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In the case of the real numbers, we have that there are precisely two square roots, one positive and one negative, except when they coincide at zero. The square root is taken to be the positive one. 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.
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. It is called a branch of the square root function, thus confusing things in a way which is traditional. We say that this is dened for < < .
This is cheating, but it cheats in a constructive and useful manner, so mathematicians don't complain that Riemann broke the rules and they won't play with him any more, they rather admire him CHAPTER 2. 14: The Square function through the Riemann Surface for pulling such a line2. If you build yourself a surface for the square function, then you project it down and squash the two sheets (cones in my picture) together to map it into C , then you can see that there is a one-one, onto, continuous map from C to the surface, S , and then there is a projection of S on C which is two-one (except at the origin).
An Introduction to Complex Analysis for Engineers by Alder.