The Geometry of C^n is Important for the Algebra of Elementary Function

On the one hand, we all “know” tha\(\sqrt {{z^2}} = z\), but on the other hand we know that this is false when z = −1. We all know that ln e x = x, and we all know that this is false when x = 2πi. How do we imbue a computer algebra system with this sort of “knowledge”? Why is it that \(\sqrt x \sqrt y = \sqrt {xy} \) is false in general y =), but \(\sqrt {1 - z} \sqrt {1 + z} = \sqrt {1 - {z^2}} \) is true everywhere? The root cause of this, of course, is that functions such as \(\sqrt {} \) and log are intrinsically multi-valued from their algebraic definition.