A Synthesis of Classical Boundary Theorems

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

Name thesis:degree_name

MS in Mathematics

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2022

Author dc:creator

• Dakhlia, Lukas A

Contributors dc:contributor

• Ryan Tully-Doyle
• Mathematics
• College of Science and Mathematics

• Fatou
• Julia
• Carathéodory
• Boundary
• Complex
• Analysis

Identifier

10.15368/theses.2022.74

OAI identifier oai:identifier

oai:digitalcommons.calpoly.edu:theses-4097