In this talk, we will focus on the following fundamental question: given an open set in the complex plane, does there exist a holomorphic (= analytic = complex-differentiable) function on it that does not extend to any larger open set? First, we will explain precisely what this means. Then we will discuss infinite products (of numbers and functions), which are the essential technical tools that allow us to approach the problem of constructing such functions.
Then we will show that, given any open set in the complex plane, there does exist a holomorphic function on it that does not extend holomorphically to any larger open set. Finally, if time permits, we will show how this situation is completely different from the one that obtains in higher-dimensional complex Euclidean space.
Dr. Anwoy Maitra is a faculty member in the Department of Mathematics at IITM.
