A Synthesis of Classical Boundary Theorems

Bounded analytic functions on the open unit disk D = {z ∈ C | |z| \u3c 1} are a fre-quent area of study in complex function theory. While it is easy to understand thebehavior of analytic functions on sequences with limit points inside D, the theorybecomes much more complicated as sequences converge to the boundary, ∂D. In thisthesis, we will explore boundary theorems, which can guarantee specific desired be-havior of these analytic functions. The thesis describes an elementary approach toproving Fatou’s Non-Tangential Limit Theorem, as well as proofs and discussion ofthe subsequent classical boundary theorems for specific points, Julia’s Theorem andthe Julia-Carathéodory Theorem. This thesis serves as a synthesis of these boundarytheorems in order to fill a gap in the overarching literature