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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 identi ed two di erent 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. One of them walks around the back of the disk. If this one sees the other side of the disk and steps through it, and if the other person goes through the other side of the disk at the same time, is it true that they must come out in the same place?

4 Describe the surface associated with the inverse function. Show that there is a one-one continuous map going in both directions between the two surfaces. It is worth pointing out that the Riemann surface can be constructed in several ways: there is nothing unique about the choice of branch cuts, for example. It is not so obvious that the Riemann surface is unique in the sense that there is always a way of deforming one into another. You don't have the background to go into this, so I shan't. But the text books often give the impression that branch cuts come automatically with the problem, whereas they are much less clear cut4 than that.

THE SQUARE ROOT: W = Z 21 p 53 We now ask for a description of the Riemann surface for z2 + 1. You might think that asking about Riemann surfaces is an idle question prompted by nothing more than a desire to draw complicated surfaces, but it turns out to be important and very practical to try to construct these surfaces. The main reason is that we shall want to be able to integrate along curves in due course, and we don't want the curve torn apart. The Riemann surface associated with the square and square root function was a surface which we pictured as sitting over the domain of the square root function, C , and which projected down to it.

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An Introduction to Complex Analysis for Engineers by Alder.

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