Table Of ContentAlgorithms and Computation
in Mathematics • Volume
12
Editors
Manuel Bronstein Arjeh M. Cohen
Henri Cohen David Eisenbud
Bernd Sturmfels
Springer-Verlag Berlin Heidelberg GmbH
John R. Shackell
Symbolic Asymptotics
~
Springer
John R. Shackell
Institute of Mathematics,
Statistics and Actuarial Science
University of Kent
Canterbury
KentCT27NZ
United Kingdom
e-mail: [email protected]
Mathematics Subject Classification (2000): 68W30, 41A60
Library of Congress Cataloging -in -Publication Data
Shackell, John R., 1943 -
Symbolic asymptotics / John R. Shackell.
p. cm. - (Algorithms and computation in mathematics, ISSN 1431-1550; v. 12)
IncIudes bibliographica1 references and index.
ISBN 978-3-642-05925-4 ISBN 978-3-662-10176-6 (eBook)
DOI 10.1007/978-3-662-10176-6
1. Asymptotic expansions. 2. Algorithms. I. Title. II. Series.
QA297.S435 2004
51l'.4-dc22 2004045315
ISSN 1431-1550
ISBN 978-3-642-05925-4
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication
of this publication or parts thereof is permitted only under the provisions of the German Copyright
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from Springer-Verlag Berlin Heidelberg GmbH.
Violations are liable to prosecution under the German Copyright Law.
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@ Springer-Verlag Berlin Heidelberg 2004
Originally published by Springer-V erlag Berlin Heidelberg New York in 2004
Softcover reprint ofthe hardcover lst edition 2004
The use of general descriptive names, registered names, trademarks, etc. in this publication does
not inlply, even in the absence of a specific statement, that such names are exempt from the relevant
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Printed on acid-free paper 46/3142at -5 43210-
This book is dedicated to the memory of the author's parents.
Ralph Ronald Shackell 1899 - 1976
Jean Shackell (nee Hatten) 1912 - 1994
Preface
This book is about calculating limits and asymptotic expansions of real func
tions. The subject has a long history and a vast literature, almost all of it
concerning power series. There are already many books about asymptotic
power series, and I am not the person best qualified to write another. So I
have concentrated on the areas where power series are no longer the right
tool. Even there I have given the most coverage to the developments that I
have personally been involved in, not because they are necessarily the most
important, but just because they are what I know most about. However I have
tried to give a reasonably comprehensive bibliography.
I have taken the stance that the book should be accessible to anyone with
a good general background in mathematics. This has caused the omission of
some topics, for example the proofs in [102). Very noticeably I have given
little coverage to Joris Van Der Hoeven's thesis and related preprints. This
work is undoubtedly important but it is also difficult, and the precise status of
some of it remains unclear at the time of writing. Those who wish to find out
more about Dr. VanDerHoeven's work are referred to his website (currently
http:/ /www.math.u-psud.fr;-vdhoeven/).
The reader who wants only a quick introduction to symbolic asymptotics
might suitably follow Chapter 1, glance through 2.1 and 2.2 as far as Algo
rithm 2, and then read 3.1, 3.2 3.3, 4.1, 4.2 and 5.1.
I have occasionally suggested exercises in the text. In many cases, an an
swer may be found in the reference cited. I have tried (mostly succeeding!) to
avoid setting as exercises, problems which I have shrunk from trying myself
but nonetheless believe to be soluble.
There are many areas of mathematics which are to a greater or lesser
extent concerned with asymptotics, and sometimes there is some overlap with
symbolic asymptotics. In particular this is the case with o-minimality, which
is briefly alluded to in Chapter 8, and non-standard analysis, which is not
hereafter mentioned at all. The book has already taken too long to write!
That it has been written at all owes much to the encouragement of col
leagues both at the University of Kent and elsewhere around the world. I
VIII Preface
would like to take this opportunity to thank them all, together with my fam
ily. I would also like to thank the staff at Springer-Verlag. There were sub
stantial periods when I made little progress with the book while I devoted
myself to mind numbing administration of the sort that is ever more taking
over the lives of British academics. Ms. Ruth Allewelt, Dr. Peters, and these
ries editor Manuel Bronstein showed considerable patience during these times.
