Applications
In complex analysis, a meromorphic function on the complex plane (or on any Riemann surface, for that matter) is a ratio f/g of two holomorphic functions f and g. As a map to the complex numbers, it is undefined wherever g is zero. However, it induces a holomorphic map (f, g) to the complex projective line that is well-defined even where g = 0. This construction is helpful in the study of holomorphic and meromorphic functions. For example, on a compact Riemann surface there are no non-constant holomorphic maps to the complex numbers, but holomorphic maps to the complex projective line are abundant.
The Riemann sphere has many uses in physics. In quantum mechanics, points on the complex projective line are natural values for photon polarization states, spin states of massive particles of spin 1/2, and 2-state particles in general. The Riemann sphere has been suggested as a relativistic model for the celestial sphere. In string theory, the worldsheets of strings are Riemann surfaces, and the Riemann sphere, being the simplest Riemann surface, plays a significant role. It is also important in twistor theory.
Read more about this topic: Riemann Sphere