News
Little mistake in Homework 3 is now fixed!Written on 18.11.22 by Sebastian Toth Unfortunetly there has been a mistake in Problem 2b) in Homework 3. We have fixed it now. Since you have already submitted your solutions we will consider this in our feedback. |
Natural Boundaries in Function Theory
Every power series with radius of convergence equal to 1 defines a holomorphic function on the unit disc having at least one singularity on the unit circle.
The aim of this course is to survey some quite classical results that relate the coefficients of a power series with its boundary behavior, in order to study
those power series for which every point of the unit circle is a singularity. Such power series are said to have a natural boundary, and they will be the main
object of the course.
The first part of the course will relate natural boundaries to gaps in the coefficients. After reviewing some direct arguments and examples, we will
see the two main results that relate natural boundaries to gaps, Fabry's gap Theorem and Polya's gap Theorem.
We will then see that many power series with no gaps have also natural boundaries. We will see some examples, including those related to quantities from
number theory. We will also relate natural boundaries with aperiodicity in the coefficients by proving Szegö's Theorem.
Lastly, we will consider random power series, that is, power series whose coefficients consist in a sequence of random variables. We will study some conditions that ensure that such random functions have a natural boundary almost surely.
The course will be self-contained, and the required prerequisites are from a standard first course on complex analysis.
Time and Place: Tuesday, 14 - 16, SR 6, E 2.4 and Thursday, 14 - 16, Zeichensaal, E 2.5,
Lectures will be held in English.
Criteria: 50% of the achievable points from the homeworks; passing of an exam
Type of exam: Oral exam
Creditpoints: 9 ECTS
Office hours: Wednesday, 16-17, Room 2.18, E 2.4
Every week there will be a new assignment sheet. Solutions of the homework problems have to be submitted at least one week after the homework has been uploaded on the homepage.
Problem sessions:
| When | Where | Tutor |
|---|---|---|
| Wednesdays, 08:15-9:45 | SR 9, E 2.4 | Sebastian Toth |
In the problem sessions solutions of the homework problems will be dicussed and presented.