An Introduction to Complex Analysis and Applications

How imaginary numbers have a real impact.

Figure 1: A representation of a Complex Function. These can be created once you understand what a complex function is. The Color represents the argument of the complex function, while the brightness/hue is represented by the magnitude. Image Citation: Creative Commons.

Introduction:

Let us start easy. What is the square root of 100? In other words, what number times itself is equal to 100? Easy, the answer is 10.

What is the square root of -1?

Well that isn’t so obvious. Since a negative number times a negative number is positive, how is it possible that we can solve for the square root of -1?

The answer is; we define it. And this isn’t just a trivial definition. Assigning this answer, i, the ‘imaginary unit’ is the beginning step of a beautiful and deep field, known as complex analysis.

So, why should you care about complex analysis? It turns out, that despite the name being ‘imaginary’, the impact of the field is most certainly real. From engineering, to applied and pure mathematics, physics and more, complex analysis continuous to show up. Moreover, there are several undeniable examples we will cover, that demonstrate that complex analysis is indeed a useful and important field.

This article doesn’t even scratch the surface of the field of complex analysis, nor does it provide a sufficient introduction to really dive into the topic. However, I hope to provide some simple examples of the possible applications and hopefully give some context.

Note: Some of these notes are based off a tutorial I ran at McGill University for a course on Complex Variables.

Are you still looking for a reason to understand complex analysis? Maybe this next examples will inspire you!

History:

Firstly, I will provide a very brief and broad overview of the history of complex analysis. I will also highlight some of the names of those who had a major impact in the development of the field. Note that this is not a comprehensive history, and slight references or possible indications of complex numbers go back as far back as the 1st Century in Ancient Greece.

Johann Bernoulli, 1702: The first reference of solving a polynomial equation using an imaginary unit.

Abraham de Moivre, 1730: Developed an equation that utilized complex numbers to solve trigonometric equations, and the equation is still used today, the De Moivre Equation.

Leonhard Euler, 1748: A True Mathematical Genius. The Euler Identity was introduced.

Augustin Louis Cauchy 1812: Introduced the actual field of complex analysis and its serious mathematical implications with his ‘memoir on definite integrals’.

Bernhard Riemann 1856: Wrote his thesis on complex analysis, solidifying the field as a subject of worthy study. Also introduced the Riemann Surface and the Laurent Series.

As for more modern work, the field has been greatly developed by Henri Poincare, Richard Dedekind and Felix Klein.

Basics:

I will first introduce a few of the key concepts that you need to understand this article.

We defined the imaginary unit i above. We also define ℂ, the complex plane. A Complex number, z, has a real part, and an imaginary part. That is, a complex number can be written as z=a+bi, where a is the real portion , and b is the imaginary portion (a and b are both real numbers). You may notice that any real number could be contained in the set of complex numbers, simply by setting b=0.

Figure 2: An image demonstrating the complex plane. Image Citation: Creative Commons.

The complex plane, ℂ, is the set of all pairs of real numbers, (a,b), where we define addition of two complex numbers as (a,b)+(c,d)=(a+c,b+d) and multiplication as (a,b) x (c,d)=(ac-bd,ad+bc). Hence, (0,1) is the imaginary unit, i and (1,0) is the usual real number, 1. Writing (a,b) in this fashion is equivalent to writing a+bi, and once we have defined addition and multiplication according to the above, we have that ℂ is a field.

Further Properties:

If z=(a,b) is a complex number, than we say that the Re(z)=a and Im(z)=b. This in words says that the real portion of z is a, and the imaginary portion of z is b.

We also define the magnitude of z, denoted as |z| which allows us to get a sense of how ‘large’ a complex number is;

Figure 3: The magnitude of a complex number , z.

If z1=(a1,b1) and z2=(a2,b2), then the distance between the two complex numers is also defined as;

Figure 4: The distance between two complex numbers.

And just like in ℝ, the triangle inequality also holds in ℂ.

