$\001 29 | 4 2 0 50 -1 0 12 0.0000 2 165 1350 1380 2157 $z=re^{i\\theta}$\001 30 | 4 0 0 50 -1 0 12 0.0000 2 165 1095 2073 1855 $e^{i\\theta}$\001 31 | 4 2 0 50 -1 0 12 0.0000 2 150 825 1759 1832 $< \\delta$\001 32 | -------------------------------------------------------------------------------- /figures/proofofdirichlet.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/proofofdirichlet.pdf 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-------------------------------------------------------------------------------- /figures/rayjordan.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7b 2 | Landscape 3 | Center 4 | Inches 5 | Letter 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #898989 11 | 0 33 #d8d8d8 12 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1875 2625 13 13 1875 2625 1888 2625 13 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 14 | 1875 2625 1875 1950 15 | 2 1 2 1 0 7 50 -1 -1 2.000 0 0 -1 0 0 2 16 | 1875 1950 1875 1650 17 | 3 3 0 1 0 7 50 -1 -1 0.000 0 0 0 11 18 | 525 3075 1050 2400 1500 2325 2925 2700 2625 2325 3375 2850 19 | 1650 2925 3075 3375 1200 3150 2025 3450 900 3300 20 | -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 21 | -0.500 -0.500 -0.500 22 | 4 0 0 50 -1 0 12 0.0000 2 165 270 1950 2700 $q$\001 23 | 4 0 0 50 -1 0 12 0.0000 2 165 720 3450 2850 $\\gamma$\001 24 | 4 0 0 50 -1 0 12 0.0000 2 135 270 1950 2175 $R$\001 25 | 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| }}}} 15 | \put(3451,-2011){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\gamma$}% 16 | }}}} 17 | \put(1951,-1336){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$R$}% 18 | }}}} 19 | \end{picture}% 20 | -------------------------------------------------------------------------------- /figures/realexp.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/realexp.pdf -------------------------------------------------------------------------------- /figures/realexp.xp: -------------------------------------------------------------------------------- 1 | /* -*-ePiX-*- */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | P F(double x, double y) 6 | { 7 | return P(x, y, exp(x)*cos(y)/10.0); 8 | } 9 | 10 | domain R1(P(-4, -6), P(0,6), mesh(10,20), mesh(10,20)); 11 | domain R2(P(0, -6), P(4,6), mesh(10,20), mesh(20,60)); 12 | 13 | int main() 14 | { 15 | picture(P(-7.1,-7.6), P(7.4,4.7), "2.5x1.9in"); 16 | 17 | begin(); 18 | //border(); 19 | 20 | camera.at(P(17,-10,15)); 21 | 22 | arrow(P(0,0,0), P(5,0,0)); 23 | label(P(5,0), P(-4,-6), "$x$", l); 24 | 25 | arrow(P(0,0,0), P(0,7.5,0)); 26 | label(P(0,7.5), P( 0,2), "$y$", t); 27 | 28 | plain (); 29 | fill(White()); 30 | surface(F, R1); // rectangular 31 | surface(F, R2); // rectangular 32 | 33 | arrow(P(0, 0, 0), P(0,0,5.0)); 34 | 35 | bbold(); 36 | plot(F,R1.slice2(0)); 37 | plot(F,R2.slice2(0)); 38 | 39 | //label(P(0,0,5.0), P(0,4), "$z$", t); 40 | 41 | tikz_format(); 42 | end(); 43 | } 44 | -------------------------------------------------------------------------------- /figures/residues.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #d9d9d9 11 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 360 630 23 23 360 630 383 630 12 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1305 1395 23 23 1305 1395 1328 1395 13 | 1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 1305 1395 360 360 1305 1395 1665 1395 14 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 2430 675 23 23 2430 675 2453 675 15 | 1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 2430 675 360 360 2430 675 2790 675 16 | 1 3 0 1 0 7 51 -1 -1 0.000 1 0.0000 360 630 360 360 360 630 720 630 17 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 18 | 1 1 1.00 60.00 120.00 19 | 1627 1845 1683 1844 20 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 21 | 1 1 1.00 60.00 120.00 22 | 3811 906 3818 876 23 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 24 | 1 1 1.00 60.00 120.00 25 | 1818 573 1807 598 26 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 27 | 1 1 1.00 60.00 120.00 28 | 2652 244 2619 244 29 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 30 | 1 1 1.00 60.00 120.00 31 | 828 974 795 1006 32 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 33 | 1 1 1.00 60.00 120.00 34 | 1589 1174 1573 1144 35 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 36 | 1 1 1.00 60.00 120.00 37 | 2724 463 2708 433 38 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 39 | 1 1 1.00 60.00 120.00 40 | 653 421 637 391 41 | 3 3 0 1 0 7 50 -1 -1 0.000 0 0 0 10 42 | 1170 630 405 1485 810 1845 3375 1665 3825 270 2160 180 43 | 1755 945 2790 1305 3060 315 1620 315 44 | -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 45 | -0.500 -0.500 46 | 4 0 0 50 -1 0 12 0.0000 2 165 555 1350 1485 $w_1$\001 47 | 4 0 0 50 -1 0 12 0.0000 2 165 555 405 720 $w_2$\001 48 | 4 0 0 50 -1 0 12 0.0000 2 165 555 2475 765 $w_3$\001 49 | 4 0 0 50 -1 0 12 0.0000 2 150 885 2250 1755 $\\Gamma$\001 50 | -------------------------------------------------------------------------------- /figures/residues.pdf: -------------------------------------------------------------------------------- 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74){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$w_3$}% 18 | }}}} 19 | \put(2251,-916){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\Gamma$}% 20 | }}}} 21 | \end{picture}% 22 | -------------------------------------------------------------------------------- /figures/rhalfcircle.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #d9d9d9 11 | 5 1 0 1 0 7 50 -1 -1 0.000 0 1 0 0 1732.500 2002.500 3285 1980 1755 450 180 1980 12 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1710 1620 23 23 1710 1620 1733 1620 13 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1710 2340 23 23 1710 2340 1733 2340 14 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 15 | 3285 1845 3285 2115 16 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 17 | 180 1845 180 2115 18 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 19 | 1 1 1.00 60.00 120.00 20 | 2745 1980 2835 1980 21 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 22 | 180 1980 3285 1980 23 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 24 | 1 1 1.00 60.00 120.00 25 | 675 1980 855 1980 26 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 27 | 1 1 1.00 60.00 120.00 28 | 811 754 781 776 29 | 4 1 0 50 -1 0 12 0.0000 2 150 270 3285 2340 $r$\001 30 | 4 1 0 50 -1 0 12 0.0000 2 150 330 180 2340 $-r$\001 31 | 4 0 0 50 -1 0 12 0.0000 2 180 1020 2700 720 $\\gamma_r$\001 32 | 4 0 0 50 -1 0 12 0.0000 2 150 330 1800 2385 $-i$\001 33 | 4 0 0 50 -1 0 12 0.0000 2 150 270 1800 1665 $i$\001 34 | -------------------------------------------------------------------------------- /figures/rhalfcircle.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/rhalfcircle.pdf 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| }}}} 19 | \put(1801,-1546){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$-i$}% 20 | }}}} 21 | \put(1801,-826){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$i$}% 22 | }}}} 23 | \end{picture}% 24 | -------------------------------------------------------------------------------- /figures/riemannmap.