Working the Complex Field
By: Jiří Lebl (website #1 https://www.jirka.org/ (personal), website #2 https://math.okstate.edu/people/lebl/ (work: OSU), email: )
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The purpose of this book is to teach a one-semester graduate course in complex analysis for incoming graduate students. It was created to teach Math 5283 at Oklahoma State University. It is a natural first semester in a two semester sequence where the second semester could be several complex variables or perhaps harmonic analysis. It could perhaps be used for a more elementary two-semester sequence if the appendix (metric spaces, some basic analysis) is covered first, and all the optional bits of the main text are also covered.
We assume basic knowledge of undergraduate analysis in the real variable, called advanced calculus in some schools. The text assumes knowledge of metric spaces and differential analysis in several variables, but if the reader is not confident on these topics or has not yet seen them, the useful results are presented (with proofs) in the appendices. With that, a basic prerequisite for the course would be at least a single semester of undergraduate analysis if the appendices are also covered or read, and if the student has seen metric spaces and mappings in \({\mathbb{R}}^2\), then the course can just start in Chapter 1. Very basic undergraduate linear and abstract algebra is also useful.
At OSU I covered Chapters 1–6 without the optional (starred) bits, and the first half of chapter 7.
Do let me know if there are typos or errors. You can email me at .
This book may be modified and customized for a specific purpose if necessary. If you do modify the book, make sure to mark it prominently as such to avoid confusion. This aspect is also important for longevity of the book. The book can be updated and modified even if I happen to drop off the face of the earth. You do not have to depend on any publisher being interested as with traditional textbooks.
Introduction
1. The Complex Plane
2. Holomorphic and Analytic Functions
3. Line Integrals and Rudimentary Cauchy Theorems
4. The Logarithm and Cauchy
5. Counting Zeros and Singularities
6. Montel and Riemann
7. Harmonic Functions
8. Weierstrass Factorization
9. Rational Approximation
10. Analytic Continuation
Appendix A. Metric Spaces
Appendix B. Results From Basic Analysis
Appendix C. Basic Notation and Terminology
There are 600 exercises in the book (483 if we ignore the appendices). There are 72 figures (52 if we ignore the appendices).
Download the book as PDF
(Version 1.1, December 18th, 2020, 304 pages)
See the errata in the current version.
Look at the change log to see what changed in the newest version.
The LaTeX source is hosted on GitHub: https://github.com/jirilebl/ca
You can get an archive of the source of the released version on github, look under https://github.com/jirilebl/ca/releases, though if you plan to work with it, maybe best to look at just the latest working version as that might have any errata or new additions. Though these might be a work in progress. Perhaps best is to let me know.
The main file is ca.tex. I compile the pdf with pdflatex. You also want to run makeindex to generate the index and makeglossaries to generate the glossary of used notation (and then rerun pdflatex a couple of times, I repeat this whole thing 4 times just to be sure: see publish.sh script).
The github 'master' version is the current working version, so it will have whatever new changes I make in my tree.
This work is dual licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License and Creative Commons Attribution-Share Alike 4.0 License. You can use, print, copy, and share this book as much as you want. You can base your own book/notes on these and reuse parts if you keep the license the same (that is, as long as you use at least one of the two licenses).