Guide to Cultivating Complex Analysis

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: )

⇒ Download the book / Buy paperback / YouTube course

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, although it has more material than can be covered in one semester. The first seven chapters 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 calculus 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.

The first time I ran this one-semester course 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.

Table of contents:

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 604 exercises in the book (487 if we ignore the appendices). There are 72 figures (52 if we ignore the appendices).

Download:

Download the book as PDF
(Version 1.5, July 19th, 2023, 304 pages)

See the errata in the current version if any.

Look at the change log to see what changed in the newest version.

Buy paperback:

Cover thumbnail image
ISBN: 979-8-6850-5792-1.
Buy a copy on Amazon for $18.00.
The book should be able to be ordered through other means than amazon as well.

YouTube course and slides (not yet complete):

I am building a set of slides and a YouTube course. You may want to watch it fullscreen. The course is not yet complete, but it is already quite far and still growing.

Link to the playlist.

The Slides: (also incomplete so far) as PDFs. Note that some of the slides have been slightly improved since I recorded the lectures, so they might not match 100%.

The LaTeX source for the slides is available on github in the slides subdirectory (see https://github.com/jirilebl/ca and see below). You should also read the README.md file in that subdirectory. You have to build them from there to get the figures right if you want to rework the slides, or you'd have to probably modify the source and copy the figures out of the figures subdirectory. Feel free to use them for your class, the license is the same as the book.

Source:

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 current working version (the master branch) to be more up to date, though at times this might be a work in progress if I am working on some new material. Perhaps best let me know if you want to make modifications.

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 slides mentioned above are in the slides subdirectory. The slides are more of a work in progress so best to get these out of the current github.

The github 'master' version is the current working version, so it will have whatever new changes I make in my tree.

License:

Creative Commons License Creative Commons License 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).

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