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I have studied a bit of complex analysis in the past, but I realized that I couldn't really get into it because I didn't really see the motivation; I was spending a lot of time trying to understand proofs, and even though the theorems seemed pretty cool, I never really got to use them. Could you please recommend me some books with applications of Complex Analysis? I don't mean applications to science or engineering, I just mean applications of the theorems in first year Complex Analysis, aka a book with good, interesting problems, not necessarily a book with formal proofs of everything (though that's a bonus).

Thank you very much.

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I would recommend the book by Freitag and Busam (Complex Analysis) as it covers also elliptic functions and basic ANT like Riemann Zeta with lots of exercises most of which have fairly detailed solutions at the end (about 60 pages of solutions). The book is classic textbook in style and sometimes a bit dry but the exercises are excellent.

If one wants to understand complex analysis in maybe a more leisurely and historically motivated way, the two books by Remmert (Theory of Complex Functions and Classical Topics in Complex Function theory) are just incomparable in exposition, motivation, how people got to think of this or that and why.

I would also give a shout to Titchmarsh classic Theory of Functions which is maybe less modern that the Ahlfors, Conway, Rudin and whatever is used today but it explains the bread and butter of the subject considerably better than modern textbook style volumes.

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I certainly enjoyed my $185$ course at Berkeley, with Hung-Hsi Wu (who is excellent). We used Lang's Complex Analysis. Graduate Texts in Mathematics series (it was an upper division undergraduate course). It's been a long time, but it must have had some good exercises and exposition in it, or I wouldn't remember it so fondly.

Another well-known text is the one by Ahlfors. Marsden and I think somebody else, though I cant remember who, have one. I think Churchhill and Brown had a book entitled Functions of a Single Complex Variable or something like that. I think Gamelin and possibly Greene, a couple of UCLA professors, have one. I haven't really looked at it though. But I know they are both good.

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    $\begingroup$ Ahlfors is nice but a bit terse, and he's famous for the expression "the details of the proof are left to the reader" in quite a few places in the book. $\endgroup$
    – kodlu
    Mar 14, 2020 at 2:04
  • $\begingroup$ @kodlu At least he doesn't spoil all the fun... I think he's an ivy league professor; probably Yale. $\endgroup$
    – user403337
    Mar 14, 2020 at 2:08
  • $\begingroup$ It was Harvard, sorry. And he died in '96. $\endgroup$
    – user403337
    Mar 14, 2020 at 2:32
  • $\begingroup$ I agree it's a great book, may not be suitable for all. $\endgroup$
    – kodlu
    Mar 14, 2020 at 4:20
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If you are more into solving problems and applying mathematics you can't go wrong with J. W. Brown and R. V. Churchill, Complex Variables and Applications (currently at its 9th edition, I guess). An alternative would be M. J. Ablowitz and A. S. Fokas, Complex Variables, great if you want to do modern applied mathematics, especially nonlinear ODEs and PDEs. Now, if you are more into pure mathematics I'd recommend R. B. Ash and W. P. Novinger Complex Variables, 2nd ed. (there is a cheap edition by Dover, but you can also legally download it, so it wouldn't hurt to take a glance). An oldie but goldie alternative would be, of course, E. C. Titchmarsh, The Theory of Functions, 2nd ed. -- it's all there, clear cut, masterly exposed. I don't like the textbooks by Ahlfors (dated, not very pedagogic, a little pedantic) or Conway (a mess), but maybe it's just me.

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  • $\begingroup$ I also thought Conway's textbook was a big mess. It was more like a reference for someone who already knowns real analysis really really well. $\endgroup$
    – Ovi
    Mar 27, 2020 at 19:33
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My first introduction was with Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables. The writing was very clear and concise, and I felt it had the right amount of motivation/explanation. Problems were at times difficult, but I enjoyed them. Note however that this book is by no means easy; it assumes an acquaintance with topology (open, closed, compact, connected etc). Also, it introduces complex integration using the language of differential forms.

Another book which I like is Stein and Shakarchi's Complex Analysis. Their writing is very clear, and the book has a collection of good problems.

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  • $\begingroup$ Its exercises (provided by Reiji Takahashi, not by Cartan himself) are really good, but I don't think its main content is suitable for self-learning. I've tried to learn complex analysis by reading the first three chapters several times, but finally gave up. It assumes that the user is very familiar with multivariate calculus (not only differential forms). Its development of Cauchy's theorem is also confusing. To date I still cannot figure out how its version of Cauchy's theorem is related to the usual homotopy or homology versions. $\endgroup$ Mar 14, 2020 at 5:41
  • $\begingroup$ @WilliamMcGonagall really? I found his treatment of Cauchy's theorem, the use of homotopy arguments, the relationship between closed and exactness, Laurent series etc all extremely clear and sufficiently general. And I mainly used this book for self-study. I also had a course which used this book, and I found it delightful (I would dread those parts of the course when we used Ahlfors lol) I guess this is a matter of opinion, because I have tried to read Ahlfors and several other books, but they were too confusing for me. I guess OP will have to decide for him/herself if the book is suitable $\endgroup$
    – peek-a-boo
    Mar 14, 2020 at 5:58
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If you're looking for a book with lots of problems and not necessarily a lot of theory, Schaum's Outline of Theory and Problems of Complex Variables; With an Introduction to Conformal Mapping and Its Applications, by Murray R. Spiegel certainly qualifies.

From the front cover: "640 problems solved step by step. Ideal for independent study!"

And the price is right, about $20 on Amazon.

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