We also define the complex conjugate of z, denoted as z*;

Figure 5: The Complex Conjugate.

The complex conjugate comes in handy. Notice that Re(z)=Re(z*) and Im(z)=-Im(z*).

A complex function can be defined in a similar way as a complex number, with u(x,y) and v(x,y) being two real valued functions.

Figure 6: An example of how we write an arbitrary complex function. Image Source: Author.

Just like real functions, complex functions can have a derivative. The Cauchy Riemann equations give us a condition for a complex function to be differentiable. Mainly, for a complex function f decomposed with u and v as above, if u and and v are real functions that have real derivatives, the Cauchy Riemann equations are a required condition;

Figure 7: Cauchy Riemann Equations. Image Source: Author.

A function that satisfies these equations at all points in its domain is said to be Holomorphic.

For example, you can easily verify the following is a holomorphic function on the complex plane ℂ, as it satisfies the CR equations at all points.

Figure 8: A holomorphic function on C. Image Source: Author.

Holomorphic functions appear very often in complex analysis and have many amazing properties. They also have a physical interpretation, mainly they can be viewed as being invariant to certain transformations.

Figure 6: A graphical interpretation of the Cauchy Riemann equations. Provided a function satisfies the CR, a vector X in the domain multiplied by a complex number z and being mapped by f, will be the same as mapping the multiplication of the vector X times z. Image Citation: Creative commons.

Euler’s Formula:

If you follow Math memes, you probably have seen the famous simplification;

Figure 9: The Famous Euler Equation. Image Source: Author.

This is derived from the Euler Formula, which we will prove in just a few steps.

Firstly, recall the simple Taylor series expansions for cos(z), sin(z) and exp(z). These are formulas you learn in early calculus; Mainly,

Figure 10: The Taylor Expansions for Sin, Cos, and e. Image Source: Author.

Hence, using the expansion for the exponential with ix we obtain;

Figure 11: Plugging in ix to the exponential expansion. Image Source: Author.

Which we can simplify and rearrange to the following

Figure 12: Rearranging and simplifying the expansion. Image Source: Author.

And that is it! That above is the Euler formula, and plugging in for x=pi gives the famous version.

Figure 13: The final result after simplification and plugging in x=pi. Image Source: Author.

Residues:

Residues are a bit more difficult to understand without prerequisites, but essentially, for a holomorphic function f, the residue of f at a point c is the coefficient of 1/(z-c) in the Laurent Expansion (the complex analogue of a Taylor series ) of f around c. These end up being extremely important in complex analysis. Several types of residues exist, these includes poles and singularities.

It turns out residues can be greatly simplified, and it can be shown that the following holds true:

Figure 14: Residue Formula. Image Source: Author.

Example:

Suppose we wanted to find the residues of f(z) about a point a=1, we would solve for the Laurent expansion and check the coefficients:

Figure 15: A function for which we want to find the residues. Image Source: Author.

Figure 16: The Laurent Expansion about a=1 of f(z). Image Source: Author.

Therefor the residue about the point a is sin1 as it is the coefficient of 1/(z-1) in the Laurent Expansion.

Contour Integration:

The above example is interesting, but its immediate uses are not obvious. Well, solving complicated integrals is a real problem, and it appears often in the real world. Fortunately, due to Cauchy, we know the residuals theory and hence can solve even real integrals using complex analysis. I won’t include all the gritty details and proofs, as I am to provide a broad overview, but full proofs do exist for all the theorems.

Figure 17: Image demonstrating a broad overview of the process of complex integration. ak are points contained within a closed path gamma, and all lie within a open section of the complex plane, U. Image Citation: Creative Commons:

Cauchy’s Integral Formula:

For a holomorphic function f, and a closed curve gamma within the complex plane, ℂ, Cauchy’s integral formula states that;

Figure 18: A result of Cauchy Integral formula around a closed path. Image Source: Author.

That is , the integral vanishes for any closed path contained within the domain.

