Applications
A common way to define functions in complex analysis proceeds by first specifying the function on a small domain only, and then extending it by analytic continuation. In practice, this continuation is often done by first establishing some functional equation on the small domain and then using this equation to extend the domain. Examples are the Riemann zeta function and the gamma function.
The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces.
The power series defined below is generalized by the idea of a germ. The general theory of analytic continuation and its generalizations are known as sheaf theory.
Read more about this topic: Analytic Continuation