Smoothing away the singularities: At first glance w has similar problems. Again, the poles can be replaced with zeroes of 1/w, which satisfies a similar equation. But the zeroes block calculation of w", so that still leaves us with a difficulty. However, in looking at the graphs of w, w' and w", I noticed that every zero of w" seemed to be a zero of w. This suggested looking at v = w"/w. It turned out that v satisfies the equation v' = ( iww' − 2v)/z. So if we set u = w', then w, u, v satisfy (w, u, v)' = (u,wv, iww' − 2v)/z). Away from the origin this is well behaved, and poles can be replaced with zeroes by passing to 1/w. Here is my attempt at using this idea to prove that w is everywhere meromorphic.
Some examples: Here are two graphs of w, made using the tricks mentioned above, on squares of sides 40 and 240. Using w it is easy to get graphs of related functions like cos(θ/2), which has the geometric significance of determining the arclength along the directions of principal curvature.
Here's looking at you, kid: Just as a point of interest, here is the derivative of w.
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