This is part one of a small series of articles meant to describe the most interesting results of complex analysis in an understandable way. Even if you have not had any course in complex function theory, you should be able to get some valuable understanding of this fantastic field. If you are studying mathematics or freshening up your skills, I believe this is also a good place to be, since I will introduce many great ways of looking at this theory.

Before exploring this fruitful and beautiful topic, let’s begin with some fundamentals.

Introduction

In the study of real-valued functions of one real variable, continuity and differentiability rely on some conditions using limits from only two directions. A direction from left to right and another from right to left. Of course, there are no other directions on the real line, since it is one-dimensional.

For instance, to check for continuity, one way is to check that the limit of the function value as the argument approaches some number, say a, from the left, is equal to the same value as the argument approaches a from the right i.e.

In the same sense, differentiability also has a two-way condition but this is a stronger condition than continuity.

That the limit can only be taken from two directions makes these properties rather weak in the sense that in real analysis we can have many malicious functions satisfying the differentiability criterion at some point on the real line but if a function satisfies this then its derivative is not necessarily differentiable in that point. In fact, it’s not even guaranteed that the limit towards that point exists!

Moreover, the weak conditions actually make our lives rather difficult in the sense that we have a limited understanding of these functions and therefore the number of powerful tools towards these is not very impressive.

However, In the world of complex numbers and the functions of them, it is a very different story as you will see in a moment. It turns out that the conditions are a lot stronger, and therefore we have a wealth of useful tools and a lot of beautiful theorems at our disposal.

Actually, there are a lot of problems in real analysis that we can only solve (as of the moment of writing) if we use techniques from complex analysis.

Before we take a dive into some interesting facts, let’s agree on a few things. In real analysis, we typically map a possibly infinite interval (or a union of these) of the real line to itself. In particular, the domains of these functions are one-dimensional.

Before turning the focus to complex functions, I will very briefly remind you about some properties of complex numbers and their geometry and topology.

The starting point is the imaginary unit. That is the number i which satisfies that i² = -1.

No real number has this property!

The complex numbers can be understood as a two-dimensional real vector spacespanned by the basis 1 and i. In other words. A complex number has the form a+bi and satisfies the usual laws that go with vector spaces e.g. the distributive law, commutative law, etc.

Therefore, there is a geometry associated with the complex numbers.

They have a magnitude and a direction (because they are vectors) and they should be visualized as lying in a plane (called the complex plane) isomorphic (meaning same algebraic structure) to ℝ².

So for instance the complex number 3+4i is the point (or vector) (3, 4) in this plane. More generally, a complex number a+bi is the point (a, b) in the complex plane.

It is nice that we have this vector view of complex numbers that is much like ℝ², but there is actually more structure than this because we can of course also multiply complex numbers together. That is, it is what mathematicians call a ringbecause vector spaces are abelian groups with respect to their addition operation.

It turns out that it is even more than a ring. Since every non-zero element (complex number) has an inverse i.e. you can divide complex numbers as well, it is a special kind of ring called a field.

When you multiply two complex numbers you use the distributive law and remember that i² = -1.

Great. Now we have an understanding of the complex plane.

In complex analysis, we typically study functions with subsets of the complex plane as domains. In your mind, you can imagine a possibly deformed disk, with zero or more holes, which is sometimes the whole complex plane.

In particular, the function’s domains are two-dimensional and that turns out to have some interesting consequences.

Holomorphic Functions

The starting point of complex analysis is the concept of complex differentiable functions also called holomorphic functions.

As mentioned in the introduction, differentiability means that a certain limit exists but when the function is defined on a two-dimensional domain there are infinitely many directions one can approach a number from. This poses a great restriction on the set of holomorphic functions and it gives these functions some great properties.

Before stating them, let’s explore this omnidirectional limit a little.

Since a complex function of one complex variable maps a complex number to a complex number we might write the input and the output as follows

To give a simple example of this, consider

That the direction of the limit doesn’t matter implies that, in particular, if we take the limit along the real axis, we will get the same result as if we take the limit along the imaginary axis.

