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Sample text

The Riemann surface associated with the square and square root function was a surface which we pictured as sitting over the domain of the square root function, C , and which projected down to it. Then we split the squaring map up so that it was made up of another map into the Riemann surface followed by the projection. Actually, the Riemann surface is just the graph of w = z2 , but instead of trying to picture it in four dimensions, we put it in three dimensions and tried not to think about the self-intersection this caused.

This ensures that in one sense every complex number has two square roots, and yet forces us to restrict the domain to ensure that we only get them one at a time. 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.

For example, if we take the region from 0 to , but without the end angles 0 and , the squaring map sends this to the whole complex plane with the positive X-axis removed. This map has an inverse, (r; ) ; (r1=2 ; 2 ) which pulls it back to the half plane above the X-axis. Another possibility is to take the half plane with positive real part, and square that. This gives us a branch cut along the negative real axis. We can then write f1 (z) = f1(r; ) = (r1=2; 2 ) for the inverse, which is called the Principal Square Root.

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