Complex graphs: Following an idea of Larry Crone, we visualize complex functions by assigning a colors to areas of the complex plane and placing the color of f(z) at z. As a guide to the color scheme, here is θ, with the identity function as a guide to the color scheme. The sharp vertical lines indicate that theta has singularities. Specifically, computation indicates that it goes off to infinity and ramifies, returning with a different value if you cycle around the singular point. In fact, it appears always to change by 4π.
Two variations on θ: This behavior suggests that we might get a single-valued function by looking at either T(z) = tan(θ(z)/4) or w(z) = exp(iθ(z)/2). In both cases you can still see evidence of the cuts in the original, because the calculation of θ was interrupted by the singularity, and the further calculations were therefore also interrupted. However, the smoothness of the colors is an indication that the problem is only in the method of calculation, not in the functions.
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