Which are the branches does Complex Analysis link?

Question

Complex Analysis is a vast topic and there are holomorphic functions/Analytic functions. I wanted to know as to why there is necessity of studying about holomorphic functions which has Cauchy's theorems, Goursat theorem, Morera's theorem and so on. Also which are the branches where Complex Analysis serves as a major prerequisite. And what are the applications of Complex Analysis in Mathematics and Physics ?

Answer 1

First, let me tell you that there are probably no limits to the number of reasons one could want to study complex analysis. For example, one can prove the fundamental theorem of algebra using complex analysis. In fact, passing to complex analysis can make your life easier. For example, if a complex function $f$ is complex differentiable, then $f'$ is also complex differentiable. This is not true for real functions!

But to give some examples of why someone would like to study holomorphic/meromorphic functions, here are a few fields who uses complex analysis regularly:

The study of complex dynamics is currently quite popular. In particular, people are interested in understanding what happens when you iterated a polynomial of the form $f(z) = z^2+c.$ This family of polynomials has many secrets and is the focus of many mathematical researchers. There are many reasons for that, it touches different parts of mathematics, such as topology, algebra and geometry. Moreover, even if the family seems to be simple enough, there are many things that are not well understood.

An other example would be low dimensional geometry. Indeed, 2-dimensional hyperbolic geometry is intrinsically related to the theory of Riemann surfaces ($1$-dimensional complex manifolds). This follows from a big theorem called Uniformization theorem. Moreover, every surface of genus at least $2$ admits infinitely many hyperbolic structures. Studying those have led to many great mathematical discoveries and a better understanding of $3$-dimensional manifolds as well. In fact, these two examples are linked and contains similar ideas that are the object of what is called Sullivan's dictionary.

The list of mathematical applications could go on and on, but let me try to also justify why people from other fields would care about holomorphic functions as well. In fact, if you go in physics, a lot of the applications would consider functions that are meromorphic (holomorphic almost everywhere, but there are a few points where your functions admits singularities). There is a wonderful theorem in complex analysis called the Residue Theorem which allow you to understand and investigate line integrals over closed curves. This have many applications in physics, in particular in Quantum field theory where integrating along paths of particles is essential.