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/riemannmap.pdf -------------------------------------------------------------------------------- /figures/riemannmap.xp: -------------------------------------------------------------------------------- 1 | /* slightly modified sample from epix samples gallery */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | double gxf, gyf, gxt, gyt; 6 | double a = -0.4; 7 | 8 | double gr; 9 | 10 | P 11 | f(double x, double y) 12 | { 13 | Complex z(x,y); 14 | Complex ii(0,1); 15 | //Complex oneplusi(1,1); 16 | Complex g; 17 | 18 | //g = (oneplusi/sqrt(2))*sqrtC(-(z-1)/(z+1)); 19 | //return -(g-ii)/(g+ii); 20 | 21 | g = -ii*(z-1)/(z+1); 22 | return -(g*g-ii)/(g*g+ii)+2.6; 23 | } 24 | 25 | P 26 | fl(double t) 27 | { 28 | return f(gxf + t*(gxt-gxf),gyf + t*(gyt-gyf)); 29 | } 30 | 31 | P 32 | fc(double t) 33 | { 34 | return f(gr*cos(M_PI*t),gr*sin(M_PI*t)); 35 | } 36 | 37 | void 38 | draw_line(double xf, double yf, double xt, double yt, int steps) 39 | { 40 | gxf = xf; 41 | gyf = yf; 42 | gxt = xt; 43 | gyt = yt; 44 | 45 | line (P(xf,yf),P(xt,yt)); 46 | 47 | plot(fl, 0, 1, steps); 48 | } 49 | 50 | void 51 | draw_toparc(double r, int steps) 52 | { 53 | gr = r; 54 | 55 | arc (P(0,0),r, 0, M_PI); 56 | 57 | plot(fc, 0, 1, steps); 58 | } 59 | 60 | void 61 | draw_dot(double x, double y) 62 | { 63 | ddot (P(x,y)); 64 | ddot (f(x,y)); 65 | } 66 | 67 | int main() 68 | { 69 | double slope; 70 | double c; 71 | double x; 72 | int i; 73 | 74 | picture(P(-1.3,-1.3), P(3.9,1.3), "3.0x1.5in"); 75 | 76 | begin(); 77 | //border(); 78 | 79 | 80 | arrow(P(1.1,0.0),P(1.5,0.0)); 81 | 82 | pen(Black(0.2)); 83 | 84 | for (i = 1; i < 8; i++) { 85 | draw_toparc(i/8.0,i*12); 86 | } 87 | 88 | for (i=1; i < 8; i++) { 89 | draw_line (0.0,0,cos(i*M_PI/8.0),sin(i*M_PI/8.0),20); 90 | } 91 | 92 | pen(Black()); 93 | 94 | draw_dot(0,sqrt(2)-1); 95 | 96 | bold(); 97 | solid(); 98 | arc(P(0,0),1,0,M_PI); 99 | line(P(-1,0),P(1,0)); 100 | circle(P(2.6,0),1); 101 | plain(); 102 | 103 | tikz_format(); 104 | end(); 105 | } 106 | 107 | -------------------------------------------------------------------------------- /figures/riemannsphere.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/riemannsphere.pdf -------------------------------------------------------------------------------- /figures/riemannsphere.xp: -------------------------------------------------------------------------------- 1 | /* -*-ePiX-*- */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | P F(double x, double y) 6 | { 7 | return sph(1,x,y); 8 | } 9 | 10 | P FLAT(double x, double y) 11 | { 12 | return P(x,y,0); 13 | } 14 | 15 | domain R1(P(-M_PI, -0.5*M_PI), P(M_PI,0), mesh(20,8), mesh(40,16)); 16 | domain R2(P(-2.0, -2.0), P(2.0,2.0), mesh(10,10), mesh(10,10)); 17 | domain R3(P(-M_PI, 0), P(M_PI,0.5*M_PI), mesh(20,8), mesh(40,16)); 18 | 19 | int main() 20 | { 21 | picture(P(-2.5,-1.00), P(2.9,1.2), "4.05x1.65in"); 22 | 23 | begin(); 24 | //border(); 25 | 26 | plain(); 27 | 28 | camera.at(P(5,10,3)); 29 | 30 | plain (); 31 | fill(White()); 32 | surface(F, R1); 33 | surface(FLAT, R2); 34 | surface(F, R3); 35 | 36 | bbold(); 37 | dot(P(0,0,1)); 38 | { 39 | double x,y; 40 | 41 | double sx,sy,sz; 42 | 43 | double t; 44 | 45 | x = 1.5; y=0.5; 46 | 47 | t = 2.0/(x*x+y*y+1.0); 48 | 49 | 50 | plain(); 51 | 52 | bbold(); 53 | line_style("- - - - "); 54 | line(P(0,0,1),P(t*x,t*y,1-t)); 55 | line(P(t*x,t*y,1-t),P(1.1*x,1.1*y,1-1.1)); 56 | solid(); 57 | line(P(t*x,t*y,1-t),P(x,y,0)); 58 | line(P(0,0,1),P(-0.1*x,-0.1*y,1-(-0.1))); 59 | 60 | bbold(); 61 | 62 | dot(P(x,y,0)); 63 | dot(P(t*x,t*y,1-t)); 64 | 65 | plain(); 66 | 67 | arrow(P(2,0,1),P(x,y,0)); 68 | arrow(P(1,0,1),P(t*x,t*y,1-t)); 69 | } 70 | 71 | label(P(0,0,1), P(-2,4), "$\\infty$", t); 72 | label(P(2,0,1), P(-2,4), "$\\xi = \\Phi(p)$", t); 73 | label(P(1,0,1), P(-2,4), "$p$", t); 74 | 75 | tikz_format(); 76 | end(); 77 | } 78 | -------------------------------------------------------------------------------- /figures/roots.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Inches 5 | Letter 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #898989 11 | 0 33 #d8d8d8 12 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1252 397 24 24 1252 397 1276 397 13 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1431 824 24 24 1431 824 1455 824 14 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1253 1252 24 24 1253 1252 1277 1252 15 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 825 1430 24 24 825 1430 849 1430 16 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 397 1253 24 24 397 1253 421 1253 17 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 220 825 24 24 220 825 244 825 18 | 1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 825 825 605 605 825 825 1425 900 19 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 397 398 24 24 397 398 421 398 20 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 825 220 24 24 825 220 849 220 21 | 2 1 0 1 0 33 50 -1 -1 0.000 0 0 -1 1 1 2 22 | 0 0 1.00 60.00 120.00 23 | 0 0 1.00 60.00 120.00 24 | 821 831 1100 270 25 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 1 2 26 | 0 0 1.00 60.00 120.00 27 | 825 375 825 975 28 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 29 | 0 0 1.00 60.00 120.00 30 | 675 825 1275 825 31 | 4 0 0 50 -1 0 12 0.0000 2 165 810 982 679 $r^{1/8}$\001 32 | -------------------------------------------------------------------------------- /figures/roots.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/roots.pdf -------------------------------------------------------------------------------- /figures/roots.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{roots.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{3947sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 | \gdef\SetFigFont#1#2#3#4#5{% 8 | \reset@font\fontsize{#1}{#2pt}% 9 | \fontfamily{#3}\fontseries{#4}\fontshape{#5}% 10 | \selectfont}% 11 | \fi\endgroup% 12 | \begin{picture}(1275,1272)(189,-622) 13 | \put(983,160){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$r^{1/8}$}% 14 | }}}} 15 | \end{picture}% 16 | -------------------------------------------------------------------------------- /figures/run-all-xp.sh: -------------------------------------------------------------------------------- 1 | #!/bin/sh 2 | # run all the .xp files into .pdf files 3 | for n in *.xp ; do 4 | if ! ./run-xp-topdf.sh $n ; then 5 | echo 6 | echo '\033[0;31m'"ERROR! run-xp-topdf $n failed"'\033[0m' 7 | echo 8 | exit 1; 9 | fi 10 | done 11 | echo 12 | echo '\033[0;32mSuccess!\033[0m' 13 | echo 14 | -------------------------------------------------------------------------------- /figures/run-xp-topdf.sh: -------------------------------------------------------------------------------- 1 | #!/bin/sh 2 | # Used from run-all-xp.sh 3 | exec elaps --pdf -p amsmath -p amsfonts -p amssymb -p newpxmath -p newpxtext --size 12 $1 4 | 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840 840 1275 375 2115 375 21 | 3 2 0 1 0 7 50 -1 -1 0.000 0 0 0 5 22 | 675 525 750 375 900 375 1050 225 1275 375 23 | 0.000 -0.500 -0.500 -0.500 0.000 24 | 3 3 0 1 32 32 50 -1 20 0.000 0 0 0 9 25 | 660 -472 336 -515 -414 -365 -1647 88 -994 1135 33 1342 26 | -500 648 -264 160 -39 -215 27 | -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 -0.500 28 | -0.500 29 | 4 1 0 50 -1 0 12 0.0000 2 135 270 -912 302 $K$\001 30 | 4 1 0 50 -1 0 12 0.0000 2 150 450 981 522 $z_2$\001 31 | 4 0 0 50 -1 0 12 0.0000 2 165 630 1336 470 $z_3=q$\001 32 | 4 2 0 50 -1 0 12 0.0000 2 165 630 875 699 $z_1=p$\001 33 | -------------------------------------------------------------------------------- /figures/rungepolepush.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/rungepolepush.pdf -------------------------------------------------------------------------------- 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| }}}} 23 | \end{picture}% 24 | -------------------------------------------------------------------------------- /figures/seqcompactiscompact.