Beginners Guide to Complex Analysis

[Remixex]

OK, so i took a course named "Oscillations and vibrations"
We began the course with an "introduction" to complex numbers, basically we raced through them in like 3 classes, we talked about how to get complex roots, adding, multiplying, Cauchy-Riemann Conditions, Cauchy's integration Theorem (quite similar to Stokes and Green theorems, at least in their conditions), and today we saw what certain operations do to functions, for example how w=z^2 just transforms it into a kaleidoscope.
Problem is i didn't quite get it, for example the teacher didn't even mention Euler's identity in the slightest, basically i need a book that teaches me, at least, basic complex analysis like a total beginner, i know about Vector analysis but not complex and i kinda need some guidance, I've been told you use complex numbers for everything in physics, so i'll need that guidance.
P.S. it's amazing how real numbers aren't enough o.0 it kinda blew my mind

 

[micromass]

Cauchy's integration Theorem (quite similar to Stokes and Green theorems, at least in their conditions),

That's because it can be proven from Stokes' Theorem.

Anyway, my favorite book is Freitag & Busam. But it might be too mathematical for your taste. Needham's Visual complex analysis is a great book with great intuitions. It might be what you're looking for, but it isn't exactly meant to be rigorous. Somewhat in between is Boas "Mathematical Methods for the Physical Sciences". Flanagan's "Complex Variables" is like Boas, but more extensive.

I'm sure one of these books will appeal to you.

 

[jasonRF]

Another option (along the same lines as Boas) is the free online math methods book from James Nearing:

http://www.physics.miami.edu/~nearing/mathmethods/

which has two relevant chapters (3 & 14) that may be helpful.

Jason