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https://hdl.handle.net/2142/23595
Description
Title
Zeros of partial sums of power series
Author(s)
Qian, Xiaoling
Issue Date
1995
Doctoral Committee Chair(s)
Rubel, Lee A.
Miles, Joseph B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
A classical result of Jentzsch states that, for a power series with radius of convergence one, every point on the unit circle is a limit point of zeros of partial sums of the power series. In this thesis, the relationship between the properties of the function and the behavior of the zeros of its partial sums is investigated. We prove that the set of limit points of xeros of partial sums of rational functions can contain an open disk outside the unit circle, while that of a power series with the unit circle as its natural boundary can be empty outside the unit circle. We also prove that, for any closed set in the complement of the open unit disk and any finite set of points inside the unit circle, there is a power series such that the set of the limit points of zeros of partial sums is the union of the two given sets and the unit circle. A condition is given on the coefficients of power series which ensures that the limit points of zeros of partial sums are contained in the closed unit disk. We obtain Jentzsch-type theorems about power series in several complex variables. Examples illustrate that Jentzsch-type theorems in several complex variables depend on the way in which the partial sums are defined and on the convergence domain of the power series.
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