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.6a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #dddddd 11 | 0 33 #707070 12 | 1 3 0 1 0 0 49 -1 20 0.000 1 0.0000 405 720 23 23 405 720 428 720 13 | 1 3 2 1 0 0 49 -1 -1 1.000 1 0.0000 405 720 549 549 405 720 954 720 14 | 1 3 2 1 0 0 49 -1 -1 1.000 1 0.0000 2070 540 549 549 2070 540 2619 540 15 | 1 3 0 1 0 0 49 -1 20 0.000 1 0.0000 2070 540 23 23 2070 540 2093 540 16 | 1 3 0 1 0 0 49 -1 20 0.000 1 0.0000 1260 675 23 23 1260 675 1283 675 17 | 1 3 2 1 0 0 49 -1 -1 1.000 1 0.0000 1241 703 549 549 1241 703 1790 703 18 | 1 3 0 1 0 0 49 -1 20 0.000 1 0.0000 2879 612 23 23 2879 612 2902 612 19 | 2 1 0 1 0 0 49 -1 -1 3.000 0 0 -1 1 0 2 20 | 0 0 1.00 60.00 120.00 21 | 405 720 180 225 22 | 3 3 0 1 0 32 52 -1 20 0.000 0 0 0 8 23 | 180 855 945 360 2160 450 2970 315 3510 720 2925 990 24 | 1710 765 444 984 25 | -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 26 | 3 3 2 1 33 0 51 -1 -1 1.000 0 0 0 4 27 | 44 127 994 541 384 1446 -921 976 28 | -1.000 -1.000 -1.000 -1.000 29 | 3 3 2 1 33 0 51 -1 -1 1.000 0 0 0 4 30 | 1776 1540 672 1055 977 135 1818 451 31 | -1.000 -1.000 -1.000 -1.000 32 | 3 3 2 1 33 0 51 -1 -1 1.000 0 0 0 4 33 | 1402 -78 1579 925 2482 987 2520 -75 34 | -1.000 -1.000 -1.000 -1.000 35 | 4 0 0 50 -1 0 12 0.0000 2 165 675 90 585 $\\delta$\001 36 | 4 0 0 50 -1 0 12 0.0000 2 180 525 325 900 $x_1$\001 37 | 4 0 33 50 -1 0 12 0.0000 2 180 1500 -775 946 $U_{\\lambda_1}$\001 38 | 4 0 0 50 -1 0 12 0.0000 2 165 345 3113 801 $K$\001 39 | 4 0 0 50 -1 0 12 0.0000 2 180 525 1114 585 $x_2$\001 40 | 4 0 0 50 -1 0 12 0.0000 2 180 525 1964 708 $x_3$\001 41 | 4 0 0 50 -1 0 12 0.0000 2 180 525 2776 539 $x_4$\001 42 | 4 0 33 50 -1 0 12 0.0000 2 180 1500 1418 1409 $U_{\\lambda_2}$\001 43 | 4 0 33 50 -1 0 12 0.0000 2 180 1500 1496 14 $U_{\\lambda_3}$\001 44 | -------------------------------------------------------------------------------- /figures/seqcompactiscompact.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/seqcompactiscompact.pdf -------------------------------------------------------------------------------- /figures/seqcompactiscompact.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{seqcompactiscompact.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{4144sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 | \gdef\SetFigFont#1#2#3#4#5{% 8 | \reset@font\fontsize{#1}{#2pt}% 9 | \fontfamily{#3}\fontseries{#4}\fontshape{#5}% 10 | \selectfont}% 11 | \fi\endgroup% 12 | \begin{picture}(4461,1847)(-938,-747) 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\put(2777,300){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$x_4$}% 26 | }}}} 27 | \put(1419,-570){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0.439,0.439,0.439}$U_{\lambda_2}$}% 28 | }}}} 29 | \put(1497,825){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0.439,0.439,0.439}$U_{\lambda_3}$}% 30 | }}}} 31 | \end{picture}% 32 | -------------------------------------------------------------------------------- /figures/sequence-convergence-metric.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/sequence-convergence-metric.pdf -------------------------------------------------------------------------------- /figures/sequence-convergence-metric.xp: -------------------------------------------------------------------------------- 1 | /* -*-ePiX-*- */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | #include 6 | 7 | double seqx[] = 8 | { 9 | 1.95, 10 | 1.3, 11 | 1.6, 12 | 0.3, 13 | 0.1, 14 | 0.50, 15 | 0.85, 16 | 1.0, 17 | 1.18, 18 | 0.85 19 | }; 20 | 21 | double seqy[] = 22 | { 23 | 1.1, 24 | 1.2, 25 | 0.8, 26 | 1.43, 27 | 0.7, 28 | 0.90, 29 | 0.75, 30 | 1.3, 31 | 0.9, 32 | 1.15 33 | }; 34 | 35 | 36 | int main() 37 | { 38 | int i; 39 | picture(P(0.0,0.55), P(2.0,1.45), "2.429x1.093in"); 40 | 41 | begin(); 42 | 43 | //border(); 44 | 45 | arrow_inset(0.9); 46 | dotted(); 47 | circle (P(1,1), 0.435); 48 | solid(); 49 | arrow(P(1,1),P(1+0.43*Cos(1),1+0.43*Sin(1))); 50 | label(P(1+0.22*Cos(1),1+0.22*Sin(1)),P(3,0),"$\\epsilon$",b); 51 | dot_size(4); 52 | ddot(P(1,1)); 53 | label(P(1, 1),P(0,-4),"$p$",b); 54 | 55 | dot_size(3); 56 | for (int i = 0; i < 10; i++) { 57 | char num[10]; 58 | ddot(P(seqx[i], seqy[i])); 59 | sprintf(num, "$x_{%d}$", i+1); 60 | label(P(seqx[i], seqy[i]),P(0,-4),num,b); 61 | } 62 | 63 | tikz_format(); 64 | end(); 65 | } 66 | 67 | -------------------------------------------------------------------------------- /figures/spheremetric.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.6a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #41534c 11 | 0 33 #555555 12 | 0 34 #888888 13 | 0 35 #dcdcdc 14 | 5 1 0 1 0 7 50 -1 -1 0.000 0 0 0 0 973.558 952.610 1062 796 1147 905 1149 913 15 | 1 3 0 1 0 7 50 -1 -1 0.000 1 0.0000 990 945 742 742 990 945 1710 1125 16 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1575 810 18 18 1575 810 1593 810 17 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 990 945 18 18 990 945 1008 945 18 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1215 450 18 18 1215 450 1233 450 19 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 20 | 990 945 1215 450 21 | 2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 22 | 990 945 1575 810 23 | 3 0 0 1 0 7 50 -1 -1 0.000 0 0 0 5 24 | 250 932 450 1125 990 1215 1530 1125 1733 923 25 | 0.000 1.000 1.000 1.000 0.000 26 | 3 0 1 1 35 7 51 -1 -1 4.000 0 0 0 5 27 | 250 913 450 720 990 630 1530 720 1733 922 28 | 0.000 1.000 1.000 1.000 0.000 29 | 4 0 0 50 -1 0 12 0.0000 2 165 315 900 1125 $0$\001 30 | 4 0 0 50 -1 0 12 0.0000 2 195 315 1530 945 $y$\001 31 | 4 0 0 50 -1 0 12 0.0000 2 165 675 1170 765 $\\theta$\001 32 | 4 0 0 50 -1 0 12 0.0000 2 165 510 135 540 $S^2$\001 33 | 4 0 0 50 -1 0 12 0.0000 2 165 315 1080 405 $x$\001 34 | -------------------------------------------------------------------------------- /figures/spheremetric.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/spheremetric.pdf -------------------------------------------------------------------------------- /figures/spheremetric.pdf_t: 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0 0 -1 0 0 2 21 | 768 675 1650 675 22 | 2 1 0 2 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 23 | 768 675 1950 1125 24 | 4 1 0 50 -1 0 12 0.0000 2 150 405 4950 900 $\\R$\001 25 | -------------------------------------------------------------------------------- /figures/splittriang.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/splittriang.pdf -------------------------------------------------------------------------------- /figures/splittriang.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{splittriang.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{3947sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 | \gdef\SetFigFont#1#2#3#4#5{% 8 | \reset@font\fontsize{#1}{#2pt}% 9 | \fontfamily{#3}\fontseries{#4}\fontshape{#5}% 10 | \selectfont}% 11 | \fi\endgroup% 12 | \begin{picture}(4899,944)(214,-308) 13 | \put(4951,-61){\makebox(0,0)[b]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\R$}% 14 | }}}} 15 | \end{picture}% 16 | -------------------------------------------------------------------------------- /figures/squarepath.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/squarepath.pdf -------------------------------------------------------------------------------- /figures/squarepath.xp: -------------------------------------------------------------------------------- 1 | /* slightly modified sample from epix samples gallery */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | int main() 6 | { 7 | double slope; 8 | double c; 9 | double x; 10 | 11 | picture(P(-0.3,-0.14), P(1.3,1.10), "1.6x1.25in"); 12 | 13 | begin(); 14 | //border(); 15 | 16 | line(P(-0.3,0),P(1.3,0)); 17 | line(P(0,-0.14),P(0,1.10)); 18 | //h_axis(3); 19 | //v_axis(3); 20 | 21 | 22 | bold(); 23 | arrow(P(0,0),P(1,0)); 24 | arrow(P(1,0),P(1,1)); 25 | arrow(P(1,1),P(0,1)); 26 | arrow(P(0,1),P(0,0)); 27 | 28 | label(P(0.05,0), P(0,-6), "$t=0$", r); 29 | label(P(0.95,0), P(0,-6), "$t=1$", r); 30 | label(P(1,1), P(4,0), "$t=2$", r); 31 | label(P(0,1), P(-4,0), "$t=3$", l); 32 | label(P(0,0.1), P(-4,0), "$t=4$", l); 33 | 34 | tikz_format(); 35 | end(); 36 | } 37 | 38 | -------------------------------------------------------------------------------- /figures/starshaped.