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An Introduction to Complex Function Theory (Undergraduate Texts in Mathematics), by Bruce P. Palka
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This book provides a rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a "short course" in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature.
- Sales Rank: #1040438 in Books
- Published on: 1995-09-07
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x 1.25" w x 6.14" l, 2.18 pounds
- Binding: Hardcover
- 560 pages
Most helpful customer reviews
2 of 2 people found the following review helpful.
Strange notation.
By mojambo
I'm most of the way though an upper-level complex analysis course where we've used this book. The book is not bad. The sections follow a nice order, and the book is much more readable compared to Ahlfors, which I have as well. I find that Ahlfors can be very unfocused, where Palka follows a more structured pattern of presenting material. Each section contains a surprisingly large number of exercises which vary from routine computation to non-trivial proofs. Many sections have plenty of figures as well, which help show the geometry behind the complex plane. Overall, I've had a nice experience using this book, and would recommend it to other students in the future.
That is not to say that Palka is without faults. Our professor rearranged the order of the book, slightly, moving Mobius transformations close to the beginning of the course. This added insight actually made a fair number of the exercises in the middle chapters (mostly concerning Cauchy's thereoms and such) easier. Palka is also prone to being overly chatty, to the point where the the content gets watered down by his verbiage.
Most aggravating, for me, is Palka's use of notation. He strays from the general path regarding most standard notation, and prefers to adopt his own. For instance, he uses "phi" for "emptyset", "~" for set difference, and several others. While these don't necessarily make any real fault in the book, it is annoying to have to translate Palka's notation. Also, the chapters are numbered with roman numerals, then individual sections and subsections start over from "1" every time. Exercise sections are numbered with the last section of the chapter--it's somewhat of a mess. A more straightforward Ch.sec.subsec would be less confusing.
Most of these remarks are merely esoteric, though, and as said, the book is overall quite sound. It's worth having for an undergraduate course as a complement to, or in lieu of, more popular books, such as Ahlfors.
6 of 6 people found the following review helpful.
Fantastic for self study
By Bryan E. Bischof
I agree that the exposition and proof's are both wordy, but for self study I found this invaluable. I took this course as a reading course, which means no lecture accompanies the course. I find most weeks, I can solve nearly all of the problems assigned by 1-3 readings of the chapter. This is in my opinion the best book I found to date for self study. However, the addition of solutions to selected exercises would be even better. I recommend the book for those wishing for a introduction to complex analysis, or those with some background and wishing to extend their background to include the material covered on most complex qualifying exams.
14 of 15 people found the following review helpful.
Good Secondary Text
By A Customer
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.
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