Gregory Zitelli

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Old videos showing a rotating Riemann sphere, the point at infinity on the top. Functions are e^z and e^(e^z-1). After the video gets going, a Mobius transformation is applied to distort the output.

Both functions have essential singularities at the point at infinity. A problem with the infinite rings is visible at the top, but the coloring scheme makes it difficult to see what's going on. The reason is that it never takes the values zero or infinity. The Mobius transforms swaps the values zero and infinity with two other antipodal points on the sphere. Since the exponential function takes on these values infinitely many times near its essential singularity, the converging dots become visible. This highlights a key behavior of an essential singularity: all points (with possibly one exception) will appear infinitely many times in the neighborhood.