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.

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