John Shackell,
Canterbury,
11 May 2004.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Zero Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Zero-equivalence of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Richardson's Uniformity Conjecture ................. 11
2.2 Zero-equivalence of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Examples ........................................ 22
2.2.2 The sets Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Modular Methods in Zero Equivalence . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Hensel Lifting.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Growth in Degrees ....................................... 31
2.4.1 Canonical Forms .................................. 32
2.5 Systems of Partial Differential Equations . . . . . . . . . . . . . . . . . . . 33
2.5.1 Notation ......................................... 33
2.5.2 Radical Differential Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.3 Computing Characteristic Sets . . . . . . . . . . . . . . . . . . . . . . 36
2.5.4 The Rosenfeld-Grabner Algorithm ................... 36
2.5.5 Consequences, Applications ......................... 38
2.5.6 Zero Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6 Finding Symmetries ..................................... 39
Hardy Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Definitions and Examples ................................. 43
3.2 Building Hardy Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Rough Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 The Map 'Y2 • . .• . • • • • . . .• . .• . • • . • . . .• • . . .• • • . . . . . . . . . . . . 54
Output Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Asymptotic Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Multiseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
X Contents
4.3 Operations on Multiseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.1 Substituting into a Power Series... . . . . . . . . . . . . . . . . . . 65
4.3.2 The Logarithm of a Multiseries. . . . . . . . . . . . . . . . . . . . . . 68
~.3.3 The Exponential of a Multiseries . . . . . . . . . . . . . . . . . . . . 69
4.3.4 Powers of a Multiseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.5 Amalgamating Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.6 Substituting One Multiseries Into Another . . . . . . . . . . . . 70
4.4 Nested Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4.1 Comparison of Nested Forms ....................... 74
4.4.2 Operations on Nested Forms and Expansions . . . . . . . . . 76
4.5 The Algebra of Star Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.1 Definitions and Elementary Properties ............... 78
4.5.2 Differentiation and Star Products . . . . . . . . . . . . . . . . . . . 81
Algorithms for Function Towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 The Exp-Log Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.1 Case 1: /i =log h, hE Fi-1. . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.2 Case 2: /i = exp h, h E Fi-1 . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.3 Handling Denominators and Other Powers ............ 87
5.1.4 Summary of the Algorithm ......................... 88
5.1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2 Asymptotic Fields ....................................... 92
5.2.1 Adding Exponentials ............................... 100
5.2.2 Adding Integrals .................................. 101
5.2.3 Algebraic Equations ............................... 110
5.3 Compositions with Meromorphic Functions ................. 116
5.4 Cartesian Representations ................................ 125
Algebraic Differential Equations ............................... 129
6.1 Nested Forms of Hardy-Field Solutions ..................... 130
6.1.1 Examples ........................................ 140
6.2 The Number of Cases .................................... 144
6.3 Reducing the Complexity ................................. 150
6.3.1 A Theorem for Sparse Differential Equations .......... 151
Inverse Functions .............................................. 155
7.1 Inverting a Nested Expansion ............................. 156
7.1.1 Summary of the Algorithm for Inversion .............. 161
7.1.2 Example ......................................... 161
7.2 Multiseries oflnverse Functions ........................... 163
7.2.1 Proof of the Iteration Formula ...................... 169
7.2.2 Asymptotic Fields and Inverse Functions ............. 173
Contents XI
Implicit Functions ............................................. 175
8.1 A Special Case .......................................... 176
8.1.1 Computation and Checking of Candidates ............ 177
8.1.2 Examples ........................................ 178
8.2 Systems of Exp-Log Equations ............................ 179
8.2.1 Zero Equivalence .................................. 180
8.2.2 Sketch of the Method .............................. 180
8.2.3 Building estimates from the Tower of Fields .......... 182
8.2.4 Extension by a Logarithm .......................... 182
8.2.5 Extension by an Exponential ....................... 183
8.2.6 Obtaining the Nested Forms ........................ 184
8.2. 7 Example 1 ........................................ 185
8.2.8 Example 2 ........................................ 187
8.2.9 Example 3 ........................................ 189
Star-Product Expansions ...................................... 191
9.1 Exp-Log Expansions ..................................... 191
9.1.1 Rewriting Exp-Log Expressions Into Standard Star
Expansion Form ................................... 193
9.1.2 Order Comparisons ................................ 196
9.2 Growth Classes in Hardy Fields ........................... 201
9.2.1 Generalizing the /'nS ..•....•...•...•...•••..•...••. 201
9.3 Generalized Star Products ................................ 204
9.3.1 Expansions ....................................... 207
9.3.2 Non-integral Iterates and Multiple Scales ............. 208
9.4 Real Iterates of Increasing Functions ....................... 209
Oscillating Functions .......................................... 215
10.1 An Interval Calculus Algorithm ........................... 216
10.1.1 Our Calculus of Intervals ........................... 217
10.2 Lim-sups and Lim-infs ................................... 220
10.3 Wandering Functions .................................... 224
10.4 Wandering Expansions ................................... 231
References ..................................................... 235
Index .......................................................... 241
