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I am currently learning complex analysis in my undergraduate studies and it's quite hard to understand for me. During my mathematical study, I learned that nothing is too hard to understand, it's just necessary to find the right source for me. When I read lecture notes or books, they feel mostly too "dry" for me and it's not so enjoyable for me to read. In the past I learned first with books like e.g. analysis I or II for dummies and afterwords I learned with proper mathematical books, which I then could enjoy and understand fully, and use as my main source.

But now as I progress to more advanced subjects, I find it hard to find books that help me. I learn best with a visual approach: the more graphics the better. And it's always good when the author explains a lot and is not assuming too many things as trivia.

I hope I made myself clear what kind of book I need and maybe someone can help me.

PS: I'm not restricted to books; if you know some good lecture notes, you are more than welcome. My native language is German, so I'm fine with sources in English and German.

J W
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Mike
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    Dr. James Cook has a series of lectures on youtube that I find particularly...nice and helpful. Wonderful professor. The link is here:https://www.youtube.com/watch?v=BzzkGZlfVmg&list=PLBY4G2o7DhF0TSossUvJ-CTKSLfOhQgb6 – Prime Jun 13 '18 at 08:53
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    I want to say thanks to all answers, the two books and the lecture on youtube are looking very promissing :-) It will probably take a long time till I can decide which answer helped me most. – Mike Jun 13 '18 at 09:25
  • If you want to read a german book, then I'd recommend 'Funktionentheorie' by Klaus Jänich. It is a short nice book, but really explained well as one is used from Jänich. – Cahn Jun 13 '18 at 09:56
  • @Marvin: I just borrowed this book this morning :-) The best german book I found so far for me, maybe short of details. – Mike Jun 13 '18 at 10:16
  • Ahlfors, Complex Analysis. Not the most visualization based ca book, but definitly the best ^^ – tired Jun 13 '18 at 18:55
  • Of all the complex analysis books I've read or taught from, the one I've felt has the best mix of rigour and visualization/clarity remains Complex Analysis with Applications to Engineering and Science by Saff and Snider. I still pull out my copy to this day when I need an elementary result. – Brevan Ellefsen Jun 13 '18 at 19:12
  • Related: https://math.stackexchange.com/questions/30749/what-is-a-good-complex-analysis-textbook-barring-ahlforss, https://math.stackexchange.com/questions/160132/complex-analysis-book – Hans Lundmark Nov 21 '21 at 19:53

8 Answers8

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If you learn best visually, I would highly recommend 'Visual Complex Analysis' by Tristan Needham, which has lots of good exercises and gives visual explanations for most of the core topics and ideas of complex analysis, plus some beautiful additional chapters on vector fields and other wonderful applications of the subject.

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    Needham would be my recommendation too. – gandalf61 Jun 13 '18 at 10:55
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    There were a lot of good recommendations which will probably all help me, but when I started to read 'Visual Complex Analysis' I knew this is exactly what I'm looking for ! – Mike Jun 13 '18 at 12:19
  • In my experience it can be a tad too advanced for undergrad, and was light on some topics that were covered during my undergrad complex analysis course. – Lonidard Jun 13 '18 at 20:51
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I suggest David C. Ullrich's Complex made simple.

José Carlos Santos
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I like Stein and Shakarchi's book on complex analysis. I used Terence Tao's blog posts on complex analysis alongside as more formal exposition. Also, I recommend Steven Miller's video lectures on complex analysis. They are very good and have the advantage of closely following the book by Stein and Shakarchi.

Manan
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2

If you liked Needham's 'Visual Complex Analysis' then check this out:

Visual Complex Functions: An Introduction with Phase Portraits by Elias Wegert. The two make a great pairing. As the title says, you get lots of really beautiful colour phase-portraits which really help visualise what's going on, with the nice side effect of blowing your mind as well.

Daniel Buck
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But now as I progress to more advanced subjects, I find it hard to find books that help me.

I find that mathematical literature stimulates my mind and the exposure to the material is good. If books aren't helping that much, I suggest looking into Dr. James Cook's Complex Analysis lectures on YouTube, or something similar. The playlist for those videos can be found here.

Prime
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I enjoyed Lang's Complex Analysis as an undergraduate. It is in the Graduate Texts in Mathematics series.

It is a beautiful exposition of a beautiful subject, and it is still accessible to a beginner. Besides which, you may as well begin to become anointed with the higher workings of the subject...

1

My favorite introduction to Complex Analysis is Folkmar Bornemann's Funktionentheorie. It is very well organized and allows you to learn basic and advanced topics very efficiently.

Elias Wegert
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0

I used Theodore Gamelin's textbook and liked it quite a bit. It didn't get too bogged down in technicalities.

tparker
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