An Introduction to Complex Function Theory (Undergraduate Texts in Mathematics)

Rating:
Summary: Wordy, but comprehensive
Review: I took this class from Dr. Palka himself back in 1994. He is obsessed with words when he speaks and naturally it comes across in his text. One of the few math books where I've occasionally been forced to utilize an English language dictionary. I am not sure whether this is a negative or a positive. It certainly didn't result in me learning any less about the subject.

Besides that Palka includes numerous calculational examples and problems covering a lot of extra material that is both useful and interesting. The lack of solutions makes it difficult for self-study, but you learn more from the problems if it's left up to you to be the final authority on whether it's right or wrong.

Anyway, I give the book a 9 for Comprehensiveness and Readability and a 6 for Succintness. But it's definitely a 10 for range and depth of problems and examples.

Rating:
Summary: Good Secondary Text
Review: I used this book for a first course in complex analysis. This book comprehensively covers the standard topics in a first course. There are also enrichment sections for those who are interested (such as proving certain definitions are equivalent to the usual definitions). The style of writing is very readable, but this is at the expense of using a lot of words and hence the author sometimes takes a long time to explain a simple idea. This book is not written concisely for this reason. Also, the book does not set out definitions as separate paragraphs nor are they numbered (this seems to be increasingly common for math texts); they are buried within the text, and are very difficult to find later on. Proofs are given out in full and explained in detail with lots of words (at the expense of length and terseness), so that readers can understand very easily. There are plenty of examples, and they demonstrate good techniques for evaluating integrals. There are many exercises and solutions are not provided. Historical facts and footnotes are seldomly found. I believe this book is also suitable for less-prepared students.

Rating:
Summary: Very readable text.
Review: I used this book for a first course in complex analysis. This book comprehensively covers the standard topics in a first course. There are also enrichment sections for those who are interested (such as proving certain definitions are equivalent to the usual definitions). The style of writing is very readable, but this is at the expense of using a lot of words and hence the author sometimes takes a long time to explain a simple idea. This book is not written concisely for this reason. Also, the book does not set out definitions as separate paragraphs nor are they numbered (this seems to be increasingly common for math texts); they are buried within the text, and are very difficult to find later on. Proofs are given out in full and explained in detail with lots of words (at the expense of length and terseness), so that readers can understand very easily. There are plenty of examples, and they demonstrate good techniques for evaluating integrals. There are many exercises and solutions are not provided. Historical facts and footnotes are seldomly found. I believe this book is also suitable for less-prepared students.

Rating:
Summary: Good Secondary Text
Review: This is a good text that covers somewhat more ground than either Ahlfors or vol. 1 of Conway, including a really nice, long section on conformal mapping. There are an astonishingly large number of exercises. The exposition is generally quite good, and the proofs are thorough and very detailed.

However, I would not want to use this as a primary text because it is too long-winded. Even the proofs have a lot of unnecessary verbiage. Put simply, it takes too damned long to read. I think the happy medium lies somewhere between the slick austerity of Rudin's analysis books and the overwritten style of Palka's book.

That said, I still think Palka's book is well worth having. You may never want to read it cover to cover, but I find it very useful to turn to when I am confused by other texts, or when I want a more detailed explanation of a given topic. Chances are that I'll find it here. Also, there are lots of good concrete examples, especially in the sections on residue techniques for integration and conformal mapping. And you will never run out of interesting exercises if you have this book around.

So what is the best primary text for complex analysis? Good question, and I don't have a good answer. Conway's book is pretty nice but unexciting. Ahlfors' is generally good but requires extremely careful reading (sometimes between the lines) to catch all the nuances; it also feels a bit old-fashioned for my tastes. The recent version of Narasimhan's book, with the large collection of expository exercises by Nievergelt, is much more efficient, sophisticated, and challenging than the others and looks like it will serve as a very good reference/second course after learning the material elsewhere.