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I'll be taking the next year off, and I intend to cover some mathematics material out of class under the supervision of an instructor. For instance, I have decided to cover both complex analysis and algebra over the summer. To that end, I am looking collect a good bunch of references that meet the following criteria:

  1. The first book should be a textbook that will introduce me to the basics; the textbook should include computations, and introduce the subject comprehensively, at the level of an undergraduate course. Also, I should also be able to use such books to prepare for the GRE.

  2. The second book should be a good supplement to the first book. By keeping the more advanced book on the side, I should be able to understand the theory deeply, and study the subject in more detail.

  3. It'd be great if someone could recommend good and approachable graduate level textbooks as well. The textbook should be approach, and a good companion to other resources for the self-study program.

I, for instance, have preliminarily decided to keep the following textbooks:

Complex Analysis

  1. Complex Variables and Applications by Churchill; and

  2. Complex Analysis Gamelin.

  3. Functions of One Complex Variable, Volume I by Conway.

Algebra

  1. Abstract Algebra by Dummit and Foote; and

  2. Basic Algebra by Anthony Knapp.

Please feel free to suggest a good bunch of textbooks which one can use with each other to study a subject more thoroughly than can be taught in a course. Ideally, as part of such a study plan, I intend to not only cover the basics also to cover the theory in greater depth. So, in addition to covering material from books in the undergraduate book(s), I am hoping for some suggestions for graduate level textbook(s) that can be used on the side with the undergraduate level books as part of an extensive self study plan.

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  • $\begingroup$ math.stackexchange.com/questions/174876/…. $\endgroup$
    – user312097
    Commented Apr 14, 2017 at 10:22
  • $\begingroup$ @A---B Perfect. In addition to books in the 'complete undergraduate bundle-pack,' I am hoping for some suggestions for graduate level textbook(s) that can be used on the side with the undergraduate level books as part of an extensive self study plan. I'll edit my post accordingly. $\endgroup$ Commented Apr 14, 2017 at 10:26
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    $\begingroup$ My recollection is that GRE math subject problems tend to be at a much lower level than most of what you'll find in an algebra or complex analysis book. Basically, they're either problems at the level of Apostol's two-volume Calculus (including linear algebra, differential equations and multivariable calculus), which could still be somewhat difficult, or they're easy questions from the very beginning of more advanced topics, like groups or metric spaces. So preparing for the GRE means knowing calculus and linear algebra well, and having a smattering of knowledge in several advanced topics. $\endgroup$
    – user49640
    Commented Apr 14, 2017 at 22:18
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    $\begingroup$ @user1147844 Since this question was asked six years ago, do you think he may have completed his year off and moved on? $\endgroup$
    – John Douma
    Commented Jun 9, 2023 at 1:25
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    $\begingroup$ "Visual Complex Analysis" by Needham is a good companion to the complex analysis books you listed. It's non-rigorous but it has a lot of figures that help greatly in building intuition. $\endgroup$ Commented Jun 9, 2023 at 1:40

3 Answers 3

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For complex analysis Complex Analysis by Bak and Newman can be a good reference for self study. Dummit and Foote book is also good. In this question I tried to answer about some good references that I think can be good for you too.

Good luck!

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Algebra (vaguely in increasing order):

  • Dummit and Foote, "Abstract Algebra"
  • M. Artin, "Algebra"
  • Rotman, "Advanced Modern Algebra"
  • Lang, "Algebra"
  • Atiyah and MacDonald, "Commutative Algebra" (towards algebraic number theory and algebraic geometry)

Complex analysis (vaguely in increasing order):

  • Churchill and Brown, "Complex Variables and Applications"
  • Ahlfors, "Complex Analysis"
  • Conway, "Functions of One Complex Variable"
  • Rudin, "Real and Complex Analysis"
  • Remmert, "Classical Topics in Complex Function Theory"
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I would add just one book to your list, given that you explicitly seek computations:

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