A Brief on Complex Analysis

I first got to learn about Complex Analysis in my third year (2008) in college when I was studying for a double major in computer engineering and mathematics. Compared to Real Analysis which is the study of (the sequences, series, continuity, differentiation, and integration) of real-valued functions, Complex Analysis is the study of complex-valued functions. Cauchy, Riemann, Mittag-Leffler, and Weierstrass are among the founders of the field that came to existence in the 1800s. Complex Analysis is applied in many branches of mathematics, engineering and physics. To name a few, algebraic geometry, analytical number theory, applied mathematics, electrical engineering, hydrodynamics, thermodynamics, quantum mechanics, and quantum gravity theories such as twistor and string theory.

Complex Analysis has remarkable results that do not have any analogies in Real Analysis. To start with, one such result is that if a complex-valued function is differentiable, i.e., analytic (equivalently holomorphic), then all its higher order derivatives exist. That is, such a function has to be infinitely differentiable! This is certainly not true in Real Analysis: Simple counterexamples are the real-valued functions x|x| and x^(5/3). An intricate counterexample is the integral of the Weierstrass function. All these three functions are differentiable only once.

Another stunning result is that given a function that is analytic in a region including its closed boundary, one can compute the values and the derivatives (the first and all higher orders) of the function for all points inside the region by just using the boundary values - thanks to Cauchy’s Integral Formulae. This is where Holomorphy comes into the picture and there is no analogy to this in Real analysis.

Yet another mind-blowing result is Picard’s theorem stating that an analytic function attains all possible complex values around a non-removable (essential) singularity infinitely often with at most a single exception. An example is the complex function exp(1/z) for which z = 0 is an essential singularity. I included a plot of exp(1/z) to illustrate the resulting behavior in the first figure below. In the figure, the hue represents the complex argument and the luminance represents the absolute value. It shows that approaching the essential singularity from different directions yields completely different patterns as the function takes on all complex values. For a comparison, I also plotted the complex function 1/z for which z = 0 is not an essential singularity in the second figure below (In more strict terms, z = 0 is called a pole of the meromorphic function 1/z.). Its plot reveals a rather uniform absolute value (white) pattern around the singularity at zero.

Moreover, Complex Analysis offers mathematical tools without which we either could not solve or we would need to do much more work to solve certain problems. For example, Liouville's theorem states that every bounded function that is analytic everywhere must be constant. We apply it to do a succinct proof of the fundamental theorem of algebra. Another example is that the Residue theorem enables us to compute certain types of definite integrals of real-valued functions, which are very difficult to compute otherwise.

As a separate note, Liouville's theorem might make you think of the Sine (Cosine) function and ask yourself: “Sine is analytic everywhere, bounded and definitely not constant. But shouldn’t this contradict Liouville's theorem?”  Well, you are right that it is contradictory to the theorem. But not due to the incorrectness of the theorem or any lacking constraints but that the complex-valued Sine (Cosine) is unbounded as opposed to the real-valued Sine (Cosine)!

Stay tuned for the upcoming article on conformal mappings and analytic continuation, and Bernhard Riemann!