2.4 Squares and Square roots: Summary

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I have gone into the business of examining the square function and the square

root function in agonising detail, because they illustrate many of the problems

and opportunities of complex functions. They show that the little sweeties

are (a) surprisingly complicated even when the real version of the function

is boringly familiar, and (b) they are not so bad we can't make sense of

them. Many hours of innocent fun can be had by exploring the behaviour of

complex functions the real versions of which are simple and uninteresting. It

is recommended that you play around with some yourself.

It makes sense to look at functions such as f(z) = z2 because we have

that C is a _eld, so we can do with C everything we could do with R. So

polynomials make sense. And so do in_nite series, as we shall see later, so

the trigonometric and exponential functions make sense, and just as we can

ask for a square root of 􀀀1, so we can ask for a logarithm of it.

Exercise 2.4.1 What would you expect to be the value of ln(􀀀1)?

This is weird stu_ by comparison with the innocent functions from R to R,

and it is a good idea to get the basics clear, which is the main reason for

doing to death the square function. We can now move on to a few more easy

functions to _nd out what they do. This should be approached in a spirit of

fun and innocence. Who knows what bizarre things we shall _nd?

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