How do you feel when see the term “holomorphic function”? It just feels like we shouldn’t be looking further into it, right? I mean, it looks like an esoteric mathematical concept that should remain in advanced textbooks. Interestingly enough, holomorphic functions are very useful in real life. Holomorphic functions are ubiquitous in the field of complex analysis. Just to clarify, “complex analysis” doesn’t refer to an analysis that’s complex or difficult. Instead, it refers to analysis of functions of complex numbers. Alright, so let’s go ahead and see how something like this can possibly be useful in real life, shall we?
What exactly is a holomorphic function anyway?
If we are being mathematical about it, we can say that a holomorphic function is a complex-valued function of complex variables that is complex differentiable in a neighborhood of every point in its domain. You see how we used the word “complex” so many times in a single sentence? Let’s break it into parts and see what lies underneath. The first part says that a holomorphic function is a complex-valued function of complex variables. It just means that the domain of this function includes complex numbers. Simple enough, right? Now the next part is tricky. We say that it’s complex differentiable. What does that mean? Is it the same as just being differentiable? Well, not exactly! The definition of differentiability is the same at a given point, but there are additional constraints as well.
Let’s take a simple example to understand what “differentiable” means. As we can see here, y = x² is differentiable everywhere because the graph is smooth and we can compute the rate of change at any given point. The graph to the right is not differentiable at x=0 because the graph is not smooth at that point, and we cannot compute the rate of change. Now, for a function to be complex differentiable, it also needs to satisfy Cauchy-Riemann equations. It’s too mathematical to be covered in this blog post, so let’s just think of it as a black box and say that a function needs to satisfy those equations in order to be called complex differentiable.
What do you mean by “neighborhood”?
If you look at the definition, we say that the function needs to be complex differentiable in a “neighborhood” of every point in its domain. In regular differentiation, we say that the function needs to be differentiable at every point. But here, we want the function to be differentiable in the neighborhood of each point. Now why do we have this requirement? If we have a complex derivative in a neighborhood of each point, it implies that holomorphic functions are actually infinitely differentiable, and are in fact equal to their own Taylor series. Wait a minute, what is this “Taylor series”? Taylor series of a given function is a way of expressing the function at every point as an infinite summation, where the n-th term in the summation is dependent on the (n-1)th derivative of the function at that point.
Is it the same as an analytic function?
People tend to use the terms “analytic function” and “holomorphic function” interchangeably, but there is actually a difference between the two. When we say something is “analytic”, we mean to describe any function (real or complex) that is differentiable at each point in its domain. This looks suspiciously similar to what we were discussing in the previous paragraph, right? The class of complex analytic functions is the same as the class of holomorphic functions. A holomorphic function is something that can be written as a convergent power series in a neighborhood of each point in its domain. From the looks of it, one thing implies the other, so you might be tempted to use the terms interchangeably. Holomorphic functions are also called conformal maps. The terminology is overlapping and we need to be careful as to how we use them.
Where is it used?
Let’s see where it comes up in real life. We will take an example of a function that is harmonic over a 2D plane. Now what does “harmonic” mean? Harmonic series is basically a sequence of multiples of a base number. For example, stringed musical instruments like guitar, violin, etc are built upon this principle. Now if we take a harmonic function and transform it via a conformal map to another plane, the transformation is also harmonic. Don’t see why it’s interesting yet? The reason we care about this is because if any function that is defined by a potential can be transformed by a conformal map, and it still remains governed by a potential. For example, electromagnetic field, gravitational field, fluid dynamics, potential flow, zero viscosity, irrotational flow, etc are all governed by a potential.
Conformal mappings are really useful for solving problems in engineering where functions exhibit inconvenient geometries. Okay what does that even mean? Well, not all functions are good looking, and we run into them all the time when we are dealing with stuff in real life. So if we choose an appropriate mapping, we can transform the inconvenient geometry into a much more convenient one. In general relativity, conformal maps are the most common type of causal transformations. Why is that? Well, these mappings are capable of describing different universes in which all the same events and interactions are still possible, but a new additional force is necessary to effect this.