1 Introduction
1.1 What is Complex Analysis?
Let us consider each word in turn:
- Complex: Here we do not mean ‘complicated’, but rather ‘to do with complex numbers’. So while the subject of real analysis looks at functions of real variables (i.e. variables that take values in \(\mathbb{R}\)) that produce real values, here we study functions of complex variables (taking values in \(\mathbb{C}\)) that produce complex values.
- Analysis: Calculus done properly. The study of limits, sequences, series, continuity of functions, differentiation, integration and related concepts.
So in this course, we will look at the convergence and limits of sequences and series of complex numbers, but our main focus will be on complex functions and their differentiation and integration.
As you might expect, complex analysis is not a small topic and so, given the time constraints, this course can cover only a small fraction of it. We will focus on functions of a single complex variable and, even given this constraint, will only be able to manage an introduction to the subject.
1.2 But I haven’t learned Real Analysis yet!
Don’t worry! We will cover the bare bones of real analysis, which is all that is needed for this course. There will be a few occasions where we take steps that use results from real analysis without a full explanation of where the result comes from, but in these cases you may find that the step seems entirely reasonable anyway.1 We will also rather casually use terms from topology, such as ‘region’ and ‘neighbourhood’ — an intuitive understanding of these will be sufficient, though I have provided a brief mathematical description of these concepts if you are interested.
1 Part of the reason for the development of mathematical analysis was that seemingly reasonable and ‘obvious’ calculational steps may be more subtle than they seem at first, or indeed turn out to be incorrect.
As you may have already inferred, we will be taking a physicist’s approach to the subject, focusing on getting to those results that are of use in the study of physics while skimming over some of the mathematical details and occasionally taking risks in our calculations that a mathematician might frown at. Those of you who are interested in understanding the mathematical underpinnings of the subject may wish to look at a standalone book (rather than a chapter of a ‘methods’ book) such as “Introduction to Complex Analysis” by H. A. Priestley.
1.3 Why learn Complex Analysis
As a physicist, your motivation for studying complex analysis is likely to be somewhat utilitarian. Perhaps you wish to learn complex analysis because you wish to learn Quantum Field Theory and the book you want to learn this from assumes knowledge of complex analysis. This is a perfectly valid reason for wishing to learn complex analysis. (And yes, as you continue your study of physics, it becomes increasingly likely that some knowledge of complex analysis will be assumed.)
Why do such books involve complex analysis? Why did the physics community feel the need to use complex analysis? Again, one answer is simply because it is useful - in particular, for calculating the value of certain integrals. Knowledge of complex analysis, combined with a little ingenuity, can reduce the calculation of a challenging looking real integral into the much simpler task of summing a few numbers, called residues.
However, having said all this, I hope that those of you who are, perhaps, a bit of a mathematician at heart, will find that complex analysis is also a particularly elegant2 subject. One might expect that the inclusion of complex numbers would make complex analysis even more complicated than real analysis, but it actually makes it simpler, cleaner and better structured in many ways. And from the introduction of the single number, \(i\), springs a structure - the set of complex numbers - with a surprising analytical structure and cohesiveness.
2 I will use the word elegant here rather that the word beautiful that is so often used to describe complex analysis. Clearly I am no artist - I am instead a rather more prosaic physicist.
Whether you are studying complex analysis due to a fascination with the structure of the complex numbers, or simply as a prerequisite for future physics study, I hope that you will enjoy the next few weeks of lectures and come to appreciate the usefulness, and perhaps the elegance, of the subject.