Complex Analysis

I’ve grown tired of using the built-in WordPress editor, mainly because it requires tweaking the colors of each TeXed formula/expression to make it look nice with the themes I have been using. I’m trying out a program called latex2wp, and I’m really liking it. It allows me to write LaTeX with my editor of choice, and then just copy/paste everything into a WordPress post when I’m done. Pretty awesome.

Okay, so we’re going to start complex analysis. Be warned, I am nothing of an analyst. I am quite certain there will be errors, and ineloquent explanations. I’m doing this for my own benefit (I really don’t remember any of this stuff), and I figured if I’m learning it, I might as well share.

Now, even as someone who does not like analysis, I really do feel like complex analysis is very beautiful. That being said, there is a hump to get over. The first parts are going to be fairly technical, and there’s really no avoiding this. I’ll do my best to write descriptively and explain the estimates and such, but it will still be technical. Once we are over the hump, we’ll have easy pickings of some beautiful theorems. Trust me on this: It’s worth it.

I should mention that in this series of posts, I’m going to mostly be following Lars Alfors’ Complex Analysis, and these notes by Mario Bonk.

The last thing before we get started is to lay down some notation. We’ll use {i} to denote a square root of {-1}. Mostly variables like {z} and {w} will denote complex numbers, and {x} and {y} will denote real numbers. I’m not willing to commit fully to this, but I’ll be sure to make it clear which variables I’m using for what. Here’s a list of some notation I’ll be using. If {z\in{\mathbb C}}, and for {x,y\in{\mathbb R}}, {z=x+iy}:

  • {{\mbox{Re}}(z)=x}, the real part of {z}.
  • {{\mbox{Im}}(z)=y}, the imaginary part of {z}.
  • {|z|=\sqrt{x^2+y^2}} is the modulus of {z}.
  • {\bar z=x-iy} is the complex conjugate of {z}.

Lastly {{\mbox{arg}} z} is the “argument,” or angle measured from the positive real axis. If, in polar form, {z=r\cdot e^{i\theta}}, then {{\mbox{arg}} z=\theta}. Of course, this is only well-defined up to adding multiples of {2\pi}. From the context it will be clear exactly what is meant. If this is worrisome, pretend I never defined it, and we’ll deal with it in due course.

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