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The point at the origin is called a branch point; I nd the whole terminology of `branches' unhelpful. It suggests rather that the Riemann surface comes in di erent lumps and you can go one way or the other, getting to di erent parts of the surface. For the Riemann surface associated with squaring and square rooting, it should be clear that there is no such thing. It certainly behaves in a rather odd way for those of us who are used to moving in three dimensions. It is rather like driving up one of those carp parks where you go upward in a spiral around some central column, only instead of going up to the top, if you go up twice you discover that, SPUNG!

The carpet needs to be made of something stretchy, like chewinggum1. When you have got back to your starting point, join up the tear you made and you have a double covering of every point under the carpet. It is worth trying hard to visualise this, chewing-gum carpet and all. Notice that there are two points which get sent to any point on the unit circle by the squaring map, which is simply an angle doubling. The same sort of thing is true for points inside and outside the disk: there are two points sent to a + ib for any a; b.

The more ways you have of talking and thinking about things, the easier it is to draw conclusions, and the harder it is to be led astray. It is also a lot more fun. The converse is also true: the inversion of a straight line is a circle through the origin. To see this, let ax + by + c = 0 be the equation of a straight line. Turn this into polars to get ar cos  + br sin  + c = 0 Now put r = 1=s to get the inversion: (a=s) cos  + (b=s) sin  + c = 0 and rearrange to get s2 + (as=c) cos  + (bs=c) sin  = 0   a= 2 c which is a circle passing through the origin with centre at b=2c .

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An Introduction to Complex Analysis for Engineers by Alder.


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