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I am learning complex analysis and currently reading Ahlfors’ Complex Analysis.

I feel like I need to do more exercises than the five problems provided in each section. Therefore, I am seeking recommendations for complex analysis books with more exercises than Ahlfors’, but still at the same level and cover all (or more) of the topics covered in Ahlfors' book.

Additionally, I am seeking recommendations for only strictly pure mathematics books without any real-world applications.

I will continue to read Ahlfors’ book alongside the recommended book because I think Ahlfors is still worth reading.

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    $\begingroup$ +1 : specifically for "I feel like I need to do more exercises that 5 problems each section", which I completely agree with. $\endgroup$ Commented Apr 27 at 17:10

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Complex Analysis by Freitag and Busam is a great textbook with plenty of exercises.

It contains chapters after the "standard" complex analysis on Elliptic Functions, Elliptic Modular Forms, and some results in analytic number theory, if those are of interest to you!

The book also has short solutions/hints for most of the exercises, so you can check your work.

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Take a look at this book: A Complex Analysis Problem Book by Daniel Alpay.

This is an excellent source of problems in Complex Analysis. One of the books fully dedicated for problems in Complex Analysis.

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The book Complex Analysis by Ahlfors is undoubtedly great (see in particular sections 2 and 3 of the reviews available at the link). I recommend Functions of One Complex Variable by John B. Conway together with a solution manual which contains solutions to most of the $452$ exercises given in the book.

The exercises are mentioned in the foreword to Conway's book:

  • [From the Preface]: Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as An Introduction to Mathematics has influenced the writing and selection of subject matter for this book.

  • The other guiding principle followed is that all definitions, theorems, etc. should be clearly and precisely stated. Proofs are given with the student in mind. Most are presented in detail and when this is not the case the reader is told precisely what is missing and asked to fill in the gap as an exercise. The exercises are varied in their degree of difficulty. Some are meant to fix the ideas of the section in the reader's mind and some extend the theory or give applications to other parts of mathematics.

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  • $\begingroup$ This book is from "Graduate Texts in Mathematics " series which have the reputation of being a very difficult to read and for only grad students or mathematicians, are you sure this book is ion the same level as Ahlfors ? $\endgroup$
    – pie
    Commented May 3 at 10:07
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    $\begingroup$ @pie: Although it is a GTM book, it is nevertheless a gentle introduction into the subject. When you go through this book you will recognize the author has a talent to guide a student smoothly through topics. Check also the exercises in the solution manual to see that they start not too hard. $\endgroup$ Commented May 3 at 11:10
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An unsurpassed reference, dealing comprehensively with both theory and exercises and thus possibly answering the above question, is surely Robert B. Burckel classical monograph [1], recently updated as [2].
In these reference(s) almost all the important results in complex analysis are proved and between each theorem/proposition/lemma many exercised are proposed, relating to them as strengthening, extensions to cases of particular interest and complements: moreover, most of these exercises have hints for their solution and each chapter is followed by a rich bibliographical section pointing the reader to original proofs and/or alternative proofs of the results in the chapter, as well as to further complements. Furthermore, the book is also clearly written and readable with pleasure.
The only weak point of the work [1] is that Prof. Burckel sadly passed away on the 10th of December 2023, leaving the second volume unfinished.

Well, my two cents.

Bibliography

[1] Robert B. Burckel, An introduction to classical complex analysis. Vol. 1., Lehrbücher und Monographien aus dem Gebiete der exakten Naturwissenschaften: Mathematische Reihe, Band 64, Basel-Stuttgart: Birkhäuser Verlag, pp. 520 (1979), MR555733, Zbl 0434.30001.

[2] Robert B. Burckel, Classical analysis in the complex plane, Cornerstones, Cham: Birkhäuser Verlag ISBN 978-1-0716-1963-6/hbk; 978-1-0716-1965-0/ebook, pp. xxix+1123 (2021), Zbl 1482.30001, MR4385454.

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"An Introduction To Complex Function Theory" by Bruce Palka.

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I liked "Basic Complex Analysis" by Jerrold E. Marsden and Michael J. Hoffman.
Each section has solved exercises presented as worked examples, then end of section and end of chapter problems.
Here is a link to the 3rd edition, mine is the second edition, 1997.

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I recommend Visual Complex Analysis by Tristan Needham (https://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469). This is a wonderful book. The following are reviews from experts:

"[Needham's] highly praised massive book Visual Complex Analysis may still be resounding in the minds of those who have read it. The original approach and the numerous graphics must have left a lasting impression." -- Adhemar Bultheel, Mathematical Association of America Reviews

"Visual Complex Analysis is a delight, and a book after my own heart. By his innovative and exclusive use of the geometrical perspective, Tristan Needham uncovers many surprising and largely unappreciated aspects of the beauty of complex analysis." --Roger Penrose

"Tristan Needham's Visual Complex Analysis will show you the field of complex analysis in a way you almost certainly have not seen before. Drawing on historical sources and adding his own insights, Needham develops the subject from the ground up, drawing us attractive pictures at every step of the way. If you have time for a year course, full of fascinating detours, this is the perfect text; by picking and choosing, you could use it for a variety of shorter courses. I am tempted to hide the book from my own students, in order to appear more clever for popping up with crisp historical anecdotes, great exercises, and pictures that explain things like that mysterious 2πi that crops up in integrals. Whether you use Visual Complex Analysis as a text, a resource, or entertaining summer reading, I highly recommend it for your bookshelf."--American Mathematical Monthly

"Delivers what its title promises, and more: an engaging, broad, thorough, and often deep, development of undergraduate complex analysis and related areas. . .A truly unusual and notably creative look at a classical subject." --American Mathematical Monthly

"One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. I'm not lamenting the loss of traditional Euclidean geometry, despite its virtues, because it too emphasised stilted formalities. But to replace our rich visual intuition by silly games with 2 x 2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's 'Visual Complex Analysis' with its elegantly illustrated visual approach. Yes, he has 2 x 2 matrices--but his are interesting." --New Scientist

"Committed to the exclusive use of geometrical arguments and content to pay the price of 'an initial lack of rigour', he has produced a radically new text. The author writes "as though [he] were explaining the ideas directly to a friend". This informal style is excellently judged and works extremely well."--Mathematical Review

"This is a book in which the author has been willing to make himself available as our teacher. His own voice enters in a rather charming way....I recommend Visual Complex Analysis, as something to read and enjoy, to share with students, and perhaps to inspire other books in which the voice of the author is vividly present to teach and explain."--American Mathematical Monthly

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I recommend the following two books:

Volkovyskii, L.I. et al., ``A collection of problems on complex analysis'', Dover Publication, New York, 1991

Krzyz, J.G., ``Problems in Complex variable theory'', American Elsevier Publ. Co., New York, 1971.

Although old, I found them very useful. In particular, the book by Krzyz has a very good section on summation of series by means of contour integration.

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