Riemann Surfaces over the Riemann Sphere
We will construct pictures that show or not over the complex z-plane, but rather over the Riemann sphere. Using the sphere allows us to cover all the z-values more "equally" and to make the branch points at more visible. Given the Riemann sphere of radius with center , we visualize as a point with the same direction as the image of on the Riemann sphere and distance from the origin . The arctan in the last formula uniquely maps the interval to a finite interval. The function RiemannSpherePoint calculates the position of the point corresponding to the spherical coordinates under the mapping of the function f onto the Riemann sphere, resulting in a "multi-skinned-Riemann egg."
The function RiemannSpherePatch generates a patch for the function f in the specified part of the domain of the spherical coordinates.
We will give some simple examples (in the sense that the functions have a branch cut structure that is easy to determine and to describe) of functions mapped onto the Riemann sphere. Let us start with the simplest function by using the functions defined above. The two branch points at 0 and are mapped into the points and on the Riemann sphere. The north pole corresponds to complex infinity. We see the symmetry between the north pole and the south pole, which is a reflection of the formula (in the last formula, corrections to the branch cut have been ignored).
The following picture shows some sheets of the Riemann surface of inside the upper half of the unit disk. For easier viewing we color the sheets differently and do not show all the polygons.
Here we show some sheets of the function .
To see the three branch points more clearly, we cut the sphere in half along the x-y-plane, the x-z-plane, and the y-z-plane.
The has the two branch points 0 and 1. They are more visible in the following half of the generalized Riemann sphere.
The has the two branch points . At these points we have an additional pole.
The next picture shows some sheets of the function . Now the north pole (as well as the south pole) shows the accumulation points of logarithmic branch points. We exclude points near the poles to avoid resolution problems.
The next picture shows some sheets of the function . On the unit circle there are five branch points (and branch cuts going out from them) and five additional branch cuts between them.
The next picture shows the function . The two branch points are 0 and 1.
One could now go on and visualize more complicated Riemann surfaces on the Riemann sphere. In the spirit of the last section, we would rewrite oder as a differential equation for the longitudinal coordinate of a spherical coordinate system according to the following formula, together with corresponding modifications in patch and RiemannSurface.
Such an extension is straightforward; we end now and leave further explorations in this direction to the interested reader.
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