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #41534c 11 | 0 33 #555555 12 | 0 34 #888888 13 | 0 35 #dcdcdc 14 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1215 855 18 18 1215 855 1233 855 15 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1800 1125 18 18 1800 1125 1818 1125 16 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 900 1260 18 18 900 1260 918 1260 17 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1800 810 18 18 1800 810 1818 810 18 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1935 405 18 18 1935 405 1953 405 19 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1170 630 18 18 1170 630 1188 630 20 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 855 405 18 18 855 405 873 405 21 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 765 630 18 18 765 630 783 630 22 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 270 765 18 18 270 765 288 765 23 | 2 1 0 1 0 35 50 -1 -1 0.000 0 0 -1 0 0 2 24 | 1215 855 1935 405 25 | 2 1 0 1 0 35 50 -1 -1 0.000 0 0 -1 0 0 2 26 | 1215 855 1170 630 27 | 2 1 0 1 0 35 50 -1 -1 0.000 0 0 -1 0 0 2 28 | 1215 855 855 405 29 | 2 1 0 1 0 35 50 -1 -1 0.000 0 0 -1 0 0 2 30 | 1215 855 270 765 31 | 2 1 0 1 0 35 50 -1 -1 0.000 0 0 -1 0 0 2 32 | 1215 855 765 630 33 | 2 1 0 1 0 35 50 -1 -1 0.000 0 0 -1 0 0 2 34 | 1215 855 900 1260 35 | 2 1 0 1 0 35 50 -1 -1 0.000 0 0 -1 0 0 2 36 | 1215 855 1800 810 37 | 2 1 0 1 0 35 50 -1 -1 0.000 0 0 -1 0 0 2 38 | 1215 855 1800 1125 39 | 2 3 2 1 0 35 53 -1 20 1.000 0 0 -1 0 0 10 40 | 585 675 900 90 1260 540 2025 270 1845 945 2250 1485 41 | 810 1305 720 990 45 720 585 675 42 | 4 0 0 50 -1 0 12 0.0000 2 165 510 1215 1080 $z_0$\001 43 | 4 0 0 50 -1 0 12 0.0000 2 150 300 1710 1305 $z$\001 44 | -------------------------------------------------------------------------------- /figures/starshaped.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/starshaped.pdf -------------------------------------------------------------------------------- /figures/starshaped.pdf_t: -------------------------------------------------------------------------------- 1 | \begin{picture}(0,0)% 2 | \includegraphics{starshaped.pdf}% 3 | \end{picture}% 4 | \setlength{\unitlength}{4144sp}% 5 | % 6 | \begingroup\makeatletter\ifx\SetFigFont\undefined% 7 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| #include "epix.h" 3 | using namespace ePiX; 4 | 5 | double derx = 0.8; 6 | double dery = 0.9; 7 | 8 | double derc; 9 | 10 | 11 | double thef(double x, double y) 12 | { 13 | return 1.4-0.15*x*x-0.15*y*y; 14 | } 15 | double thedf(double x, double y) 16 | { 17 | return derc-0.3*derx*(x-derx)-0.3*dery*(y-dery); 18 | } 19 | 20 | P F(double x, double y) 21 | { 22 | return P(x, y, thef(x,y)); 23 | } 24 | 25 | P DF(double x, double y) 26 | { 27 | return P(x, y, thedf(x,y)); 28 | } 29 | 30 | double Bd(2); 31 | P pt1(-0.5,-0.5), pt2(1.5, 1.5); 32 | domain R(pt1, pt2, mesh(15,15), mesh(30,30)); 33 | 34 | domain DR(P(derx-0.5,dery-0.5), P(derx+0.5,dery+0.5), mesh(8,8), mesh(8,8)); 35 | 36 | int main() 37 | { 38 | derc = thef(derx, dery); 39 | 40 | picture(P(-Bd,-Bd), P(Bd,Bd), "2.5x2.5in"); 41 | 42 | begin(); 43 | 44 | camera.at(P(6.0/0.8,1.9/0.8,3.0/0.8)); 45 | 46 | arrow(P(0,0,0), P(Bd,0,0)); 47 | label(P(Bd,0), P(-4,-2), "$x_1$", l); 48 | 49 | arrow(P(0,0,0), P(0,Bd,0)); 50 | label(P(0,Bd), P( 2,-2), "$x_2$", br); 51 | 52 | line(P(0,0,0),P(0,0,thef(0,0))); 53 | 54 | dashed(); 55 | 56 | line(P(derx,0,0),P(derx,dery,0)); 57 | line(P(0,dery,0),P(derx,dery,0)); 58 | line(P(derx,dery,0),P(derx,dery,thef(derx,dery))); 59 | dot(P(derx,dery,0)); 60 | 61 | solid(); 62 | 63 | plain (); 64 | fill(White()); 65 | surface(F, R); 66 | 67 | arrow(P(0, 0, thef(0,0)), P(0,0,1.7)); 68 | label(P(0,0,1.7), P(0,4), "$y$", t); 69 | 70 | label(P(0,1.3,1.5), P(0,4), "$y=f(x_1,x_2)$", t); 71 | 72 | surface(DF, DR); 73 | 74 | dot(F(derx,dery)); 75 | 76 | bold(); 77 | 78 | double hx = 0.25; 79 | double hy = 0.9; 80 | 81 | arrow(P(derx, dery, 0), P(derx+hx,dery+hy,0)); 82 | label(P(derx+hx/2.0, dery+hy/2.0, 0), P(0,-4), "$h$", b); 83 | 84 | plain(); 85 | line(P(derx, dery, thef(derx,dery)), P(derx+hx,dery+hy,thef(derx,dery))); 86 | line(P(derx, dery, thef(derx,dery)), P(derx+hx,dery+hy,thedf(derx+hx,dery+hy))); 87 | 88 | bold(); 89 | arrow(P(derx+hx,dery+hy,thef(derx,dery)), P(derx+hx,dery+hy,thedf(derx+hx,dery+hy))); 90 | 91 | dot(P(derx+hx, dery+hy, thedf(derx,dery))); 92 | label(P(derx+hx, dery+hy, (thedf(derx,dery)+thedf(derx+hx,dery+dery))*0.5), P(4,0), "$Ah$", r); 93 | 94 | tikz_format(); 95 | end(); 96 | } 97 | -------------------------------------------------------------------------------- /figures/threediscs.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7a 2 | Landscape 3 | Center 4 | Inches 5 | Letter 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #898989 11 | 0 33 #d8d8d8 12 | 1 3 0 1 0 33 50 -1 -1 0.000 1 0.0000 840 840 840 840 840 840 1680 840 13 | 1 3 0 1 0 33 50 -1 -1 0.000 1 0.0000 840 840 600 600 840 840 1440 840 14 | 1 3 0 1 0 33 50 -1 -1 0.000 1 0.0000 840 840 720 720 840 840 1560 840 15 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 840 840 12 12 840 840 852 840 16 | 2 1 0 1 0 33 50 -1 -1 0.000 0 0 -1 1 1 2 17 | 0 0 1.00 60.00 120.00 18 | 0 0 1.00 60.00 120.00 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-------------------------------------------------------------------------------- 1 | /* slightly modified sample from epix samples gallery */ 2 | #include "epix.h" 3 | #include 4 | #include 5 | using namespace ePiX; 6 | 7 | 8 | double epsilon = 0.3; 9 | 10 | double f(double t) 11 | { 12 | return (-t*(t-2.6)*(t-3)/1.9+1.95)*0.4+0.4; 13 | } 14 | double fu(double t) 15 | { 16 | return f(t)+ epsilon; 17 | } 18 | double fl(double t) 19 | { 20 | return f(t)-epsilon; 21 | } 22 | double fn(double t) 23 | { 24 | return f(t)+0.9*epsilon*cos(10*t*sqrt(t)); 25 | } 26 | 27 | int main() 28 | { 29 | int i; 30 | picture(P(-0.45,-0.03), P(3,1.6), "3.6x1.6in"); 31 | 32 | begin(); 33 | 34 | //border(); 35 | h_axis(P(0,0),P(3,0),1); 36 | v_axis(P(0,0),P(0,1.6),1); 37 | 38 | bold(); 39 | plot(f, 0.0, 3.0, 40); 40 | plain(); 41 | plot(fn, 0.0, 3.0, 200); 42 | dashed(); 43 | plot(fu, 0.0, 3.0, 40); 44 | plot(fl, 0.0, 3.0, 40); 45 | solid(); 46 | 47 | //label (P(0,0.0), P(0,-4), "$a$", b); 48 | //label (P(3,0.0), P(0,-4), "$b$", b); 49 | 50 | label (P(-0.1,fl(0)), P(0,0), "$f-\\epsilon$", l); 51 | label (P(-0.1,f(0)), P(0,0), "$f$", l); 52 | label (P(-0.1,fu(0)), P(0,0), "$f+\\epsilon$", l); 53 | 54 | arrow_inset(0.9); 55 | label (P(1.1,1.35), P(0,0), "$f_n$", r); 56 | arrow (P(1.1,1.35),P(0.8,fn(0.8))); 57 | 58 | tikz_format(); 59 | end(); 60 | } 61 | 62 | -------------------------------------------------------------------------------- /figures/varphiplot.