Introduction

In this chapter we introduce the methods by which certain types of definite integral may be evaluated. Similar methods may be used to sum certain types of infinite series. The approach has many applications, and will be considered again in Chapter 16, in applications to Fourier transforms, and in Chapter 17, on Laplace transforms. We begin by establishing the Residue theorem, which relates a contour integral to the residues of the integrand at its various singularities. Then we explore how various types of real integral can be transformed into contour integrals, and then evaluated by an analysis of their singularities. Finally we take a brief look at the summation of series by residue methods.

Mathematica can play various roles in this part of the theory related to the evaluation of integrals by the calculus of residues. It can just be there to help with the algebra in calculating residues. You can use the functions Residue and NResidue to work out the residues directly. Finally you can use Integrate and NIntegrate to do a direct calculation of the answer. In this last case considerable care is required. The symbolic treatment of general integrals is an evolving (black) art and the results, mostly in the way they are displayed and the full details of conditions for the results to hold, will vary from version to version of the software. This matters particularly when the integrand contains parameters.

Introduction

In this section we give a more precise characterization of complex functions and review their basic properties. We also introduce some formal concepts, such as neighbourhoods and open sets, in order to lay the foundations for a discussion of continuity and differentiability. We shall then make a first definition of basic functions such as the exponential and trigonometric functions, and their inverses, by referring back to real definitions. This will be revisited in Chapter 9 from a power series perspective. We shall also look at the concept of branch points, and the extended complex plane or ‘Riemann sphere’.

We shall also explore various ways of visualizing complex functions using Mathematica. We can build various routines for looking at functions. The first one we will consider takes a two-dimensional point of view, where functions are regarded as mappings taking one region of the complex plane to another. The second regards the function as a pair of functions of two real variables, and we show how to use Mathematica's three-dimensional plotting routines to view simultaneously both the modulus and argument of complex functions. Then we shall develop some plot routines tailored to bring out the folded structure of certain complex functions. Note that, in this chapter, the output of all Mathematica computations is set to appear in TraditionalForm. If you are using Mathematica technology beyond version 5.2, you should explore the options provided in your current version for managing graphics. See also the on-line supplement and enclosed CD.

Why this book?

Since 1985, I have been fortunate to have taught the theory of complex variables for several courses in both the USA and the UK. In the USA I lectured a course on advanced calculus for engineers and scientists at MIT, and in the UK I have given tutorials on the subject to undergraduate students in mathematics at both Cambridge and Oxford. Indeed, draft versions of this text have been inflicted on my students at Balliol and, more recently, at St. Catherine's over the last fourteen years. Few topics have given me such pleasure to teach, given the rich yet highly accessible structure of the subject, and it has at times formed the subject of my research, notably in its development into twistor theory, and latterly in its applications to financial mathematics. A parallel thread of my work has been in the applications of computer algebra and calculus systems, and in particular Mathematica®, to diverse topics in applied mathematics. This book is in part an attempt to use Mathematica to illuminate the topic of complex analysis, and draws on both these threads of my experience.

The book attempts also to inject some new mathematical themes into the topic and the teaching of it. These themes I feel are, if not actually missing, under-emphasized in most traditional treatments. It is perfectly possible for students to have had a formal training in mathematics that leaves them unaware of many key and/or beautiful topics.

Introduction

You have already read about how to motivate the introduction of complex numbers by the need to solve quadratic equations, and have seen how to solve higher order polynomial equations both through ‘pen and paper’ analysis and with the help of Mathematica. In the previous chapter you looked at Newton—Raphson iteration. This is not the only way of defining an iterative solution method, and there is another approach called ‘cobwebbing’ which is the subject of this chapter.

You are now in a position to perform a basic investigation of some of the most fascinating topics in modern mathematics: period doubling and transitions to chaos. This topic can be introduced by considering simple quadratic or cubic functions. However, rather than solving a simple quadratic or cubic equation, you are now going to be concerned with applying a function over and over again, given a starting value. Under certain circumstances, this has the effect of finding the solutions to the original equation, but in other situations you will be led to the solutions of other polynomial equations. Hence the need for a complex view.

There are many good reasons for you to investigate these topics. First, you should appreciate the emergence of complexity and beauty from the iteration (repeated application) of a simple quadratic or cubic map. Second, you should appreciate that there is some value in doing ‘experimental mathematics’. However, here and elsewhere in this book we shall be concerned with appreciating the special role that complex numbers play.

Introduction

Complex functions have an elegant interpretation in terms of mappings of the complex plane into itself. We explored this briefly in Chapter 8. Now we wish to study the geometrical aspects in rather more detail. Our plan is as follows. First, we shall literally play with Mathematica to get a feel for what some simple mappings do to simple regions. Next we shall look at the property of ‘conformality’ – that holormorphic functions, when interpreted as mappings, preserve angles between curves at most points. Then we shall explore the relationship between the geometry of circles and lines and a special class of mappings called Möbius transforms.

