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I took a course in Complex Analysis last term and really enjoyed it. The level though was very basic, the last bit that we did was to prove the Cauchy Residue Theorem. We haven't even touched upon important aspects such as for example Analytic Continuation.

I'd love to take this further on my own, and I'm curious to know:

Which aspect, topic or concept from Complex Analysis do you think is the most important or interesting to know about.

I would be more interested in applications of concepts from Complex Analysis within pure mathematics, rather than applications to science, but all thoughts are helpful!

Also, in case you have a favourite book about Complex Analysis, please post it!

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I don't think this is a good way to use this site, honestly. You should probably pick good complex analysis books, those which treat the subject beyond an introduction, and read their indices and —preferably!— some of their text; I would recommend the amazing book by Reinhold Remmert, for its depth, breath and historical information. – Mariano Suárez-Alvarez♦ Jan 21 '12 at 23:43
Thks for your comment! I totally agree one should look at textbooks to get a clearer view, the reason I am asking for other people's opinion is that they often help to see a big area of mathematics from different angles before starting to tackle it, more than a textbook can (because most of them try to be impartial and do not suggest one theorem or concept to be more important to another, which I think is a good thing when it comes to a textbook!) It helps to get started, I think. – harlekin Jan 22 '12 at 1:43
No textbook is impartial: the author has made choices of subjects to include... – Mariano Suárez-Alvarez♦ Jan 22 '12 at 2:08
true .. may I ask, how do you decide which book to use or which topic within an area of mathematics to start studying, without having a clearly defined research problem (say) or course that guides you in your decision? In particular, how do you decide on this given that you have finished the "introductory phase" ? I hope this question is not inappropriate since you suggested not to use this site for this, please ignore my comment in case you don't want to be asked such things! Tks – harlekin Jan 22 '12 at 2:23
It seems like you know enough Complex Analysis to look into some of the applications to number theory. I can't recommend any particular book off the top of my head, however. – Thomas Andrews Jan 22 '12 at 2:52

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