Tasty Bits of Several Complex Variables

This book is a polished version of my course notes for Math 6283, Several Complex Variables, given in Spring 2014, Spring 2016, Spring 2019, and Fall 2023 semesters at the Oklahoma State University. There is more material than can fit in a one semester class to allow for several different versions of the course. In fact, I did a different selection each semester I taught it. Quite a few exercises of various difficulty are sprinkled throughout the text, and I hope a reader is at least attempting or thinking about most of them. Many are required later in the text. The reader should attempt exercises in sequence; earlier exercises can help or even be required to solve later ones.

The prerequisites are a decent knowledge of vector calculus, basic real analysis, and a working knowledge of complex analysis in one variable. Measure theory (Lebesgue integral and its convergence theorems) is useful, but it is not essential except in a couple of places later in the book. The first two chapters and most of the third are accessible to beginning graduate students after one semester of a standard single-variable complex analysis graduate course. From time to time, basic knowledge of differential forms is useful, and in the chapter on varieties we use some basic ring theory from algebra. By design, it can replace the second semester of complex analysis, with the first semester perhaps taught with my one-variable book GCCA.

This book is not intended as an exhaustive reference. It is simply a whirlwind tour of several complex variables. See the end of the book for a list of books for reference and further reading. There are also appendices for a list of one-variable results, an overview of differential forms, some basic algebra, measure theory, and other bits and pieces of analysis.

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.