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Twenty-One Lectures on Complex Analysis

Alexander Isaev
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Miklós Bóna
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When I saw that the book consists of a very short 194 pages, yet it covers most topics that we find in competing textbooks, my first questions were “Just what is missing? What did the author leave out to make the book so concise?”

The theorems are rigorously proved, and there is no lack of exercises (though solutions for some of them would have been most welcome). What is missing is context. There are very few sentences between two lemmas, propositions, theorems, or proofs. My favorite example comes on page 88, where the author says, “We will need the following important fact,” and it turns out that the important fact is a theorem, which has parts (a), (b), (c), and (d), some of which have further components, and which takes half a page to state.

There are very few applications to other parts of mathematics. Neither is any historical background given.

This leads to the interesting question of whether I like this bare-bones approach or not. On the balance, I believe that Complex Analysis is not the best subject for this concept; it is simply too advanced and multifaceted for it. I can see the benefit of a book like this for some students in Calculus, namely those students who only want to learn the basic facts and are not interested in anything else. By the time students get to a Complex Analysis course, however, they have probably made some commitment to Mathematics, and want to understand not just the material itself, but also how it fits in a bigger picture. This is where a good book can step in, explaining facts in a deeper and more thorough way than it is possible to do in lectures.

Miklós Bóna is Professor of Mathematics at the University of Florida.

See the table of contents in the publisher's webpage.