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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 sur ng, and you need to. This surface ought to extend to in nity 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.

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