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Additional info for An Introduction to Complex Analysis for Engineers
The situation for the complex plane is not nearly so neat, and the reason is that as we go around the circle, looking for square roots, we go continuously from one solution to another. Start o at 1 + i0 and you will surely agree that the obvious value for its square root is itself. Proceed smoothly around the unit circle. To take a square root, simply halve the angle you have gone through. By the time you get back, you have gone through 2 radians, and the preferred square root is now 1 + i0. So whereas the two solutions formed two branches in the case of the reals, and you could only get from one to the other by passing through zero, for C there are continuous paths from one solution to another which can go just about anywhere.
This may not be an altogether plausible model of third year students. The book  by Copson is rather old fashioned but well organised. Jameson's book, , is short and more modern and is intended for those with more of a taste for rigour. Phillips, , gets through the material eciently and fast, I liked Kodaira, , for its attention to the topological aspects of the subject, it does it more carefully than I do, but runs into the fundamental problems of rigour in the area: it is very, very dicult.
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. Looking just at the unit circle to see where that goes: imagine a loop made of chewing gum circling a can of beans.