By Alder.

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We clearly pass from one plane to another at a branch cut, which can be anywhere, and then we go back again a full circuit later. p Now I shall do the same thing with z2 + 1. But before tackling this case, 54 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS a short digression. 1 In the television series `Sliders', the hero generated a disk shaped region which identied two dierent universes. Suppose there are two people intending to slide into a new universe and they see this disk opening into a tunnel in front of them.

For this form also, it is easy to conrm that inversion takes circles to circles, where a straight line is just a rather extremal case of a circle. Malcolm Hood told me this one. These representations are sneaky and probably cheating, but it is telling you something important, namely, some representations for things will make some problems dead easy, and others make it horribly dicult. Thinking about this early on can save you a lot of grief. 5 Can you see why a parametric representation of the circle of the form z = a + r cos + i(b + r sin ) could be a serious blunder in trying to show that inversions take circles to circles?

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? 5 The function f (z) = z1 The real function f (x) = 1=x is a perfectly straightforward function which is dened everywhere except, of course, at x = 0.