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/varphiplot.pdf -------------------------------------------------------------------------------- /figures/varphiplot.xp: -------------------------------------------------------------------------------- 1 | /* slightly modified sample from epix samples gallery */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | double gxf, gyf, gxt, gyt; 6 | double a = -0.4; 7 | 8 | double gr; 9 | 10 | P 11 | f(double x, double y) 12 | { 13 | return P(((x-a)*(1-a*x)-a*y*y)/(a*a*y*y+(1-a*x)*(1-a*x)) + 2.6, 14 | (((1-a*x)*y+a*(x-a)*y))/(a*a*y*y+(1-a*x)*(1-a*x))); 15 | } 16 | 17 | P 18 | fl(double t) 19 | { 20 | return f(gxf + t*(gxt-gxf),gyf + t*(gyt-gyf)); 21 | } 22 | 23 | P 24 | fc(double t) 25 | { 26 | return f(gr*cos(2*M_PI*t),gr*sin(2*M_PI*t)); 27 | } 28 | 29 | void 30 | draw_line(double xf, double yf, double xt, double yt, int steps) 31 | { 32 | gxf = xf; 33 | gyf = yf; 34 | gxt = xt; 35 | gyt = yt; 36 | 37 | line (P(xf,yf),P(xt,yt)); 38 | 39 | plot(fl, 0, 1, steps); 40 | } 41 | 42 | void 43 | draw_circle(double r, int steps) 44 | { 45 | gr = r; 46 | 47 | circle (P(0,0),r); 48 | 49 | plot(fc, 0, 1, steps); 50 | } 51 | 52 | void 53 | draw_dot(double x, double y) 54 | { 55 | ddot (P(x,y)); 56 | ddot (f(x,y)); 57 | } 58 | 59 | int main() 60 | { 61 | double slope; 62 | double c; 63 | double x; 64 | 65 | picture(P(-1.3,-1.3), P(3.9,1.3), "3.0x1.5in"); 66 | 67 | begin(); 68 | //border(); 69 | 70 | arrow(P(1.1,0.0),P(1.5,0.0)); 71 | 72 | pen(Black(0.2)); 73 | 74 | draw_circle(0.2,20); 75 | draw_circle(0.4,40); 76 | draw_circle(0.6,60); 77 | draw_circle(0.8,80); 78 | 79 | draw_line (-1.0,0.0,1.0,0.0,20); 80 | draw_line (0.0,-1.0,0.0,1.0,20); 81 | draw_line (-1.0/sqrt(2),-1.0/sqrt(2),1.0/sqrt(2),1.0/sqrt(2),20); 82 | draw_line (-1.0/sqrt(2),1.0/sqrt(2),1.0/sqrt(2),-1.0/sqrt(2),20); 83 | 84 | pen(Black()); 85 | 86 | draw_dot(a,0.0); 87 | 88 | draw_line (-0.5,-0.3,0.5,-0.3,10); 89 | draw_line (-0.5,0.3,0.5,0.3,10); 90 | draw_line (0.5,-0.3,0.5,0.3,10); 91 | draw_line (-0.5,-0.3,-0.5,0.3,10); 92 | 93 | draw_line (-0.5,-0.3,0.5,0.3,10); 94 | draw_line (-0.5,0.3,0.5,-0.3,10); 95 | 96 | draw_line (-0.5,0.3,-0.6,0.6,10); 97 | 98 | draw_line (-0.5,-0.3,-0.6,-0.6,10); 99 | draw_line (-0.5,-0.3,-0.4,-0.6,10); 100 | 101 | draw_line (0.5,-0.3,0.6,-0.6,10); 102 | draw_line (0.5,-0.3,0.4,-0.6,10); 103 | 104 | draw_line (0.5,-0.3,0.85,0.0,10); 105 | draw_line (0.5,0.3,0.85,0.0,10); 106 | 107 | bold(); 108 | solid(); 109 | circle(P(0,0),1); 110 | circle(P(2.6,0),1); 111 | plain(); 112 | 113 | 114 | tikz_format(); 115 | end(); 116 | } 117 | 118 | -------------------------------------------------------------------------------- /figures/weiergeneral.fig: -------------------------------------------------------------------------------- 1 | #FIG 3.2 Produced by xfig version 3.2.7b 2 | Landscape 3 | Center 4 | Metric 5 | A4 6 | 100.00 7 | Single 8 | -2 9 | 1200 2 10 | 0 32 #41534c 11 | 0 33 #555555 12 | 0 34 #888888 13 | 0 35 #e9e9e9 14 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1395 1440 18 18 1395 1440 1413 1440 15 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 1963 1105 9 9 1963 1105 1972 1105 16 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 2233 1200 9 9 2233 1200 2242 1200 17 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 2395 1007 9 9 2395 1007 2404 1007 18 | 1 3 0 1 0 0 50 -1 20 0.000 1 0.0000 2070 1354 9 9 2070 1354 2079 1354 19 | 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1917 995 40 | 0.000 -0.500 0.000 41 | 3 2 0 1 0 7 50 -1 -1 0.000 0 1 0 3 42 | 0 0 1.00 60.00 120.00 43 | 2242 471 2369 581 2727 649 44 | 0.000 -0.500 0.000 45 | 4 0 0 50 -1 0 12 0.0000 2 165 990 2531 1124 $\\{ a_n \\}$\001 46 | 4 1 0 50 -1 0 12 0.0000 2 135 270 908 680 $U$\001 47 | 4 1 0 50 -1 0 12 0.0000 2 135 270 1710 1212 $D$\001 48 | 4 0 0 50 -1 0 12 0.0000 2 135 270 1468 1504 $0$\001 49 | 4 0 0 50 -1 0 12 0.0000 2 165 1080 1543 367 $\\partial U$\001 50 | 4 1 0 50 -1 0 12 0.0000 2 180 990 2178 376 $\\{ p_n \\}$\001 51 | -------------------------------------------------------------------------------- /figures/weiergeneral.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/weiergeneral.pdf -------------------------------------------------------------------------------- /figures/weiergeneral.pdf_t: 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\put(1469,-665){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$0$}% 20 | }}}} 21 | \put(1544,472){\makebox(0,0)[lb]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\partial U$}% 22 | }}}} 23 | \put(2179,463){\makebox(0,0)[b]{\smash{{\SetFigFont{12}{14.4}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\{ p_n \}$}% 24 | }}}} 25 | \end{picture}% 26 | -------------------------------------------------------------------------------- /figures/xyxsqysq.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/xyxsqysq.pdf -------------------------------------------------------------------------------- /figures/xyxsqysq.xp: -------------------------------------------------------------------------------- 1 | /* -*-ePiX-*- */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | P F(double x, double y) 6 | { 7 | return P(x, y, x*y/(0.000000000000000000001+x*x+y*y)); 8 | } 9 | 10 | P FF(double r, double th) 11 | { 12 | double x = r*Cos(th); 13 | double y = r*Sin(th); 14 | return F(x,y); 15 | } 16 | 17 | double MAX(1.5), Bd(2); 18 | P pt1(-MAX,-MAX), pt2(MAX, MAX); 19 | domain R1(P(-MAX,-MAX), P(-2.5*MAX/12.5,MAX), mesh(10,25), mesh(10,75)); 20 | domain R2(P(-2.5*MAX/12.5,-MAX),P(2.5*MAX/12.5,MAX), mesh(5,25), mesh(10,75)); 21 | domain R3(P(2.5*MAX/12.5,-MAX), P(MAX,MAX), mesh(10,25), mesh(10,75)); 22 | 23 | domain RR(P(0,0), P(1.5, 2*M_PI), mesh(8,40), mesh(16,120)); 24 | 25 | int main() 26 | { 27 | picture(P(-Bd,-Bd), P(Bd,Bd), "2.5x2.5in"); 28 | 29 | begin(); 30 | 31 | camera.at(P(6,2,5)); 32 | 33 | arrow(P(-Bd,0,0), P(Bd,0,0)); 34 | label(P(Bd,0), P(-4,-2), "$x$", l); 35 | 36 | arrow(P(0,-Bd,0), P(0,Bd,0)); 37 | label(P(0,Bd), P( 2,-2), "$y$", br); 38 | 39 | plain (); 40 | fill(White()); 41 | surface(F, R1); // rectangular 42 | surface(F, R2); // rectangular 43 | surface(F, R3); // rectangular 44 | //surface(FF, RR); // polar 45 | 46 | arrow(P(0, 0, 0.3), P(0,0,1.5)); 47 | label(P(0,0,1.5), P(0,4), "$z$", t); 48 | 49 | tikz_format(); 50 | end(); 51 | } 52 | -------------------------------------------------------------------------------- /figures/zsqplot.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/figures/zsqplot.pdf -------------------------------------------------------------------------------- /figures/zsqplot.xp: -------------------------------------------------------------------------------- 1 | /* slightly modified sample from epix samples gallery */ 2 | #include "epix.h" 3 | using namespace ePiX; 4 | 5 | double gxf, gyf, gxt, gyt; 6 | double a = -0.4; 7 | 8 | double gr; 9 | 10 | P 11 | f(double x, double y) 12 | { 13 | return P(x*x-y*y + 2.6, 2*x*y); 14 | } 15 | 16 | P 17 | fl(double t) 18 | { 19 | return f(gxf + t*(gxt-gxf),gyf + t*(gyt-gyf)); 20 | } 21 | 22 | P 23 | fc(double t) 24 | { 25 | return f(gr*cos(2*M_PI*t),gr*sin(2*M_PI*t)); 26 | } 27 | 28 | void 29 | draw_line(double xf, double yf, double xt, double yt, int steps) 30 | { 31 | gxf = xf; 32 | gyf = yf; 33 | gxt = xt; 34 | gyt = yt; 35 | 36 | line (P(xf,yf),P(xt,yt)); 37 | 38 | plot(fl, 0, 1, steps); 39 | } 40 | 41 | void 42 | draw_arc(double in, double en, double r, int steps) 43 | { 44 | gr = r; 45 | 46 | arc (P(0,0),r,in*2*M_PI,en*2*M_PI); 47 | 48 | plot(fc, in, en, steps); 49 | } 50 | 51 | void 52 | draw_dot(double x, double y) 53 | { 54 | ddot (P(x,y)); 55 | ddot (f(x,y)); 56 | } 57 | 58 | int main() 59 | { 60 | double slope; 61 | double c; 62 | double x; 63 | 64 | picture(P(-0.3,-1.3), P(3.9,1.3), "2.5x1.5in"); 65 | 66 | begin(); 67 | //border(); 68 | 69 | plain(); 70 | 71 | //pen(Black(0.2)); 72 | 73 | double r; 74 | for (r = 0.1; r <= 1.1001; r += 0.1) { 75 | draw_arc(-0.125,0.125,r,r*80); 76 | } 77 | 78 | arrow(P(1.6,0),P(2.3,0)); 79 | 80 | double th; 81 | for (th = -0.125; th <= 0.1250001; th+= 0.125/4.0) { 82 | draw_line(0,0,1.1*cos(2*M_PI*th),1.1*sin(2*M_PI*th),2); 83 | } 84 | 85 | tikz_format(); 86 | end(); 87 | } 88 | 89 | -------------------------------------------------------------------------------- /getstats.sh: -------------------------------------------------------------------------------- 1 | #!/bin/sh 2 | 3 | echo 4 | echo -e "Chapters (including intro): \t"`grep '^[\]chapter\**{.*label{ch:' ca.tex | wc -l` 5 | echo -e "Appendices: \t"`grep '^[\]chapter{.