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Complex Analysis Book

  1. Apr 6, 2010 #1
    I'm going to be taking the graduate complex analysis this coming Fall and I've not taken the undergraduate version of the course. It will be a challenge but something that my advisers told me will be surely doable. Anyway, aside from the textbook used for the course, can anyone recommend a supplementary graduate level textbook and perhaps a lighter, undergraduate level textbook? I'm going to be doing some reading on the subject this summer to prepare for the course, and some solid supplementary texts would be a great aid.
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  3. Apr 6, 2010 #2


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    Introduction to Complex Analysis by Priestley is a good undergraduate level book to work through if you're new to the material.
  4. Apr 6, 2010 #3
  5. Apr 7, 2010 #4
    Ahlfors is the best book on complex analysis that I've been able to find. Ahlfors develops the subject from scratch and relies as little as possible on other areas of mathematics such as multivariable calculus, but the reading level is nonetheless much higher than this might lead you to expect. I think a lot of people are caught off guard and end up giving the book poor reviews because of this.

    Ahlfors develops integration theory using a beautiful and extraordinarily elementary characterization of homology in the plane due to Emil Artin. This simplifies everything a lot and increases the elegance tenfold.

    Ahlfors develops in great detail the geometry of fractional linear transformations before studying integration theory and the more analytic areas of the subject. Mastery of fractional linear transformations is a very useful skill to have in complex analysis, and most textbooks don't emphasize this enough. The graduate course in complex analysis that I'm taking started on day one with these transformations and moved on to develop Lobachevsky's geometry in the unit disc and the upper half-plane.

    Ahlfors also contains a lot of tidbits of wisdom. He describes removable singularities as nothing more than points at which we lack information. One review I read of this book really criticized this description, but I find it highly suggestive and useful.

    It may be helpful to supplement Ahlfors with something like Basic Complex Analysis by Marsden and Hoffman. This book is written at a much lower level in the sense that almost no details are ever omitted and it goes to very extreme lengths to explains techniques for solving problems by working through tons of examples. Ahlfors gives almost no examples and expects you to have enough understanding to figure out the problems on the spot.

    Conway is a common graduate book used for the subject. It's a decent book, but I don't like it because the writing is exceptionally dull. It has no spirit.

    Lang is also a decent book in terms of coverage, but Lang's command of English is so poor I can hardly bring myself to read anything by him. I've never seen so many comma splices before in one book.

    The writing style of Ahlfors is not exactly exciting, but it is at least nearly free of grammatical errors, and throughout the book the writing carries a confident, elegant tone.
  6. Apr 7, 2010 #5


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    I think I have said this in another thread (and also replying after zpconn), but I am using Lang's book in a course right now and quite like it. It covers pretty much. It's a bit strange though; the first few chapters are at a low level, and repeats all kinds of stuff that you should know from (real) analysis (limits, series, compactness, etc.). Also, there are a lot of trivial exercises. But after a while the level gets higher, and the exercises require more thinking. Lang's writing is concise and to the point, a bit informal sometimes.

    I think zpconn's comment on his English is funny. I kind of see what he means, Lang's English is not perfect. Maybe it's because I'm no native speaker, but it doesn't really bother me. Comma slicing is in particular hard for me to spot, because in my language it's not an error.

    I still have to check out Ahlfors and Conway, which seem to be the only (?) alternatives in terms of rigour and elegance.
  7. Apr 7, 2010 #6


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    Which is...? :smile:
  8. Apr 7, 2010 #7
    there are so many good books on complex analysis (or complex variables) you can't really go wrong with any of them (imho). in addition to the ones already mentioned i would add the ones by brown/churchill & bak/newman. they're both about as basic as they come, especially brown/churchill. & of course schaum's is also a good supplement. hille & markushevich do more advanced ones. & for problems to solve, there are the collections by volkovyskii/lunts/aramanovich, krzyz & knopp.
    Last edited: Apr 7, 2010
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