## Real and Complex Analysis, by C. Apelian and S. Surace, now published December 19, 2009

Posted by Akhil Mathew in analysis, General, math education.
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(This material is reposted from here.)

The book Real and Complex Analysis, by Christopher Apelian and Steve Surace, was recently released.

It’s mainly for an introductory upper-level undergraduate course in real and complex analysis, especially at small liberal-arts colleges.  In this post, I’ll describe this book and how I was involved in its production.

At the start of my freshman year, my analysis teacher, Professor Surace, asked me to check over the drafts of a book he and his colleague (and my former teacher) Prof. Apelian were working on. It was the textbook for the course. At the time, if I remember correctly, there were six chapters: on the real and complex spaces, basic topology, limits, continuity, convergence of functions, and derivatives. The complex analysis part of the book was in its infancy (e.g., there was only a rudimentary outline of one chapter, which had been written some time back and was typeset in Word—they wrote it well before before they had switched to LaTeX).

Anyway, since one of my hobbies that year was playing around with LaTeX and trying to figure out all the cool formatting tricks used in books, and since they hadn’t yet done much to change the (somewhat bland) default book style, I pointed out a few tweaks that, in my opinion, would make it look better. They agreed, and I ended up being put in charge of the layout. It turned out that I would need to learn a lot more about LaTeX though (or at least, learn to look up things a lot more). As you can see from the sample pages, the authors—or to be precise, one of them :-)—had fairly detailed ideas of how the book should appear. I don’t know if I learned LaTeX properly, but I sure learned a lot of hacks.

I also ended up sketching the figures, which enabled me to pick up the useful (and definitely nontrivial) skill of using Adobe Illustrator. In fact, I’ve used it in making some of these blog posts.  (Though I would recommend interested passers-by also to consider, say, Tikz depending on your aims; I’m not familiar with it, but it has the benefit of being free.)

I don’t know all the details of how the book got started–I think the idea goes back many years, and certainly before I started helping them. The authors, who were faculty members at Drew University, a small liberal arts college, found it inconvenient to have separate courses for real and complex analysis. So they taught them together, i.e. covered topics such as continuity and convergence for real and complex functions simultaneously. They couldn’t find a textbook to use though, so they decided to write one.

I had noticed that a lot of basic complex analysis textbooks (e.g. Ahlfors, Hille) covered much logically redundant material. Then again, at least when I first got interested in analysis, it was Cauchy’s theorem and the residue theorem—not rigorizing integration—that piqued my curiosity. So I first learnt about compactness and continuity from a book on complex analysis—if I remember correctly, an old volume by Herb Silverman that I found in a public library.

(One could of course ask—why not just talk about metric spaces, or topological spaces more generally? But the curriculum at Drew University does not typically cover that, and this is an introductory textbook.)

I also learned that while having two authors definitely does not halve the time for completion, it can improve the quality.

The book also has a website, here. Soon I am going to write up partial solutions to the exercises to be posted there.  It’s now been more than three years that I’ve been helping out, and as fun as it’s been, I’m glad to see the book in print!