*label{ap:' ca.tex | wc -l` 6 | echo -e "Sections: \t"`grep '^[\]section\**{' ca.tex | wc -l` 7 | echo -e "Sections (excluding appendix): \t"`cat ca.tex | sed '/^[\]appendix/q' | grep '^[\]section\**{' | wc -l` 8 | echo 9 | echo -e "Figures: \t"`grep '^[\]begin{myfig' ca.tex | wc -l` 10 | echo -e "Figures (excluding appendix): \t"`cat ca.tex | sed '/^[\]appendix/q' | grep '^[\]begin{myfig' | wc -l` 11 | echo 12 | echo -e "Exercises: \t"`grep '^[\]begin{exercise' ca.tex | wc -l` 13 | echo -e "Exercises (excluding appendix):\t"`cat ca.tex | sed '/^[\]appendix/q' | grep '^[\]begin{exercise' | wc -l` 14 | echo 15 | echo -e "Footnotes: \t"`grep '[\]footnote{' ca.tex | wc -l` 16 | echo -e "Footnotes (excluding appendix):\t"`cat ca.tex | sed '/^[\]appendix/q' | grep '[\]footnote{' | wc -l` 17 | 18 | echo 19 | -------------------------------------------------------------------------------- /publish.sh: -------------------------------------------------------------------------------- 1 | #!/bin/sh 2 | # This is really overly anal, but at least 4 runs seem to be needed with 3 | # the ogx, so better be safe and run it 5 times, also automake in glossaries 4 | # doesn't seem to always work 5 | for n in 1 2 3 4; do 6 | pdflatex ca 7 | makeindex ca 8 | makeglossaries ca 9 | done 10 | pdflatex ca 11 | # We are now using imakeidx and [automake] in glossaries 12 | # so no need to run these separately (FIXME: doesn't seem to work 13 | # reliably, that's why the run above, I've added makeindex 14 | # just for good measure there) 15 | -------------------------------------------------------------------------------- /resizepdftocrownquatro.sh: -------------------------------------------------------------------------------- 1 | #!/bin/sh 2 | gs \ 3 | -sOutputFile="$1"-cq.pdf \ 4 | -sDEVICE=pdfwrite \ 5 | -r720 \ 6 | -g5356x6976 \ 7 | -dCompatibilityLevel=1.5 \ 8 | -dNOPAUSE \ 9 | -dBATCH \ 10 | -dPDFFitPage \ 11 | "$1".pdf 12 | 13 | #-sPAPERSIZE=a4 \ 14 | -------------------------------------------------------------------------------- /slides/.gitignore: -------------------------------------------------------------------------------- 1 | *.aux 2 | *.toc 3 | *.snm 4 | *.log 5 | *.nav 6 | *.out 7 | *.vrb 8 | *.rubbercache 9 | *.fdb_latexmk 10 | *.fls 11 | *.synctex.gz 12 | intro.pdf 13 | 1.1-p1.pdf 14 | 1.1-p2.pdf 15 | 1.1-p3.pdf 16 | 1.2-p1.pdf 17 | 1.2-p2.pdf 18 | 1.3.pdf 19 | 1.4-p1.pdf 20 | 1.4-p2.pdf 21 | 2.1.1.pdf 22 | 2.1.2.pdf 23 | 2.2.1.pdf 24 | 2.2.2.pdf 25 | 2.2.3.pdf 26 | 2.3-p1.pdf 27 | 2.3-p2.pdf 28 | 2.3-p3.pdf 29 | 2.4.1-2.4.2.pdf 30 | 2.4.3.pdf 31 | 2.4.4.pdf 32 | 3.1-p1.pdf 33 | 3.1-p2.pdf 34 | 3.1-p3.pdf 35 | 3.2.1.pdf 36 | 3.2.2.pdf 37 | 3.2.3.pdf 38 | 3.2.4.pdf 39 | 3.3.1.pdf 40 | 3.3.2.pdf 41 | 3.3.3.pdf 42 | 3.3.4.pdf 43 | 3.4.1.pdf 44 | 3.4.2.pdf 45 | 3.5.1.pdf 46 | 3.5.2.pdf 47 | 4.1.1.pdf 48 | 4.1.2.pdf 49 | 4.2.pdf 50 | 4.3-p1.pdf 51 | 4.3-p2.pdf 52 | 4.4-p1.pdf 53 | 4.4-p2.pdf 54 | 5.1.pdf 55 | 5.2.1.pdf 56 | 5.2.2.pdf 57 | 5.2.3.pdf 58 | 5.2.4.pdf 59 | 5.3-p1.pdf 60 | 5.3-p2.pdf 61 | 5.4.1.pdf 62 | 5.4.2.pdf 63 | 5.4.3.pdf 64 | 5.5.pdf 65 | 5.6.pdf 66 | 6.1.1-6.1.2.pdf 67 | 6.1.3.pdf 68 | 6.2.pdf 69 | 6.3.1.pdf 70 | 6.3.3.pdf 71 | 7.1.1.pdf 72 | 7.1.2.pdf 73 | 7.2.1.pdf 74 | 7.2.2.pdf 75 | 7.2.3-7.2.4.pdf 76 | slides.zip 77 | -------------------------------------------------------------------------------- /slides/2.1.1.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt,aspectratio=169]{beamer} 2 | 3 | % All the boilerplate is in ccaslides.sty 4 | % Note that this also pulls in a custom vogtwidebar.sty 5 | \usepackage{ccaslides} 6 | 7 | \author{Ji\v{r}\'i Lebl} 8 | 9 | \institute[OSU]{% 10 | Departemento pri Matematiko de Oklahoma {\^S}tata Universitato} 11 | 12 | \title{Cultivating Complex Analysis:\\% 13 | Holomorphic functions (2.1.1)} 14 | 15 | \date{} 16 | 17 | \begin{document} 18 | 19 | \begin{frame} 20 | \titlepage 21 | \end{frame} 22 | 23 | \begin{frame} 24 | 25 | \section{} 26 | Consider a polynomial 27 | $P(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$. 28 | 29 | \medskip 30 | \pause 31 | 32 | Polynomials are nice, but there aren't many of them. 33 | 34 | \medskip 35 | \pause 36 | 37 | We can't even solve a simple equation like: $f'=f$. 38 | 39 | \medskip 40 | \pause 41 | 42 | Fix $z_0 \in \C$ and expand $P(z)$: 43 | 44 | \[ 45 | P(z) = c_0 + c_1 (z-z_0) + c_2 {(z-z_0)}^2 + \cdots + c_n {(z-z_0)}^n , 46 | \] 47 | \pause 48 | or 49 | \[ 50 | P(z_0+h) = c_0 + c_1 h + c_2 h^2 + \cdots + c_n h^n . 51 | \] 52 | \pause 53 | Then 54 | \[ 55 | \lim_{h \to 0} \frac{P(z_0+h) - P(z_0)}{h} = 56 | \lim_{h \to 0} \frac{P(z_0+h) - c_0}{h} = c_1 . 57 | \] 58 | \pause 59 | $P(z_0+h)$ is approximated (locally) by $c_0 + c_1 h$ 60 | up to an error that vanishes faster than $h$. 61 | 62 | \medskip 63 | \pause 64 | 65 | \textbf{Key point:} The limits are ``as a \emph{complex} $h$ goes to $0$.'' 66 | 67 | \end{frame} 68 | 69 | \begin{frame} 70 | We want functions with such a derivative. That is, functions approximated 71 | by $c_0 + c_1 h$: 72 | \[ 73 | f(z_0+h) = \underbrace{f(z_0)}_{c_0} + \underbrace{\xi h}_{c_1 h} + o(\sabs{h}) 74 | \qquad \text{for some } \xi \in \C . 75 | \] 76 | 77 | \medskip 78 | \pause 79 | 80 | \begin{definition} 81 | Let $U \subset \C$ be open. 82 | A function 83 | $f \colon U \to \C$ 84 | is \emph{complex differentiable} at $z_0 \in U$ if 85 | the limit 86 | \[ 87 | f'(z_0) \overset{\text{def}}{=} 88 | \lim_{h \to 0} \frac{f(z_0+h) - f(z_0)}{h} 89 | \qquad \text{exists.} 90 | \] 91 | \pause 92 | We call $f'(z_0)$ the \emph{complex derivative} of $f$ 93 | at $z_0$. Sometimes $\frac{df}{dz}$ is used. 94 | \pause 95 | \medskip 96 | 97 | $f \colon U \to \C$ is 98 | \emph{holomorphic} 99 | if it is complex differentiable at every point. 100 | \end{definition} 101 | 102 | \end{frame} 103 | 104 | \begin{frame} 105 | 106 | We proved above: 107 | 108 | \begin{proposition} 109 | If $P(z)$ is a polynomial, then $P \colon \C \to \C$ is holomorphic. 110 | \end{proposition} 111 | \pause 112 | 113 | An exercise: 114 | 115 | \begin{proposition} 116 | If $U \subset \C$ is open and $f \colon U \to \C$ is holomorphic, then $f$ 117 | is continuous. 118 | \end{proposition} 119 | 120 | \end{frame} 121 | 122 | \end{document} 123 | -------------------------------------------------------------------------------- /slides/2.4.3.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt,aspectratio=169]{beamer} 2 | 3 | % All the boilerplate is in ccaslides.sty 4 | % Note that this also pulls in a custom vogtwidebar.sty 5 | \usepackage{ccaslides} 6 | 7 | \author{Ji\v{r}\'i Lebl} 8 | 9 | \institute[OSU]{% 10 | Departemento pri Matematiko de Oklahoma {\^S}tata Universitato} 11 | 12 | \title{Cultivating Complex Analysis:\\% 13 | The exponential (as power series) (2.4.3)} 14 | 15 | \date{} 16 | 17 | \begin{document} 18 | 19 | \begin{frame} 20 | \titlepage 21 | \end{frame} 22 | 23 | \begin{frame} 24 | Previously we defined 25 | \[ 26 | e^{x+iy} 27 | = e^x (\cos y + i \sin y) 28 | \] 29 | and we showed that $e^z$ is holomorphic. \pause Let's define it a different 30 | way. 31 | 32 | \medskip 33 | \pause 34 | 35 | \begin{proposition} 36 | The power series 37 | \begin{equation*} 38 | f(z) = \sum_{n=0}^\infty \frac{1}{n!} z^n , 39 | \end{equation*} 40 | is the unique convergent power series at the origin 41 | such that $f(0)=1$ and $f'=f$. 42 | \pause 43 | 44 | Moreover, the series converges on $\C$ and 45 | $f(z) = e^z$. 46 | \end{proposition} 47 | 48 | \pause 49 | 50 | \textbf{Proof:} 51 | Consider a convergent series $f$ satisfying $f(0)=1$ and $f'=f$: 52 | \[ 53 | f(z) = \sum_{n=0}^\infty c_n z^n . 54 | \] 55 | \pause 56 | $f(0)=1$ implies that $c_0 = 1$. 57 | 58 | \end{frame} 59 | 60 | \begin{frame} 61 | $f' = f$, so \pause 62 | \[ 63 | f'(z) = 64 | \sum_{n=1}^\infty n c_n z^{n-1} = 65 | \sum_{n=0}^\infty (n+1) c_{n+1} z^{n} 66 | \pause 67 | = 68 | f(z) = \sum_{n=0}^\infty c_n z^n . 