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I'm looking for a specific recommendation for a textbook on complex analysis. I very much like the outline of Bruce Palka's An Introduction to Complex Function Theory. The textbook is an ideal resource which can be used for an upper-level undergraduate course/beginning graduate course. Some of the topics that catch my eye:

  1. Chapter 1: The Complex Number System,
  2. Chapter 2: The Rudiments of Plane Topology
  3. Chapter 3: Analytic Functions
  4. Chapter 4: Complex Integration
  5. Chapter 5: Cauchy's Theorem and its Consequences
  6. Chapter 6: Harmonic Functions
  7. Chapter 7: Sequences and Series of Analytic Functions
  8. Chapter 8: Isolated Singularities of Analytic Functions
  9. Chapter 9: Conformal Mapping

The textbook, however, is quite verbose, and chatty. I don't mind reading such textbook, but I suppose reading Palka's textbook will consume a lot of time. Moreover, there are a very large number of interdependent exercises, and it would be very hard to do all of them.

Therefore, I'm looking for alternative books that covers all the aforementioned topics in detail and has good exercises. Please note that I'm looking for such books as Brown and Churchill's Complex Variables and Applications.

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First book: Theory of Complex Functions, Reinhold Remmert.

This book does not assume real analysis, only mathematical analysis is sufficient. The author gives a concise introduction in complex analysis, but there are only a few exercises available in the book. Meanwhile, there is no complex geometry involved, so this is really an undergraduate complex analysis book.

Second book: Complex Analysis in One Variable, Raghavan Narasimhan/Yves Nievergelt.

The first chapter of this book covers the whole semester of complex analysis course in undergraduate. There are a bunch of exercises available in the part II of this book. There are some complex manifold issues addressed in this book. It also contains a chapter that briefly introducing the topic of several complex variables.

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Try these:

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I like Gamelin's Complex Analysis. It's geared at the same level, covers the topics you mentioned, and is readable without being verbose.

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  • $\begingroup$ I have read a bit of Gamelin's textbook. I did not enjoy the presentation of the textbook. Most of the theorems weren't proved rigorously, and some of the material was scattered in different chapters; for example, the topics on harmonic functions etc. More importantly, I didn't particularly like the exercises in the textbook. If you have another recommendation in mind, please do share it. $\endgroup$ – Junaid Aftab Apr 6 '18 at 16:47
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I find "Complex Analysis" by Bak & Donald to be very readable.

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