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 to hard to understand, it's just necessary to find the right source for me. When I read skrypts 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 it 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 to much 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 skript you are more than welcome. My native language is german, so I'm fine with sources in english and german.

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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:youtube.com/… – Prime Jun 13 at 8:53
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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. – Mikemathics Jun 13 at 9:25
  • 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. – Marvin Jun 13 at 9:56
  • @Marvin: I just borrowed this book this morning :-) The best german book I found so far for me, maybe short of details. – Mikemathics Jun 13 at 10:16
  • Ahlfors, Complex Analysis. Not the most visualization based ca book, but definitly the best ^^ – tired Jun 13 at 18:55
up vote 15 down vote accepted

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 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 ! – Mikemathics Jun 13 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 at 20:51

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.

  • +1 for Stein and Shakarchi – mathphys Jun 13 at 10:36

I suggest David C. Ullrich's Complex made simple.

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.

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...

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

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.

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.

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