Zeros of Sections of Power Series
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1002
AI bert Edrei Edward B. Saff Richard S. Varga
Zeros of Sections of Power Series
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Authors
Albert Edrei Department of Mathematics, Syracuse University Syracuse, New York 13210, USA Edward B. Saff Center for Mathematical Services, University of South Florida Tampa, Florida 33620, USA Richard S. Varga Institute for Computational Mathematics Kent State University, Kent, Ohio 44242, USA
AMS Subject Classifications (1980): 30C15, 30015, 30E15 ISBN 3-540-12318-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12318-0 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
contribution of Albert Edrei is lovingly dedicated to
Lydia.
The contribution of Edward Saff is lovingly dedicated to LoJte:tta. •
The contribution of Richard Varga is lovingly dedicated to EJ.d.heJt.
Acknowledgments We sincerely wish to thank Mr. Amos J. Carpenter (now at the Department of Computer Science, St. Joseph's College, Rensselaer, IN 47978) who performed the necessary calculations for Figures 1-12.
These calculations were done on a
VAX-ll/780 and graphed by a Calcom plotter (both at Kent State university). We also express our gratitude to Faith Carver, Esther Clark, Anna Lucas and Mary McGill, who took great care in the typing of our manuscript.
Table of Contents
Acknowledgments 1.
Introduction • .
1
2.
Statements of our results
7
3.
1.
Mittag-Leffler functions of order A > 1
II.
Functions of genus zero whose zeros are all negative
19
III.
Problems for further study
23
7
Discussion of our numerical results
26 29-40
Figures 1 - 12 4.
Outline of the method
41
5.
Notational conventions
43
6.
Properties of the Mittag-Leffler function of order 1 < A < 00 • • • • • • •
44
7.
Estimates for Gm(w) and Qm(w)
49
8.
A differential equation
53
9.
Estimates for J
m
(w) near the circumference
Iw I =
1
60 62
10.
Existence and uniqueness of the Szego curve
11.
Crude estimates for
12.
Proof of Theorem 5
70
13.
Proof of Theorem 1
70
14.
Proof of Theorem 2
72
15.
The circular portion of the Szego curve (Proof of Theorem 3) .
77
16.
Proof of Theorem 4
80
Ium(w)
I
and 1Qm(w)
I
63
VII'
17.
Proof of Theorem 6
82
18.
Properties of £-functions; proof of assertion I of Theorem 7 •••••• • ••.
87
19.
£-functions of genus zero are admissible in the sense of Hayman .••. . • • . .
91
20.
The functions Um(w), Qm(w), Gm(w) associated with £-functions of genus z
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