Complex analysis: a review

Recently, I pay two days to review the most basic one variable complex analysis. I have to say that, in fact, I have learned this topic in 2012 (OK, I have learned system in 2011 even earlier but I still have to review them in this year), but in that time, I find that difficult to understand this topic and I got even the worst result during my undergraduate in this course. However, for most people, complex analysis is easier than real analysis. Really?

I think that maybe the difficulty is that, the real analysis is more quantitative but complex analysis has more structure. If we have well understood the differential calculus and some topology, this topic will be obvious. I recommend this Note as a reference.

0. Angle, multiple value function, logarithm
The first difficulty is to define the logarithm. The angle of a complex is clear, but why we need complex logarithm to make it sense by striping one line? In fact, we always hope to make a function continuous, but since the angle is a multiple value function, to achieve we have to pay something. Then the log is well defined and so is the power.

1. Differential, analytic, confromal, holomorphic function
This four words represent in fact the same sense in complex analysis. The derivative can be understood as the form on C and if we know the sense in Banach space, we know that it is just a linear approximation in distance. But in complex situation, it has other senses, like the Cauchy-Riemann equation, the conformal map keeps the angle and the real part and imaginary part are all harmonic, so we have all the property from harmonic function.

Thanks to the Cauchy integral, we pass the differential to analytic. So the regularity is not C^1, C^2 but C^infinity. What an amazing result! So we would like to know how to prove that a function is analytic? A strong weapon is Motel lemma, who says that if the integral along any triangle is 0, this function is analytic. We can develop other methods from this, like passing the limit from a series of functions.

2. Singularities and Laurent series
The opposite side of analytic is singularity. What the other side of analytic? There are three type, removable singularity,  pole and essential singularity. The removable singularity says that we can redefine the value on the singularity. The pole is the case that limit infinity so its inverse is 0. The essential singularity says the value around it is dense in C! Notice that all the case discussed above is the isolated singularity. Well, to define something in any form makes no sense.

Then the Laurent series give a criteria for these singularities. If it has only positive part, it is analytic. If it has some negative part, it is pole. But if it has infinite negative term, it is an essential singularity.

3. More about integral - homotopy and residue theorem
A theorem, which is always correct says that along two homotopic trajectories the integral is the same. This strong property makes the connection between the complex analysis and topology. So we have a stronger version of Cauchy integral and even the residue theorem. That is when we make the integral, we do this only in the part around the singularity.

We apply this technique also in counting the zeros and the argument of angle.

4. Reimann mapping theorem
Last but not the least, a powerful theorem, any simply connected  domain have a conformal mapping to disk.

Why I revise complex analysis? Recently, the probability theorem has a strong connection with the complex analysis. I believe that these basic elements will help me understand those delicate theory.