69 | \] 70 | \pause 71 | Coefficients of power series are unique so $c_n = (n+1) c_{n+1}$. 72 | 73 | \pause 74 | By induction, $c_n = \frac{1}{n!}$ so 75 | \(\displaystyle f(z) = \sum_{n=0}^\infty \frac{1}{n!} z^n \) 76 | 77 | \pause 78 | And this series converges in $\C$ (exercise). 79 | 80 | \pause 81 | \medskip 82 | 83 | Both $f$ and the exponential are holomorphic and equal to their derivatives: 84 | \pause 85 | \[ 86 | \frac{d}{dz} \left[ \frac{f(z)}{\exp(z)} \right] 87 | = 88 | \frac{f'(z)\exp(z) - f(z) \exp'(z)}{{\bigl(\exp(z)\bigr)}^2} 89 | \pause 90 | = 91 | \frac{f(z)\exp(z) - f(z) \exp(z)}{{\bigl(\exp(z)\bigr)}^2} 92 | = 0. 93 | \] 94 | \pause 95 | Hence, $f(z) = C \exp(z)$ for some constant $C$. 96 | \pause 97 | As $f(0) = \exp(0) = 1$, conclude $C=1$. 98 | \qed 99 | \end{frame} 100 | 101 | 102 | \end{document} 103 | -------------------------------------------------------------------------------- /slides/2.4.4.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt,aspectratio=169]{beamer} 2 | 3 | % All the boilerplate is in ccaslides.sty 4 | % Note that this also pulls in a custom vogtwidebar.sty 5 | \usepackage{ccaslides} 6 | 7 | \author{Ji\v{r}\'i Lebl} 8 | 9 | \institute[OSU]{% 10 | Departemento pri Matematiko de Oklahoma {\^S}tata Universitato} 11 | 12 | \title{Cultivating Complex Analysis:\\% 13 | The identity theorem (2.4.4)} 14 | 15 | \date{} 16 | 17 | \begin{document} 18 | 19 | \begin{frame} 20 | \titlepage 21 | \end{frame} 22 | 23 | \begin{frame} 24 | \begin{theorem}[Identity] 25 | Suppose $U \subset \C$ is a domain, 26 | and $f \colon U \to \C$ analytic. 27 | \pause 28 | If $Z_f = \bigl\{ z \in U : f(z) = 0 \bigr\}$ 29 | has a limit point in $U$, then $f$ is identically zero. 30 | \pause 31 | In other words, all points of $Z_f$ are isolated unless $f \equiv 0$. 32 | \end{theorem} 33 | 34 | \pause 35 | 36 | \begin{definition} 37 | The points in the set $Z_f$ are called the \emph{zeros} of $f$. 38 | \end{definition} 39 | 40 | \pause 41 | \medskip 42 | 43 | Common application: 44 | 45 | \pause 46 | 47 | \emph{If the function is zero on a nonempty open subset, then $f \equiv 0$.} 48 | 49 | \end{frame} 50 | 51 | \begin{frame} 52 | 53 | \textbf{Proof:} 54 | Suppose $f$ is not identically zero. 55 | 56 | \medskip 57 | \pause 58 | 59 | $Z_f$ is closed (in $U$, of course) as $f$ is continuous. 60 | 61 | \medskip 62 | \pause 63 | 64 | Must show that points of $Z_f$ are isolated. 65 | 66 | \medskip 67 | \pause 68 | 69 | WLOG $0 \in U$, $0 \in Z_f$, $0$ not in the interior of $Z_f$. 70 | 71 | \pause 72 | 73 | \[ 74 | f(z) = \sum_{n=0}^\infty c_n z^n . 75 | \] 76 | \pause 77 | As $f(0) = 0$, $c_0 =0$. 78 | \pause 79 | \medskip 80 | 81 | Let $k$ be the smallest $k$ such that $c_k \not=0$ (exists since $f$ 82 | is not identically zero nearby). 83 | 84 | \pause 85 | 86 | \[ 87 | f(z) = z^k \sum_{n=k}^\infty c_n z^{n-k} = z^k g(z) . 88 | \] 89 | $g(z)$ is a convergent power series and $g(0) = c_k \not= 90 | 0$. 91 | 92 | \pause 93 | \medskip 94 | 95 | $g$ is continuous and so $g(z) \not= 0$ in a whole neighborhood of $0$. 96 | 97 | \pause 98 | \medskip 99 | 100 | $z^k$ is only zero at $0$ $\Rightarrow$ $0$ is an isolated zero of $f$. 101 | 102 | \end{frame} 103 | 104 | \begin{frame} 105 | 106 | So points of $Z_f$ are either interior points or isolated points. 107 | 108 | \medskip 109 | \pause 110 | 111 | Let $Z_f'$ be the interior points (nonisolated points). 112 | 113 | \medskip 114 | \pause 115 | 116 | $Z_f'$ is closed (in $U$) ($Z_f$ is closed 117 | and points in $Z_f'$ cannot approach isolated points). 118 | 119 | \medskip 120 | \pause 121 | 122 | $Z_f'$ is open and closed and $U$ is connected $\Rightarrow$ either $U=Z_f'$ or $Z_f' = \emptyset$. 123 | \qed 124 | 125 | \end{frame} 126 | 127 | \begin{frame} 128 | A useful idea from the proof: 129 | 130 | \medskip 131 | \pause 132 | 133 | If $f(z)$ is a power series at $a$ and $f(a)=0$ (and $f\not\equiv 0$), 134 | we can factor out some power of $z-a$: 135 | \pause 136 | \begin{equation*} 137 | f(z) = {(z-a)}^k g(z) , 138 | \end{equation*} 139 | where $g(z)$ is a power series at $a$ such that $g(a) \not= 0$. 140 | \end{frame} 141 | 142 | 143 | \end{document} 144 | -------------------------------------------------------------------------------- /slides/5.2.4.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt,aspectratio=169]{beamer} 2 | 3 | % All the boilerplate is in ccaslides.sty 4 | % Note that this also pulls in a custom vogtwidebar.sty 5 | \usepackage{ccaslides} 6 | 7 | \author{Ji\v{r}\'i Lebl} 8 | 9 | \institute[OSU]{% 10 | Departemento pri Matematiko de Oklahoma {\^S}tata Universitato} 11 | 12 | \title{Cultivating Complex Analysis:\\% 13 | Meromorphic functions (5.2.4)} 14 | 15 | \date{} 16 | 17 | \begin{document} 18 | 19 | \begin{frame} 20 | \titlepage 21 | \end{frame} 22 | 23 | \begin{frame} 24 | \begin{definition} 25 | A holomorphic function $f \colon U \setminus S \to \C$ with poles on a 26 | discrete set $S \subset U$ is said to be \emph{meromorphic}. 27 | \end{definition} 28 | 29 | \pause 30 | 31 | If $p \in S$ is a pole, set 32 | $f(p) = \infty$ to get a function 33 | \[ 34 | f \colon U \to \C_{\infty} . 35 | \] 36 | 37 | \pause 38 | The extended $f$ is continuous. 39 | 40 | \pause 41 | \medskip 42 | 43 | In fact, $\nicefrac{1}{f}$ is holomorphic at a pole $p$ 44 | \quad 45 | (defn.\ of ``$f \colon U \to \C_\infty$ is holomorphic at $p$''). 46 | 47 | \pause 48 | \medskip 49 | 50 | A \emph{meromorphic function} is 51 | ``a holomorphic function $f \colon U \to \C_{\infty}$.'' 52 | 53 | \pause 54 | \medskip 55 | 56 | \textbf{Technicality:} Should we consider the constant $\infty$ 57 | a meromorphic function? 58 | 59 | \pause 60 | In this course, we do not. 61 | 62 | \pause 63 | \medskip 64 | 65 | To emphasize we will often say 66 | ``Let 67 | $f \colon U \to \C_\infty$ be meromorphic.'' 68 | 69 | \end{frame} 70 | 71 | \begin{frame} 72 | 73 | One could define functions on $U \subset \C_\infty$ (like we did with LFTs). 74 | 75 | \pause 76 | \medskip 77 | 78 | If $\infty \in U$, 79 | \quad 80 | $f \colon U \to \C_\infty$ is holomorphic at $\infty$ if 81 | $f(\nicefrac{1}{z})$ is holomorphic at $0$. 82 | 83 | \pause 84 | \medskip 85 | 86 | So an LFT is a biholomorphic mapping $\C_\infty \to \C_\infty$. 87 | 88 | \pause 89 | \medskip 90 | 91 | \textbf{Exercise:} 92 | Show that a holomorphic $f \colon \C_\infty \to \C_\infty$ has 93 | at most finitely many poles and finitely many zeros. 94 | 95 | \pause 96 | \medskip 97 | 98 | \textbf{Exercise:} 99 | Show that a holomorphic $f \colon \C_\infty \to \C_\infty$ is 100 | either constant or onto. 101 | 102 | \pause 103 | \medskip 104 | 105 | \textbf{Exercise:} 106 | Show that a holomorphic $f \colon \C_\infty \to \C_\infty$ is a 107 | rational function (a polynomial divided by a polynomial). 108 | 109 | \pause 110 | \medskip 111 | 112 | \textbf{Exercise:} 113 | Show that an injective holomorphic $f \colon \C_\infty \to \C_\infty$ is an 114 | LFT\@. 115 | \end{frame} 116 | 117 | \end{document} 118 | -------------------------------------------------------------------------------- /slides/README.md: -------------------------------------------------------------------------------- 1 | Slides for a YouTube course 2 | =========================== 3 | 4 | These are slides for a youtube course, but the idea is that these could also be 5 | just taken and used in other classes. They will be slightly updated and 6 | improved even after the youtube videos are done, so they may be slightly 7 | different than what appeared in the youtube course. The copyright license is 8 | the same as the book, so feel free to use these in your classes and modify them 9 | in the same way as the book itself. 