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As an applied science student, I've been taught math as a tool. And although I've been studying a lot throughout the years, I always felt like I am missing depth. Then I read geodude's answer on this post, that cited these beautiful quotes:

You might want to do calculus in $\mathbb{R}$, but the functions themselves naturally live in $\mathbb{C}$

Even in $\mathbb{R}$, and in the most practical and applied problems, you can hear distant echos of the complex behavior of the functions. It's their nature, you can't change it.

And although pieces of complex analysis are well known even to the most applied scientist (e.g Euler's identity), these quotes really helped me understand why my math knowledge is so shallow. It seems I share the same worries with other engineers: (What's the best way for an engineer to learn "real" math?) and I've found many beautiful and informative answers about diving deeper into mathematics, but none of them (as far as I could spot) addressed complex analysis. And as I think I am lost in the labyrinth of math knowledge, I ask this question:

How can one that has an basic knowledge of real analysis approach complex analysis? What do I start? Are there any books you would recommend?

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    $\begingroup$ For self-study, I advise you the very didactic book "Visual Complex Analysis" of T. Needham $\endgroup$ – Jean Marie Sep 4 '20 at 15:14
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    $\begingroup$ Complex Variables and Applications by Brown and Churchill might be a good place to start if you're coming from a decent knowledge of real analysis but feel your general mathematical understanding is low. Although I'll confess complex analysis is not really my area, as it were, so there may be better entry level texts. $\endgroup$ – SeraPhim Sep 4 '20 at 15:14
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    $\begingroup$ Complex Variables by Athanassios S. Fokas, Mark J. Ablowitz. $\endgroup$ – Pixel Sep 9 '20 at 8:34
  • $\begingroup$ Thank you all so much. $\endgroup$ – Dimitris Sep 9 '20 at 8:48
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I found the book by Gamelin "Complex Analysis" together with "Visual Complex Analysis" by T. Needham to be a good combination.

Also if you read German, then I recommend chapters 1-4 from "Funktionentheorie" Busam/Freitag (not sure if it exists in English).

There is also "Complex Analysis" by Stein which people recommend, but I have not read.

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Complex Analysis by Ahlfors is a masterpiece! But whatever book you read, you must read with not only your mind but also with your heart and soul!! You must feel the subject and only then Complex Analysis will stick.

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Assuming you are interested in applications (given your background), my favorite book for applications of complex analysis is Fundamentals of Complex Analysis: with Applications to Engineering and Science by E.B. Saff and A.D. Snider. Their coverage of residue theory, in particular, is more extensive than in most of the other texts I have seen, with many examples and exercises.

Regardless of the text you choose (there are many excellent books), I hope you will carry out your plan to study complex analysis. I think it is one of the most beautiful areas of mathematics.

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