By H. A Lauwerier

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I recommend map. It is shorter, and all important and much used terms should be short. I shall defer to tradition and call them complex functions much of the time. This is shorter than `map from C to C ', which is necessary in general because you do need to tell people where you are coming from and where you are going to. 2 The function w = z2 We can get some idea of what the function w = z2 does by the same process. 4 and also made it smaller so you could see the `after' picture at the same scale.

So a path along the real axis from 1 towards 1 in the universe f slips smoothly into the universe f at the origin. You can retrace your path exactly. If you start o in Universe f at +1 going left, then you slide over into universe f Actually they've been ready for well over a century. Riemann discussed this sort of thing in 1851. It took a while to get down to the level of popular television. 3. THE SQUARE ROOT: W = Z 21 57 at the origin. So in this case, it doesn't matter which way you go into the `gate', you wind up in the same universe- there are only two.

If you don't know what this means, you have never done any net surng, and you need to. This surface ought to extend to innity radially; rather than being made from two disks, it should be made from two copies of the complex plane itself, with the gluings as described. It is known as a Riemann Surface. 3 The Square Root: w = 1 z2 The square root function, f (z) = z 12 is another function it pays to get a handle on. 3. THE SQUARE ROOT: W = Z 21 47 the square root of a a number you get the number back.