DOI: https://doi.org/10.15368/theses.2022.74
Available at: https://digitalcommons.calpoly.edu/theses/2496
Date of Award
6-2022
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Ryan Tully-Doyle
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
Bounded analytic functions on the open unit disk D = {z ∈ C | |z| < 1} are a fre-
quent area of study in complex function theory. While it is easy to understand the
behavior of analytic functions on sequences with limit points inside D, the theory
becomes much more complicated as sequences converge to the boundary, ∂D. In this
thesis, we will explore boundary theorems, which can guarantee specific desired be-
havior of these analytic functions. The thesis describes an elementary approach to
proving Fatou’s Non-Tangential Limit Theorem, as well as proofs and discussion of
the subsequent classical boundary theorems for specific points, Julia’s Theorem and
the Julia-Carathéodory Theorem. This thesis serves as a synthesis of these boundary
theorems in order to fill a gap in the overarching literature.