Preface

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These notes are intended to be of use to Third year Electrical and Electronic

Engineers at the University ofWestern Australia coming to grips with

Complex Function Theory.

There are many text books for just this purpose, and I have insu_cient time

to write a text book, so this is not a substitute for, say, Matthews and Howell's

Complex Analysis for Mathematics and Engineering,[1], but perhaps a

complement to it. At the same time, knowing how reluctant students are to

use a textbook (except as a talisman to ward o_ evil) I have tried to make

these notes su_cient, in that a student who reads them, understands them,

and does the exercises in them, will be able to use the concepts and techniques

in later years. It will also get the student comfortably through the

examination. The shortness of the course, 20 lectures, for covering Complex

Analysis, either presupposes genius ( 90% perspiration) on the part of the

students or material skipped. These notes are intended to _ll in some of the

gaps that will inevitably occur in lectures. It is a source of some disappointment

to me that I can cover so little of what is a beautiful subject, rich in

applications and connections with other areas of mathematics. This is, then,

a sort of sampler, and only touches the elements.

Styles of Mathematical presentation change over the years, and what was

deemed acceptable rigour by Euler and Gauss fails to keep modern purists

content. McLachlan, [2], clearly smarted under the criticisms of his presentation,

and he goes to some trouble to explain in later editions that the book

is intended for a di_erent audience from the purists who damned him. My

experience leads me to feel that the need for rigour has been developed to

the point where the intuitive and geometric has been stunted. Both have a

part in mathematics, which grows out of the conict between them. But it

seems to me more important to penetrate to the ideas in a sloppy, scru_y

but serviceable way, than to reduce a subject to predicate calculus and omit

the whole reason for studying it. There is no known means of persuading a

hardheaded engineer that a subject merits his time and energy when it has

been turned into an elaborate game. He, or increasingly she, wants to see two

elements at an early stage: procedures for solving problems which make a

di_erence and concepts which organise the procedures into something intelligible.

Carried to excess this leads to avoidance of abstraction and consequent

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loss of power later; there is a good reason for the purist's desire for rigour.

But it asks too much of a third year student to focus on the underlying logic

and omit the geometry.

I have deliberately erred in the opposite direction. It is easy enough for the

student with a taste for rigour to clarify the ideas by consulting other books,

and to wind up as a logician if that is his choice. But it is hard to _nd in

the literature any explicit commitment to getting the student to draw lots

of pictures. It used to be taken for granted that a student would do that

sort of thing, but now that the school syllabus has had Euclid expunged, the

undergraduates cannot be expected to see drawing pictures or visualising surfaces

as a natural prelude to calculation. There is a school of thought which

considers geometric visualisation as immoral; and another which sanctions it

only if done in private (and wash your hands before and afterwards). To my

mind this imposes sterility, and constitutes an attempt by the bureaucrat to

strangle the artist. 1 While I do not want to impose my informal images on

anybody, if no mention is made of informal, intuitive ideas, many students

never realise that there are any. All the good mathematicians I know have a

rich supply of informal models which they use to think about mathematics,

and it were as well to show students how this may be done. Since this seems

to be the respect in which most of the text books are weakest, I have perhaps

gone too far in the other direction, but then, I do not o_er this as a text

book. More of an antidote to some of the others.

I have talked to Electrical Engineers about Mathematics teaching, and they

are strikingly consistent in what they want. Prior to talking to them, I

feared that I'd _nd Engineers saying things like 'Don't bother with the ideas,

forget about the pictures, just train them to do the sums'. There are, alas,

Mathematicians who are convinced that this is how Engineers see the world,

and I had supposed that there might be something in this belief. Silly me.

In fact, it is simply quite wrong.

The Engineers I spoke to want Mathematicians to get across the abstract

ideas in terms the students can grasp and use, so that the Engineers can

subsequently rely on the student having those ideas as part of his or her

1The bureaucratic temper is attracted to mathematics while still at school, because it

appears to be all about following rules, something the bureaucrat cherishes as the solution

to the problems of life. Human beings on the other hand _nd this su_ciently repellant

to be put o_ mathematics permanently, which is one of the ironies of education. My own

attitude to the bureaucratic temper is rather that of Dave Allen's feelings about politicians.

He has a soft spot for them. It's a bog in the West of Ireland.

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thinking. Above all, they want the students to have clear pictures in their

heads of what is happening in the mathematics. Since this is exactly what

any competent Mathematician also wants his students to have, I haven't felt

any need to change my usual style of presentation. This is informal and

user-friendly as far as possible, with (because I am a Topologist by training

and work with Engineers by choice) a strong geometric avour.

I introduce Complex Numbers in a way which was new to me; I point out

that a certain subspace of 2_2 matrices can be identifed with the plane R2 ,

thus giving a simple rule for multiplying two points in R2 : turn them into

matrices, multiply the matrices, then turn the answer back into a point. I

do it this way because (a) it demysti_es the business of imaginary numbers,

(b) it gives the Cauchy-Riemann conditions in a conceptually transparent

manner, and (c) it emphasises that multiplication by a complex number is a

similarity together with a rotation, a matter which is at the heart of much

of the applicability of the complex number system. There are a few other

advantages of this approach, as will be seen later on. After I had done it this

way, Malcolm Hood pointed out to me that Copson, [3], had taken the same

approach.2

Engineering students lead a fairly busy life in general, and the Sparkies have

a particularly demanding load. They are also very practical, rightly so, and

impatient of anything which they suspect is academic window-dressing. So

far, I am with them all the way. They are, however, the main source of

the belief among some mathematicians that peddling recipes is the only way

to teach them. They do not feel comfortable with abstractions. Their goal

tends to be examination passing. So there is some basic opposition between

the students and me: I want them to be able to use the material in later

years, they want to memorise the minimum required to pass the exam (and

then forget it).

I exaggerate of course. For reasons owing to geography and history, this

University is particularly fortunate in the quality of its students, and most

of them respond well to the discovery that Mathematics makes sense. I hope

that these notes will turn out to be enjoyable as well as useful, at least in

retrospect.

But be warned:

2I am most grateful to Malcolm for running an editorial eye over these notes, but even

more grateful for being a model of sanity and decency in a world that sometimes seems

bereft of both.

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' Well of course I didn't do any at _rst ... then someone suggested I try just

a little sum or two, and I thought \Why not? ... I can handle it". Then

one day someone said \Hey, man, that's kidstu_ - try some calculus" ... so I

tried some di_erentials ... then I went on to integrals ... even the occasional

volume of revolution ... but I can stop any time I want to ... I know I can.

OK, so I do the odd bit of complex analysis, but only a few times ... that

stu_ can really screw your head up for days ... but I can handle it ... it's OK

really ... I can stop any time I want ...' ( tim@bierman.demon.co.uk (Tim

Bierman))

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