These first two videos show a rotating Riemann sphere, with the point at infinity on the top. The functions displayed are e^z (left) and e^(e^z-1) (right). After the video gets going, a continuous Mobius transformation is applied which distorts the outputs of the functions.

Both functions have essential singularities at the point at infinity. For the exponential function, we can see that there is a problem with the infinite rings closing in on the point, however our coloring scheme makes it difficult to see exactly what is going on. The reason is that the exponential function (as a meromorphic function on C) never takes on the values zero or infinity. In order to give a clearer picture of what is happening, we apply a Mobius transformation which 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, we can see a number of dots converging to the point at infinity after the Mobius transform is applied. This highlights a key behavior of an essential singularity: all points (with possibly one exception) will appear infinitely many times in the neighborhood.

The function on the right is similar. It never takes on the values zero or infinity, but after a Mobius transformation we can see that it is taking on other points at locations which converge to the essential singularity.

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