Let’s see what this means.

where δ and η run along the real axis. These two limits need to be the same, yielding

When we expand f in the real functions u and v as above so that f = u+iv, and plug that into the differential equation we just found, we get a system of partial differential equations known as the Cauchy-Riemann equations.

Our derivation above shows that every holomorphic function must satisfy these equations, putting a strict requirement on them. The converse is almost true. But we need some arguments involving continuity for that to work.

It also turns out that in a point where a given holomorphic function’s derivative is not zero, the function is angle preserving. This is called a conformal map and is an important and nice trade of these functions.

Properties of Holomorphic Functions

The first sign that these functions have some interesting properties is the following. It turns out that if a function is holomorphic i.e. complex differentiable at some domain containing some point, say p, then it is complex differentiable infinitely many times at p.

This can be stated a little more compactly, namely that a complex differentiable function’s derivative is complex differentiable.

This is certainly not true for real functions in general.

Analytic Functions

This also means that holomorphic functions are analytic, that is, they have a converging power series around any point in their domain with some radius of convergence. In fact, the converse is also true. Any analytic function is holomorphic. Therefore, we sometimes call them analytic functions instead of holomorphic functions. The two definitions are mathematically equivalent even though we mean something different by the two expressions. We keep the two definitions since there is also a concept of real analytic functions which, as a definition, is not equivalent to real differentiability.

As mentioned above, a holomorphic function has a power series expansion. Power series have what is known as a radius of convergence. You should try to imagine a disk in the complex plane as the domain of such a series and that disk's radius is its radius of convergence. This means that there is at least one point on the boundary of the disk that makes the series diverge.

As you will see in the next part, there are ways around those annoying points, again because of the two-dimensionality of the complex plane!

Invisible Symmetries

Before plunging ourselves into the implications of holomorphy, I want to share a particularly nice treat with you, namely the geometry of taking roots. And I am not talking about roots of polynomials but rather square roots, cube roots, etc.

From school, we all know about the square root. That the square root of say 4 is 2. But what is this operation really? If an equivalent question to “what is the square root of 4?” is “what number squared is 4?” then certainly the question is ill-posed, because there are of course two numbers satisfying this.

By convention, we mean the positive square root, but there are two square roots of 4, namely 2 and -2.

How about the cube root? Well, we all learned at some point that there is exactly one cube root of a real number. But in fact, this is a truth with modifications. It is true that there is only one real cube root of a real number, but actually, there are two other roots hiding in another dimension! There are three distinct cube roots of any number except 0.

In fact, there are n distinct n’th roots of any complex (including real of course) number!

As always, the additive identity 0 is a special number in the multiplicative world. If we count what is known as multiplicity, then the above also holds for 0, otherwise, we have to exclude 0 from the statement.

But what’s even more amazing, these n roots of a complex number z lie symmetrically on the circumference of a circle with radius equal to the positive n’th root of the length of z and center at 0.

This explains why there is only one real cube root and two real square roots of a real number, because if they have to be placed symmetrically around a circle, then one root determines the positions of the rest. For instance, one of the square roots of 1 is 1, but then since there are exactly two square roots and the other root has to be placed symmetrically around the unit circle, it has to be -1.

In the same fashion, one cube root of 1 is 1, but then the other two roots lie at the vertices of the equilateral triangle inscribed in the unit circle with one vertex at 1.

In general, the n roots of 1 form the vertices of a regular n-gon (a polygon with n sides), and they all lie on the unit circle. These are called roots of unity.

The n roots of unity actually form an abelian group (with respect to multiplication) and have in general many interesting properties but that’s for another article.

This already shows that by extending the real numbers by another dimension, a lot of mysteries that were hidden in the dark now become clear and this is just the beginning of our enlightenment.

