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: )
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. 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.
1. Holomorphic functions in several variables
2. Convexity and pseudoconvexity
3. CR Geometry
4. The dbar-problem
5. Integral kernels
6. Complex analytic varieties
There are 304 exercises in the book of various difficulty.
Download the book as PDF
(Version 3.1, May 21st, 2019, 182 pages)
Version 3.x has changed exercise numbers, and changed some exercises, (unavoidable, but hopefully the improvements are worth it, see the change log to see what's new).
(If you require the old version 2.4 for the old exercise numbers you can download the PDF)
See the errata in the current version.
Look at the change log to see what changed in the newest version.
Buy a copy at lulu.com for $9.36 (the discounted price on lulu). The downside of buying on lulu is that shipping is not as fast. It will take a while before 3.1 will appear on Google, Amazon, Barns and Noble, etc., so the following links might not work yet. The lulu link above should work. The ISBN is 978-0-359-64225-0 and you can also buy it on Amazon. The list price is 15.60, but (at least once they will "stock it") it will ship faster. These versions always take a few weeks before a new update appears once it is updated.
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.
The source is now 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 scv.tex. I compile the pdf with pdflatex. You also want to run makeindex to generate the index (I generally run pdflatex scv three times, then makeindex scv, and then finally pdflatex scv again)..
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).