Riemann Sphere

Riemann Sphere
Fields: Algebra, Analysis, and Dynamic Systems
Image Created By: Unknown
Website: wikipedia

Riemann Sphere

In complex analysis and dynamics, the Riemann Sphere is often used to simplify problems and analysis. It is sometimes denoted as $\textbf{C} \bigcup \{ \infty \}$, the union of the complex numbers with a point called infinity.

A More Mathematical Explanation

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The Riemann Sphere is a stereographic projection of the unit sphere onto [...]

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The Riemann Sphere is a stereographic projection of the unit sphere onto the complex plane. To visualize the sphere, first picture the complex plane, which has complex dimension one, even though we generally draw it in the same way we draw the two dimensional plane, $\textbf{R}^2.$ Now, imagine a ball with half above the plane and half below. Wrap the part of the plane that is outside the sphere around the top, and bump the part of the plane that is inside the sphere out so that it covers the bottom of the sphere. We have identified each point in the complex plane with a point on the sphere.

Equivalently, we can take a line connecting any point on the plane to the top pole of the sphere. Each line intersects the sphere at precisely one location, determining the identification of points on the plane with points on the sphere. In addition to providing us with all of the numbers on the complex plane, these two methods of visualization allow us to do is identify the point $z=\infty$ as a particular point. We can therefore say that a function converges to infinity rather than diverges, so we can call the point $z=\infty$ a fixed point for the sake of simplicity.

We also can say that a value of the form $\frac{z}{0}=\infty$ instead of saying that it is undefined. This means that all rational functions, which are functions $R(x)=\frac{P(x)}{Q(x)}$ where P(x) and Q(x) are polynomials with distinct roots, are defined for all values on the Riemann Sphere.

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