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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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