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I'm an engineering student but I self-study pure mathematics. I am looking for a Complex Variables Introduction book (to study before complex analysis). I have the Brown and Churchill book but I was told that's for engineers and physicist mostly, not for mathematicians. I also looked for Fisher and Flanigan, but they don't seem to have as many topics as Brown. I wonder which book is best for the subject or if one of the two previously mentioned will do to master most of the topics of complex variables as a mathematician. Thanks.

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closed as off-topic by Stopple, RP_, Timothy Chow, Andy Putman, Ben McKay May 25 at 8:16

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    $\begingroup$ This is not a research level question.. please see math.stackexchange.com/questions/30749/… $\endgroup$ – Praphulla Koushik May 24 at 16:33
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    $\begingroup$ You might look at Tristan Needham, Visual Complex Analysis. $\endgroup$ – Joseph O'Rourke May 24 at 17:24
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    $\begingroup$ math.stackexchange.com/q/30749/43208 I recall at least one very strong review of the book by Ralph Boas. $\endgroup$ – Todd Trimble May 24 at 18:58
  • $\begingroup$ I'm very confused by your main question, which asks for a book on complex variables which one should study before a course in complex analysis. I think you are under the wrong impression that the complex numbers have a vast theory that, once understood, you'll be ready to tackle complex analysis. At least historically, this is not true, with analysis and complex numbers converging very early. I think you should just open any book in complex analysis and read the first one or two chapters, and you'll be ready for complex analysis. $\endgroup$ – Andy Sanders May 24 at 23:14
  • $\begingroup$ Above, by "any book in complex analysis," I mean you should open all the books people have recommended and read the first chapter or two. To add a few more books to list, have a look at the books by: Greene, Gamelin, and Lang. These are three separate books, google is your friend in finding them. If you understand the first chapter or two of these books, you will be more prepared than most students who enter a complex analysis class for the first time. $\endgroup$ – Andy Sanders May 24 at 23:22
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There are many good books, but the choice depends on your background and on your needs and on your taste. For what purpose do you study complex variables? Do you like geometry or formulas?

If your aim is to use complex variables (for example in engineering and physics problems) Whittaker and Watson is an excellent choice. It is somewhat outdated, but contains most of the things useful in applications. By far more than modern texts. And I have to warn you that this is a difficult reading, but it has an enormous number of exercises.

The standard textbook for mathematicians (US graduate students) is Ahlfors. An excellent choice for the very beginning (mathematician) is Cartan (translated into English from the French, no exercises). One very good recent one is by Don Marshall. The last 3 are oriented on pure mathematicians, while Whittaker Watson is universal, can be read by engineers and mathematicians with equal profit. Remmert has an excellent book 2 volumes, both translated into English: 1. Theory of complex functions and 2. Classical topics in complex function theory. What makes these books special is the great attention to history of the subject.

Ahlfors and WW are very different in contents, which reflects the change of fashion in mathematics. Ahlfors is more geometric, while WW is full of formulas.

On the very minimum level an old but excellent little book by Knopp can be recommended. He has 2 separate small books of exercises.

By difficulty I can order these books as follows:

Knopp < Cartan < Marshall = Remmert << Ahlfors << WW.

If you read foreign languages, I can also recommend Hurwitz-Courant which does not exist in English. It is of the same epoch as WW but written from a completely different point of view. It begins on a very basic level, but ends with more advanced material then all other texts that I mention (the things which are covered nowadays under the title Riemann surfaces, and not included in CV textbooks anymore). For this reason it does not fit into the ordering I wrote above. But the first part can be considered as a superb minimal introduction to the subject, written by one of the greatest masters of it (Hurwitz). There are very good, corrected and amended editions: the German (by Rohrl) and in Russian (by Evgrafov). It has no exercises.

There is a completely different approach to the self study of complex variables. Take a good problem book and solve problems. (Keep a companion text besides and look into it when necessary).

By "good problem book" on the subject I mean the above mentioned Knopp, and also Volkovyski (Translated from the Russian by Dover), and Polya Szego. The difficulty ranking is

Knopp < Volkovyski <<< Polya-Szego.

Volkovyskii is especially recommended: first it is very large, and second, every chapter has a short background. So you can really use it without a textbook. In any case, solving problem is a very important part of self-study. You cannot claim that you understood something, until you solve a couple of problems.

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  • $\begingroup$ Has any serious effort been considered to translate Hurwitz-Courant into English? I would love to read it eventually. $\endgroup$ – Brevan Ellefsen May 25 at 0:06
  • $\begingroup$ @Brevan Ellefsen: to the best of my knowledge: no. You will have to learn German to read it. $\endgroup$ – Alexandre Eremenko May 25 at 3:52
  • $\begingroup$ It is highly unlikely that it will be translated. There is no market for it: it cannot be used as a textbook in the USA because it has no exercises, and the exposition of advanced topics is probably considered out of date by many. $\endgroup$ – Alexandre Eremenko May 25 at 4:15
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The Complex Analysis Project is a rich-in-content textbook on this topic which realises modern approach with Maple. Also see William T. Shaw, Complex Analysis with MATHEMATICA®

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