1.7 Conclusions

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I have gone over the fundamentals of Complex Numbers from a somewhat

di_erent point of view from the usual one which can be found in many text

30 CHAPTER 1. FUNDAMENTALS

books. My reasons for this are starting to emerge already: the insight that

you get into why things are the way they are will help solve some practical

problems later.

There are lots of books on the subject which you might feel better about

consulting, particularly if my breezy style of writing leaves you cold. The

recommended text for the course is [1], and it contains everything I shall do,

and in much the same order. It also contains more, and because you are

doing this course to prepare you to handle other applications I am leaving

to your lecturers in Engineering, it is worth buying for that reason alone.

These notes are very speci_c to the course I am giving, and there's a lot of

the subject that I shan't mention.

I found [4] a very intelligent book, indeed a very exciting book, but rather

densely written. The authors, Carrier, Krook and Pearson, assume that you

are extremely smart and willing to work very hard. This may not be an

altogether plausible model of third year students. The book [3] by Copson

is rather old fashioned but well organised. Jameson's book, [5], is short

and more modern and is intended for those with more of a taste for rigour.

Phillips, [6], gets through the material e_ciently and fast, I liked Kodaira,

[7], for its attention to the topological aspects of the subject, it does it more

carefully than I do, but runs into the fundamental problems of rigour in

the area: it is very, very di_cult. McLachlan's book, [2], has lots of good

applications and Esterman's [8] is a middle of the road sort of book which

might suit some of you. It does the course, and it claims to be rigorous,

using the rather debatable standards of the sixties. The book [9] by Jerrold

Marsden is a bit more modern in approach, but not very di_erent from the

traditional. Finally, [10] by Ahlfors is a classic, with all that implies.

There are lots more in the library; _nd one that suits you.

The following is a proposition about Mathematics rather than in Mathematics:

Proposition 1.7.1 (Alder's Law about Learning Maths) Confusion propagates.

If you are confused to start with, things can only get worse.

You will get more confused as things pile up on you. So it is necessary to get

very clear about the basics.

The converse to Mike Alder's law about confusion is that if you sort out the

1.7. CONCLUSIONS 31

basics, then you have a much easier life than if you don't.

So do the exercises, and su_er less.

32 CHAPTER 1. FUNDAMENTALS

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