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Extra resources for An Introduction to Complex Analysis for Engineers
11 It should conrm your expectations based on a little thought. Suppose I had asked what happens to the unit disk under the map f (z) = z2 ? You should be able to see fairly quickly that it goes to the unit disk, but in a rather peculiar way: far from being the identity map, the perimeter is stretched out to twice its length and wrapped around the unit circle twice. Some people nd this hard to visualise, which gives them a lot of trouble; fortunately you are engineers and good at visualising things.
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! you are back where you started. Such behaviour in a car park would worry anyone except Dr.
The two points i get sucked in towards zero. Because of the slit in the plane, this is now a continuous map, although we haven't dened it on the points we threw out. There is also a perfectly respectable map f which sends z to f (z). This has exactly the same domain and range space, C with a vertical slit in it, between i and i, and it has the same range space, C with a horizontal slit in it, between 1 and 1. It is just f followed by a rotation by 180o. 3. THE SQUARE ROOT: W = Z 21 p 53 We now ask for a description of the Riemann surface for z2 + 1.
An Introduction to Complex Analysis for Engineers by Alder.