A book for complex analysis

Question

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.

Answer 1

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.

Answer 2

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

Answer 3

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.

Answer 4

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.

Answer 5

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...

Answer 6

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

Answer 7

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.

Answer 8

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.