Cauchy’s Residue Theorem states that every function that is holomorphic inside a disk is completely determined by values that appear on the boundary of the disk.

Figure 19: Cauchy’s Residue Theorem. Image Source: Author.

Figure 20: Cauchy Integral Formula. I is a sum of the winding numbers, which we won’t worry about in this article. The integral around a closed path is a sum of the residues about singularities contained in the path (multiplied by 2pi and a winding number). Image Source: Author.

Example:

Suppose we wanted to solve the following line integral;

Figure 21: An integral we wish to solve over some closed disk on the complex plane. Image Source: Author.

Figure 22: The formula for solving this integral involves finding its residues on C. Image Source: Author.

Since it can be easily shown that f(z) has a single residue, mainly at the point z=0 it is a ‘pole’, we can evaluate to find this residue is equal to 1/2. Thus, the above integral is simply pi times i.

Solving Real Integrals:

Suppose you were asked to solve the following integral;

Figure 23: A hard real valued integral.

Using only regular methods, you probably wouldn’t have much luck. It turns out, by using complex analysis, we can actually solve this integral quite easily. If you want, check out the details in this excellent video that walks through it. But the long short of it is, we convert f(x) to f(z), and solve for the residues. Then we simply apply the residue theorem, and the answer pops out;

Figure 24: The answer to the above integral.

Proving the Fundamental Theory of Algebra:

Proofs are the bread and butter of higher level mathematics. While it may not always be obvious, they form the underpinning of our knowledge.

The Fundamental Theory of Algebra states that every non-constant single variable polynomial which complex coefficients has atleast one complex root.

Using complex analysis, in particular the maximum modulus principal, the proof can be done in a few short lines.

Engineering:

Indeed complex numbers have applications in the real world, in particular in engineering.

Electrical:

Complex numbers show up in circuits and signal processing in abundance. They are used in the Hilbert Transform, the design of Power systems and more.

Nuclear:

Complex analysis is used in advanced reactor kinetics and control theory as well as in plasma physics.

Theoretical Physics:

They also show up a lot in theoretical physics. Do you think complex numbers may show up in the “theory of everything”?

Quantum Mechanics:

Complex analysis is used to solve the CPT Theory (Charge, Parity and Time Reversal), as well as in conformal field theory and in the Wick’s Theorem.

Complex variables are also a fundamental part of QM as they appear in the Wave Equation.

String Theory:

Indeed, Complex Analysis shows up in abundance in String theory. In particular they help in defining the conformal invariant. I don’t quite understand this, but it seems some physicists are actively studying the topic.

Data Science:

Finally, Data Science and Statistics. The field for which I am most interested. I have yet to find an application of complex numbers in any of my work, but I have no doubt these applications exist. Some applications have already been made, such as using complex numbers to represent phases in deep neural networks, and using complex analysis to analyse sound waves in speech recognition.

Summary:

It is worth being familiar with the basics of complex variables. There are already numerous real world applications with more being developed every day. Despite the unfortunate name of ‘imaginary’, they are in by no means ‘fake’ or not legitimate. Complex analysis shows up in numerous branches of science and engineering, and it also can help to solidify your understanding of calculus. While we don’t know exactly what next application of complex analysis will be, it is clear they are bound to show up again. Maybe even in the unified theory of physics?

Thanks for reading!

Questions, comments or suggestions?

Sources:

[1] Hans Niels Jahnke(1999) A History of Analysis

[2] H. J. Ettlinger (1922) Annals of Mathematics

[3]Peter Ulrich (2005) Landmark Writings in Western Mathematics 1640–1940

[4] Umberto Bottazzini (1980) The higher calculus’. A history of real and complex analysis from Euler to Weierstrass.

[5] James Brown (1995) Complex Variables and Applications

[6] M Spiegel , S Lipschutz , J Schiller , D Spellman (2009) Schaum’s Outline of Complex Variables, 2ed

[7] R. B. Ash and W.P Novinger(1971) Complex Variables.