Introduction of Hurwitz’s Formula

In this talk, we will present Hurwitz's Formula for a holomorphic mapping between compact Riemann Surfaces. This formula relates the genus of each surface with the degree and ramification of the mapping and is an example of how topology and complex analysis interacts. The outline of the presentation includes (1) introducing some basic definitions such as the definition of Riemann Surfaces and holomorphic maps between two Riemann Surfaces, (2) introducing definitions that related to the Hurwitz's Formula, (3) stating the Hurwitz's Formula and proving it, and (4) giving examples that apply the Hurwitz's Formula.