**Contemporary Mathematics**

Volume: 373;
2005;
371 pp;
Softcover

MSC: Primary 03; 34; 58; 40; 39;

Print ISBN: 978-0-8218-3419-0

Product Code: CONM/373

List Price: $103.00

AMS Member Price: $82.40

MAA Member Price: $92.70

**Electronic ISBN: 978-0-8218-7963-4
Product Code: CONM/373.E**

List Price: $103.00

AMS Member Price: $82.40

MAA Member Price: $92.70

# Analyzable Functions and Applications

Share this page *Edited by *
*O. Costin; M. D. Kruskal; A. Macintyre*

The theory of analyzable functions is a technique used to study a wide class of
asymptotic expansion methods and their applications in analysis, difference and
differential equations, partial differential equations and other areas of
mathematics.

Key ideas in the theory of analyzable functions were laid out by Euler, Cauchy,
Stokes, Hardy, E. Borel, and others. Then in the early 1980s, this theory took
a great leap forward with the work of J. Écalle. Similar techniques and
concepts in analysis, logic, applied mathematics and surreal number theory
emerged at essentially the same time and developed rapidly through the
1990s. The links among various approaches soon became apparent and this body of
ideas is now recognized as a field of its own with numerous applications.

This volume stemmed from the International Workshop on Analyzable Functions and
Applications held in Edinburgh (Scotland). The contributed articles, written by
many leading experts, are suitable for graduate students and researchers
interested in asymptotic methods.

#### Readership

Graduate students and research mathematicians interested in asymptotic methods.

# Table of Contents

## Analyzable Functions and Applications

- Contents v6 free
- Preface vii8 free
- A singularly perturbed Riccati equation 110 free
- On global aspects of exact WKB analysis of operators admitting infinitely many phases 1120
- Asymptotic differential algebra 4958
- Formally well-posed Cauchy problems for linear partial differential equations with constant coefficients 8796
- Non-oscillating integral curves and O-minimal structures 103112
- Asymptotics and singularities for a class of difference equations 113122
- Topological construction of transseries and introduction to generalized Borel summability 137146
- Addendum to the hyperasymptotics for multidimensional Laplace integrals 177186
- Higher-order terms for the de Moivre-Laplace theorem 191200
- Twisted resurgence monomials and canonical-spherical synthesis of local objects 207216
- 1. Introduction: Object Analysis and Object Synthesis 210219
- 2. Reminders about moulds, resurgent functions, alien derivations 217226
- 3. Object Analysis: six basic examples 235244
- 3.1 Example 1: shift-like diffeomorphism 235244
- 3.2 Example 2: Euler-like differential equation 236245
- 3.3 Example 3: monocritical linear differential system 236245
- 3.4 Example 4: monocritical non-linear differential system 237246
- 3.5 Example 5: polycritical linear differential system 237246
- 3.6 Example 6: polycritical non-linear differential system 238247

- 4. The reverse problem: Object Synthesis 239248
- 4.1 Standard or hyperlogarithmic resurgence monomials and monics 239248
- 4.2 Semi-formal synthesis in Example 1 241250
- 4.3 Semi-formal synthesis in Example 2 242251
- 4.4 Semi-formal synthesis in the remaining examples 243252
- 4.5 Inadequacy of the standard resurgence monomials for the purpose of synthesis 244253
- 4.6 First intimations of "antipodality" 246255
- 4.7 The need for one free parameter at least 248257

- 5. Methods for non-canonical Object Synthesis 249258
- 6. Four closely related challenges 252261
- 6.1 The main obstacle: faster-than-lateral growth 253262
- 6.2 Challenge 1 : Searching for well-behaved averages 255264
- 6.3 Challenge 2 : Searching for well-behaved alien derivations 259268
- 6.4 Challenge 3 : Searching for well-behaved resurgence monomials 262271
- 6.5 Challenge 4 : Searching for the proper notion of ramified exponential growth 264273
- 6.6 Proofs 266275

- 7. Well-behaved uniformising averages 269278
- 8. Well-behaved alien derivations 278287
- 8.1 Reminder about the standard alien derivations 278287
- 8.2 Diffusion-induced alien derivations 278287
- 8.3 Scale-invariance 279288
- 8.4 Standard and organic alien derivations as limit-cases 279288
- 8.5 Proofs and comments 280289
- 8.6 Tables of averages and derivations 282291
- 8. 7 Pinpointing the difference between "good" and "bad" 285294

- 9. Proper notion of ramified-exponential growth 286295
- 10. Well-behaved resurgence monomials 288297
- 11. Applications to canonical Object Synthesis 295304
- 12. Hyper-, peri-, para-logarithmic monomials and monies: total closure 306315

- Matching and singularities of canard values 317326
- On the renormalization method of Chen, Goldenfeld, and Oono 337346
- Generalizing surreal numbers 347356
- Two examples of resurgence 355364