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I have looked at the post about Complex Analysis books but that is just a list of books and what level they are at and who they would be geared towards.

What would be a good book for someone who has been through about 80% Complex Analysis by Serge Lang?

The post I have looked at are:

  1. Complex Analysis Book
  2. Books on complex analysis
  3. Complex analysis book for Algebraic Geometers
  4. Are there books introducing to Complex Analysis for people with algebraic background?

Edit 1:

How does The Theory of Functions by Titchmarsh fit in? By this, I mean compared to the books suggested in the two answers.

Edit 2: How does Complex Analysis by Ahlfors fit in as well?

Community
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dustin
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2 Answers2

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I'm not a big fan of Lang's complex analysis book-I consider it the weakest by far of all his textbooks. But since that's what you're using, you're really asking for a recommendation for an advanced course on complex analysis.

The most intensive and yet readable textbook I know on the subject is Complex Analysis in One Variable by University of Chicago master Raghavan Narasimhan and Yves Nievergelt. It is complex analysis for the serious analyst, from rapid coverage of the basics of analytic functions, covering spaces and Runge's theorem through the basics of functions of several complex variables and the elements of complex manifolds. This is unquestionably a graduate level text-it requires a good working knowledge of both real analysis at the level of Rudin or Pugh,a basic knowledge of abstract and linear algebra and topology. In short,it's a serious book for advanced students. I think you'll find it very helpful.

Another possible good text is Function Theory on Planar Domains by Steven Fisher, which covers a second course in complex analysis in a much more geometric manner, focusing on the Dirichlet problem and Hardy spaces.If you're interested in the geometric aspects of function theory, this will be a better choice for you. Best of all,it's in Dover and really cheap!

Those are the 2 best ones I know for a second course.Hope it helped.

Mathemagician1234
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It's a broad question because there are so many different topics you could cover. Why not try Sanford Segal's book "Nine Introductions in Complex Analysis" published by North-Holland. It's purpose is give nine different paths after completing a first course in complex analysis, so that you can better decide what you'd like to study after.

You might also try to pick and choose from Siegel's three volume series "Topics in Complex Function Theory". Finally I suggest you also look at the beautiful theory of complex manifolds.