This chapter is the foundation for several that follow. In particular, in Chapter 19 we shall explore the application of conformal mapping to problems in physics in 2-dimensional regions. Chapter 23 will explore how some of this material may be generalized to higher dimensions. Chapter 21 will look at how conformal maps, and the Schwarz—Christ-offel transformation in particular, can be managed numerically. Chapter 23 will also reveal the real physics underlying the Möbius transform when it is seen in terms of Einstein's theory of special relativity.

Recall of visualization tools

Our first goal is to use Mathematica to explore some simple mappings. We shall do so by loading the ComplexMap Package and making a pair of additional functions, CartesianMap and PolarMap.

Introduction

In our studies so far we have been concerned with the complex plane interpreted as a two-dimensional Euclidean plane – when the concept of distance has been needed, it has always been the standard Euclidean notion expressed by Pythagoras' theorem. There are concepts of distance other than the standard Euclidean one. Indeed, this notion is at the heart of modern geometrical physics, and finds expression in both the non-positive-definite notions of distance of special relativity, and the non-flat metrics of general relativity.

In this chapter we shall meet the hyperbolic plane, which is perhaps the simplest non-Euclidean geometry. We shall not be able, in one chapter, to do full justice to this concept – indeed, excellent entire books have already been written about it (Coxeter, 1965; Stahl, 1993). What we shall do is explore a little of the geometry through the process of tiling the hyperbolic plane (see the Bibliography for papers by Coxeter and Levy on this particular matter also).

This chapter is based substantially (the sections on triangles and the ‘ghosts and birdies’ tiling) on a project carried out by a former colleague, V. Thomas, for the BBC Open University Production Centre. Gratitude is expressed to A.M. Gallen and, latterly, the Open University for permission to use this material, to Professor R. Penrose F.R.S. (‘R. Penrose’ for short) for several helpful suggestions, and to V.

Introduction

In the first chapter you saw why you need imaginary and complex numbers, by considering the solution of simple quadratic equations. In this chapter you will see how we set up complex numbers in general, and establish their basic algebraic and geometrical properties.

We shall assume that you have some understanding of what is meant by a real number. The exact nature and depth of this understanding will not materially affect the discussion thoughout most of this book, and this is not a book about the fundamentals of real analysis. We should, however, take a moment to remind ourselves what a ‘real’ number is, before we start defining ‘imaginary’ and ‘complex’ numbers. Students of pure mathematics should remind themselves of the details of these matters — there is really nothing for it but to go for a proper mathematical definition, and experience has shown that one needs to be slightly abstract in order to get it right, in the sense that the resulting definition contains all the numbers ‘we need’. For a full exposition, complete with proofs, you should consult a text on real analysis, such as that by Rudin (1976). For our purposes it will mostly be sufficient to regard real numbers as being all the points on a line (which we call the real axis) extending to infinity in both directions.

Introduction

You may already have seen how to solve polynomial equations numerically in Chapter 3, using the NSolve function, or FindRoot. How in general can we solve an equation, polynomial or otherwise, numerically? There are many schemes for doing this, with one or perhaps many variables. Given that we cannot solve most polynomials, or indeed other equations, in an exact analytical form, we need to consider a numerical treatment.

Let's look now at the most important such scheme. It leads naturally to the consideration of the solution of polynomial equations by iteration of rational functions, and this chapter is a brief introduction to this theory. Entire books can and have been written about both the art and mathematics of this. In the view of the author there is none better than that by Beardon (1991), which should be consulted by anyone serious about exploring the matter thoroughly. This chapter contains only introductory analytical comments on the matter, and for the most part we shall focus on exploring the art with Mathematica!

So here we take a novel route, looking at how the business of equation solving, which was the motivation for introducing complex numbers in the 16th century, becomes a whole new area of interest when we combine complex numbers with a computer system.

Note that in this chapter we shall be producing moderately complicated graphics. A machine running at 1 GHz or better is recommended for interactive use of the more complicated examples presented. If you are using an old machine just lower the plot resolution.

Introduction

The existence of functions such as Nest and NestList, together with extensive visualization tools, ensure that Mathematica is a natural system for investigating the iteration of mappings. One favourite is the logistic map that was considered in Chapter 5. Complex numbers play a very natural role, from two quite different points of view. First, we wish to understand why maps such as the logistic map behave the way they do, without relying merely on numerical simulation. Second, we wish to extend the use of numerical simulation to mappings of the complex plane.

