Analytical Continuation and Natural Domains of Holomorphic Functions of One Variable

Abstract

In this thesis, we will talk about analytic continuations of holomorphic functions. We will compare real and complex settings. We know that holomorphic functions can be represent as a Taylor series in their domains. We will find the region outside of the domains with the help of analytic continuation where these functions are holomorphic. We will introduce the bump function in order to prove the analytic continuations of real smooth functions is not unique. On the other hand, we will see that analytic continuations of holomorphic functions is unique along a chosen curve. In general, it is not true that analytic continuation is unique along two different curves. It will happen when domain is simply connected and if function can be continued along every curves with common initial and end points in its domain. We will prove a core result of analytic continuation known as Monodromy Theorem. We will also see singular points of holomorphic functions and some results on extensions of holomorphic functions on bounded domains.