Prerequisites to learn Complex Analysis

Question

I really wanna learn Complex Analysis but I don't know where to start.

Basically I can do olympiad problems, but I don't know calculus that well, so I would appreciate it if someone can post a list of topics (preferably in chronological order, if that makes sense) for me to learn before I am absolutely 100% ready to start learning the complex analysis, which I've read about and think is A-M-A-Z-I-N-G.

Thank you for your time everyone! :D

Answer 1

Sequences and series of numbers and of vectors

Derivative in one variable

Integration in one variable

Integration with parameters (i.e. of the form $f(x) = \int_D \phi(x,t)\, dt$)

Sequences and series of functions

Uniform vs. pointwise convergence

Derivative in several variables

Line integrals

Topology of metric spaces

I'm told that the Stewart text is good for learning calculus for the first time.

If you're looking for a challenge, Buck's Advanced Calculus is rigorous. He even introduces complex variables at the end.

If you want a fun introduction without too much rigor, there are some popular books on complex variables, such as An Imaginary Tale: The Story of $\sqrt{-1}$.

Answer 2

This goes along with the excellent advice you have gotten so far. Learning calculus is, of course, an important step. Then you might consider learning real analysis. It is usually the transition to rigorous math, which is, in it's entirety, as you say amazing.

This set of lecture notes by Fields Medal winner Vaughan Jones (like the Nobel Prize in math) are fabulous. A master, giving great insight as to how to think about things, starting from a position of no prior experience. So it's great for self-learning. They are free for downloading. The last part is a rigorous presentation of a good deal of calculus - which will be invaluable in CA studies.

https://sites.google.com/site/math104sp2011/lecture-notes

I am sure once you actually start complex analysis you can get excellent recommendation for study materials here. But I might suggest Flanigan.

http://www.amazon.com/Complex-Variables-Dover-Books-Mathematics/dp/0486613887

It rigorous, but makes things very clear with the added feature of many examples and pictures, not to be underestimated as an aid to visualization (redundancy intended).

Good luck.

Answer 3

The most useful topics to cover, which relate directly to complex analysis are:

• integration (1 variable)
• differentiation (1 variable)
• and basic topology

Yaseen