Tasty Bits of Several Complex Variables

A whirlwind tour of the subject

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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This book is a polished version of my course notes for Math 6283, Several Complex Variables, given in Spring 2014, Spring 2016, and Spring 2019 semesters at Oklahoma State University. It should be suitable for a semester-long topics course or for self-study as an introduction to the subject.

The prerequisites are decent knowledge of vector calculus, basic real analysis, and a working knowledge of complex analysis in one variable (e.g. GCCA). It should be accessible to beginning graduate students after a complex analysis course, and perhaps even very advanced undergraduates.

Overall it should be enough material for a semester-long course, though that of course depends on the speed of the lecturer. There are quite a few exercises sprinkled throughout the text, and I hope that the reader is at least attempting all of them.

It is not meant as an exhaustive reference. It is simply a whirlwind tour of several complex variables. To find a list of books useful for reference and further reading, see the end of the book.

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:


1. Holomorphic functions in several variables
2. Convexity and pseudoconvexity
3. CR Geometry
4. The \(\bar{\partial}\)-problem
5. Integral kernels
6. Complex analytic varieties

There are 304 exercises in the book of various difficulty.

Download:

Download the book as PDF
(Version 3.4, December 23rd, 2020, 182 pages)

See the errata in the current version.

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

Buy paperback:

Buy a copy at lulu.com for $15.60. The downside of buying on Lulu is that shipping is not as fast (the upside is that I get more money). The ISBN is 978-0-359-64225-0 and you can also buy it on Amazon or Barnes and Noble, etc. The list price is 15.60, but Amazon will ship faster than Lulu (but I only get like 50 cents). Beware however that due to some interaction of Lulu and Amazon, the Amazon version can be quite out of date. Even if everything goes to plan, the version on Amazon, Barns and Noble, etc., always takes a few weeks before a new update appears.

You can also buy a larger (letter size) coil-bound copy at lulu.com for $12.59. I use this when teaching. It is also good for studying from: it can be kept open at a page easily, it has larger font for those late night study sessions, and it has larger margins to write notes into. Lulu doesn't sell these versions elsewhere though, you have to buy it from them.

Source:

The source is hosted on GitHub: https://github.com/jirilebl/scv

You can get an archive of the source of the released version on github, look under https://github.com/jirilebl/scv/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).

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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