10 | 11 | The aspect ratio 16:9 is by default set to make it work better with youtube or 12 | online streaming to take advantage of the whole screen. Perhaps for 13 | traditional projectors it might be good to change this back to 4:3, 14 | but perhaps you might have to modify some of the slides if you do that. 15 | 16 | The figures are imported from a subdirectory ../figures/ relative to the 17 | current directory, so if you want to build these in a different directory 18 | structure, make sure to copy the relevant figures and adapt the subimport 19 | and includegraphics commands. 20 | 21 | This is a longer term project so it will probably take a while for this to be 22 | completely filled in. 23 | -------------------------------------------------------------------------------- /slides/essential-sing.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jirilebl/ca/0c89feac1a1fe9c352e806d97e30b0b79f881c5e/slides/essential-sing.png -------------------------------------------------------------------------------- /slides/make-zip.sh: -------------------------------------------------------------------------------- 1 | #!/bin/sh 2 | echo Building 3 | 4 | for n in *.tex ; do 5 | echo ============================= 6 | p=`basename $n .tex`".pdf" 7 | echo $n '-->' $p 8 | #rubber -d --synctex $n 9 | latexmk -pdf -synctex=1 "$n" 10 | done 11 | 12 | echo ============================= 13 | echo Packaging 14 | 15 | rm -f slides.zip 16 | zip slides.zip *.pdf 17 | 18 | -------------------------------------------------------------------------------- /slides/vogtwidebar.sty: -------------------------------------------------------------------------------- 1 | % 2 | % \widebar from Stackexchange by Hendrik Vogt 3 | % https://tex.stackexchange.com/questions/16337/can-i-get-a-widebar-without-using-the-mathabx-package 4 | % 5 | \makeatletter 6 | \let\save@mathaccent\mathaccent 7 | \newcommand*\if@single[3]{% 8 | \setbox0\hbox{${\mathaccent"0362{#1}}^H$}% 9 | \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}% 10 | \ifdim\ht0=\ht2 #3\else #2\fi 11 | } 12 | %The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath: 13 | \newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna} 14 | %If there's a superscript following the bar, then no negative kern may follow the bar; 15 | %an additional {} makes sure that the superscript is high enough in this case: 16 | \newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}} 17 | %Use a separate algorithm for single symbols: 18 | \newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}} 19 | \newcommand*\wide@bar@[3]{% 20 | \begingroup 21 | \def\mathaccent##1##2{% 22 | %Enable nesting of accents: 23 | \let\mathaccent\save@mathaccent 24 | %If there's more than a single symbol, use the first character instead (see below): 25 | \if#32 \let\macc@nucleus\first@char \fi 26 | %Determine the italic correction: 27 | \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}% 28 | \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}% 29 | \dimen@\wd\tw@ 30 | \advance\dimen@-\wd\z@ 31 | %Now \dimen@ is the italic correction of the symbol. 32 | \divide\dimen@ 3 33 | \@tempdima\wd\tw@ 34 | \advance\@tempdima-\scriptspace 35 | %Now \@tempdima is the width of the symbol. 36 | \divide\@tempdima 10 37 | \advance\dimen@-\@tempdima 38 | %Now \dimen@ = (italic correction / 3) - (Breite / 10) 39 | \ifdim\dimen@>\z@ \dimen@0pt\fi 40 | %The bar will be shortened in the case \dimen@<0 ! 41 | \rel@kern{0.6}\kern-\dimen@ 42 | \if#31 43 | \overline{\rel@kern{-0.6}\kern\dimen@\macc@nucleus\rel@kern{0.4}\kern\dimen@}% 44 | \advance\dimen@0.4\dimexpr\macc@kerna 45 | %Place the combined final kern (-\dimen@) if it is >0 or if a superscript follows: 46 | \let\final@kern#2% 47 | \ifdim\dimen@<\z@ \let\final@kern1\fi 48 | \if\final@kern1 \kern-\dimen@\fi 49 | \else 50 | \overline{\rel@kern{-0.6}\kern\dimen@#1}% 51 | \fi 52 | }% 53 | \macc@depth\@ne 54 | \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar 55 | \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}% 56 | \macc@set@skewchar\relax 57 | \let\mathaccentV\macc@nested@a 58 | %The following initialises \macc@kerna and calls \mathaccent: 59 | \if#31 60 | \macc@nested@a\relax111{#1}% 61 | \else 62 | %If the argument consists of more than one symbol, and if the first token is 63 | %a letter, use that letter for the computations: 64 | \def\gobble@till@marker##1\endmarker{}% 65 | \futurelet\first@char\gobble@till@marker#1\endmarker 66 | \ifcat\noexpand\first@char A\else 67 | \def\first@char{}% 68 | \fi 69 | \macc@nested@a\relax111{\first@char}% 70 | \fi 71 | \endgroup 72 | } 73 | \makeatother 74 | -------------------------------------------------------------------------------- /vogtwidebar.sty: -------------------------------------------------------------------------------- 1 | % 2 | % \widebar from Stackexchange by Hendrik Vogt 3 | % https://tex.stackexchange.com/questions/16337/can-i-get-a-widebar-without-using-the-mathabx-package 4 | % 5 | \makeatletter 6 | \let\save@mathaccent\mathaccent 7 | \newcommand*\if@single[3]{% 8 | \setbox0\hbox{${\mathaccent"0362{#1}}^H$}% 9 | \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}% 10 | \ifdim\ht0=\ht2 #3\else #2\fi 11 | } 12 | %The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath: 13 | \newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna} 14 | %If there's a superscript following the bar, then no negative kern may follow the bar; 15 | %an additional {} makes sure that the superscript is high enough in this case: 16 | \newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}} 17 | %Use a separate algorithm for single symbols: 18 | \newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}} 19 | \newcommand*\wide@bar@[3]{% 20 | \begingroup 21 | \def\mathaccent##1##2{% 22 | %Enable nesting of accents: 23 | \let\mathaccent\save@mathaccent 24 | %If there's more than a single symbol, use the first character instead (see below): 25 | \if#32 \let\macc@nucleus\first@char \fi 26 | %Determine the italic correction: 27 | \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}% 28 | \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}% 29 | \dimen@\wd\tw@ 30 | \advance\dimen@-\wd\z@ 31 | %Now \dimen@ is the italic correction of the symbol. 32 | \divide\dimen@ 3 33 | \@tempdima\wd\tw@ 34 | \advance\@tempdima-\scriptspace 35 | %Now \@tempdima is the width of the symbol. 36 | \divide\@tempdima 10 37 | \advance\dimen@-\@tempdima 38 | %Now \dimen@ = (italic correction / 3) - (Breite / 10) 39 | \ifdim\dimen@>\z@ \dimen@0pt\fi 40 | %The bar will be shortened in the case \dimen@<0 ! 41 | \rel@kern{0.6}\kern-\dimen@ 42 | \if#31 43 | \overline{\rel@kern{-0.6}\kern\dimen@\macc@nucleus\rel@kern{0.4}\kern\dimen@}% 44 | \advance\dimen@0.4\dimexpr\macc@kerna 45 | %Place the combined final kern (-\dimen@) if it is >0 or if a superscript follows: 46 | \let\final@kern#2% 47 | \ifdim\dimen@<\z@ \let\final@kern1\fi 48 | \if\final@kern1 \kern-\dimen@\fi 49 | \else 50 | \overline{\rel@kern{-0.6}\kern\dimen@#1}% 51 | \fi 52 | }% 53 | \macc@depth\@ne 54 | \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar 55 | \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}% 56 | \macc@set@skewchar\relax 57 | \let\mathaccentV\macc@nested@a 58 | %The following initialises \macc@kerna and calls \mathaccent: 59 | \if#31 60 | \macc@nested@a\relax111{#1}% 61 | \else 62 | %If the argument consists of more than one symbol, and if the first token is 63 | %a letter, use that letter for the computations: 64 | \def\gobble@till@marker##1\endmarker{}% 65 | \futurelet\first@char\gobble@till@marker#1\endmarker 66 | \ifcat\noexpand\first@char A\else 67 | \def\first@char{}% 68 | \fi 69 | \macc@nested@a\relax111{\first@char}% 70 | \fi 71 | \endgroup 72 | } 73 | \makeatother 74 | --------------------------------------------------------------------------------