You explored the first point in Chapter 5, and the second in Chapters 4 and 6, for simple polynomial maps. Now you will consider some other mappings of the complex plane into itself. These mappings are non-holomorphic (in the sense that will be defined properly in Chapter 10 – for now it suffices to realize that this means the functions involve complex conjugation in an essential way). Normally, when considering the theory of complex numbers, one's interest is quite rightly focused on the analytical properties of holomorphic or meromorphic functions. I hope the constructions described here will suggest that there is much that is both beautiful and interesting in complex structures that are not holomorphic. My approach is based on that of Field and Golubitsky (1992), and the use of Mathematica on this topic was first given by the author (Shaw, 1995).

Introduction

In this chapter we shall explore the notion of a ‘transform’ of a function, where an integral mapping is used to construct a ‘transformed’ function out of an original function. The continuous Fourier transform is one of a family of such mappings, which also includes the Laplace transform and the discrete Fourier transform. The Laplace transform will be discussed in Chapter 18. Numerical methods for the discrete Fourier transform and for the inversion of Laplace transforms will be given in Chapter 21.

What is the point of such transforms? Perhaps the most important lies in the solution of linear differential equations. Here the operation of a transform can convert differential equations into algebraic equations. In the case of an ordinary differential equation (ODE), one such transform can produce a single algebraic condition that can be solved for the transform by elementary means, leaving one with the problem of inversion – the means by which the transformed solution is turned into the function that is desired. In the case of a partial differential equation (PDE), for example in two variables, one transform can be used to reduce the PDE into an ODE, which may be solved by standard methods, or, perhaps, by the application of a further transform to an algebraic condition. Again one proceeds to a solution of the transformed problem. One or more inversions is required to obtain the solution.

Introduction

In real-world applications if is often the case that a purely analytical approach to transform calculus is insufficient. This chapter explores two techniques that extend the utility of transform methods by allowing a purely numerical treatment. We can explore numerical methods for both Fourier and Laplace transforms – in principle any method could be applied to either type of transform, since one is a rotation of the other. However, in practice, two types of problem appear to be of most importance:

(1) An essentially numerical approach to forward and backward Fourier transforms;

(2) The inversion of a complicated Laplace transform given in analytical form, for which there is no known analytical inverse.

An entire book could be written about the first topic, which is at the heart of many problems in applied mathematics, physics and engineering. It is of particular importance for signal and image processing. We shall briefly explore Mathematica's Fourier and InverseFourier functions. It should also be clear than any Fourier transform could be worked out numerically by the use of NIntegrate. In this chapter this approach will be explored in detail only for Laplace transforms.

The second topic is particularly important for generating solutions to partial differential equations. As we have seen in Chapters 18 and 19, certain partial differential equations may be simplified by transforming with respect to one or more of the independent variables, leading to a solvable algebraic or ordinary differential equation.

Introduction

In this chapter we introduce the concept of differentiation of a complex function of a complex variable. There are several ways of approaching this topic, and we shall consider at least two. Given that a complex number may be regarded as a pair of real numbers, we need to make very clear the distinction between differentiability of a function that has two real arguments and differentiability of a function with a single complex argument, and shall take as our starting point a review of the differentiation of functions of two real variables. This approach has the merit of generalizing in a straight-forward way to functions of many real or complex variables. We shall also consider another simple approach to differentiation based on the limit of a ratio. This latter approach is perhaps more familiar if you have taken a course in one-variable real calculus, but does not generalize to functions of several real or complex variables. A key result that we will establish is that a complex function is differentiable in the complex sense if (a) it is differentiable when considered as two real functions of two real variables and also (b) the Cauchy—Riemann equations (partial differential equations) apply. These equations link the real and imaginary parts of the function. After proving some basic results about complex differentiability, e.g., the product, quotient and chain rules, we then derive one of the principal results of basic complex analysis — that power series are differentiable within their radius of convergence.

Introduction

The goal of this Chapter is to provide you with a very basic understanding of how Laplace's equation (in three space variables) and the scalar wave equation (in three space and one time dimension) can be solved using holomorphic methods. This chapter will build on the methods developed in Chapter 23 in a very direct way, and you are recommended to read that chapter now before proceeding further here.

The material on dimension three is, in part, a much simplified version of part of a series of lectures given by Nigel Hitchin (now Professor, F.R.S.) in Oxford in the early 1980s. The picture presented here will not give you anything like the full geometrical picture underlying the results, which we shall develop by elementary methods. Part of the theory of the intrinsic three-dimensional approach is developed in a paper published by Hitchin (1982). The four-dimensional picture is covered most comprehensively by its principle architect, Professor Sir Roger Penrose, F.R.S., in Penrose and Rindler (1984a, b).

Laplace's equation in dimension three

Perhaps the most natural place to start is with the solution of Laplace's equation in three variables. In Chapter 23 we found, in Eq. (23.85), a natural holomorphic representation of a point in (possibly complex) three-dimensional space, arising as a degenerate case of a holomorphic null curve. This representation is in terms of a special type of quadratic holomorphic function.