It almost seems (excuse me for my philosophical opinion that I inflict upon you here) that the imaginary dimension is necessary in order to really understand many of the real concepts like taking roots or multiplying by -1 (you can take a look at this story for a discussion on that).

It’s like it was there all the time and we struggled to see in the dark, in a lower dimension — but we only saw the shadows of the real things making it impossible to understand them properly. However, the extra dimension gave us the light that we so desperately needed.

Contour Integration

This is where things get really interesting.

When we integrate real functions then we do so over an interval (possibly infinite interval) of the real line and here again because of the 1-dimensionality when given the endpoints we don’t have much choice about the interval other than the direction.

This is not the case for complex integrals. When we have two endpoints in the complex plane (which might be the same point) then we have infinitely many curves (or union of certain connected curves called contours) we could integrate over since the domain is 2-dimensional.

This might seem daunting (and hard to visualize) but it is not as bad as it sounds. In fact, for holomorphic functions the following amazing thing is true.

Cauchy’s Integral Theorem

When two contours (you can think of contours as curves in the complex plane) are chosen between the same two endpoints, then if we can continuously deform the one contour into the other (i.e. if they are homotopic — see below gif), then the integrals along them are equal.

This intuitively means that if there are no holes in the domain between the contours, then it doesn’t matter which contour we choose.

This has some interesting consequences by pure topological reasons.

It means for example that if the two endpoints of a contour coincide (so that we have a closed contour) then if the space that the contour inscribes is contractable i.e. homotopic to a point (constant curve), then the integral over that contour is zero. This is known as Cauchy’s Theorem after Augustin-Louis Cauchy.

So if the function is holomorphic on that domain, then the integral is zero.

for f a holomorphic function on a domain D containing γ.

Meromorphic Functions and the Argument Principle

Recall that holomorphic functions have no poles on their domain i.e. they don’t blow up and become infinite at any points. They are very nice to work with but sometimes you want to work with functions that are holomorphic in all of their domain except for a countable amount of points.

These functions are called meromorphic. You can think of them as a generalization of fractions of polynomials. The great thing about them is that they have holomorphic properties but are allowed to have poles or singularities as well. Some of the most important functions in mathematics are meromorphic functions including the Riemann Zeta Function, The Gamma Function, 1/z, etc.

Some of the interesting questions regarding these functions are of course related to the poles and zeroes of these and it turns out that the contour integralcombined with the logarithmic derivative gives us a tool to calculate just that.

If f is a meromorphic function inside and on a closed contour C, and f has no zeros or poles on C, then

Where Z is the number of zeros and P is the number of poles in C.

It is very interesting to see that the integral only cares about topology and not geometry! The area and position of the curve don’t matter, but that it is closed and that the function has poles and zeros in the subset inscribed by the contour of the complex plane — that matters!

This result is related to a theory called the calculus of residues, which is closely related to an extension of Taylor series called Laurent series but that is for another article.

You can read more about Taylor series here and explore some nice properties of the logarithmic derivative together with me here.

Before ending this part, let’s recap what we have learned.

Because of the difference between geometry and topology in 1 dimension and in 2 dimensions, there are tighter restrictions on complex functions than on real functions if they need to satisfy some conditions involving limits.

That in turn leads to strong properties of the functions satisfying these conditions, making holomorphic functions extremely useful in solving problems in both complex and real analysis.

Moreover, it turns out that the field of topology becomes useful because complex integrals known as contour integrals depend only on the contour's homotopy equivalence classes and not the paths themselves.

This led to beautiful discoveries by Augustin-Louis Cauchy and can be used to calculate zeros and poles of meromorphic functions which we shall investigate soon.

In the next part we will continue to study the beauty of holomorphic and meromorphic functions because as you will see, they have some properties that have applications to quantum physics not quite understood yet. It has to do with giving values to divergent series and to extract deep information out of functions even outside their domain of definition.

Many who have had an opportunity of knowing any more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination.
~ Sofia Kovalevskaya