Asymptotic analysis is extensively used to explore systems for which exact
solutions are not available or are difficult to manipulate. In most cases of interest,
however, classical asymptotics does not provide global information; in particular it
does not yield information at values of the parameters which are not large, small, or
otherwise special. The information provided by classical expansions is not complete;
every classical asymptotic series represents a large class of functions. Furthermore,
classical expansions are not closed under all common operations. Larger structures
are needed to represent faithfully functions occurring in applications.
Transseries and LE series, two very related structures, are formally asymptotic
expansions of ordinal length of power series, exponentials, and logs. They form
a field closed under most usual operations, including algebraic ones, functional
ones like composition and functional inversion, and analytical ones including dif-
ferentiation and integration. As a consequence, they have the natural richness to
represent a large class of functions including quite general solutions of differential
and difference equations and some types of partial differential equations. In the
theory of analyzable functions, transseries are precisely defined, studied, and can
be relatively generally "summed" to actual functions. Transseries contain global
information about the functions they represent.
Key ideas in the direction of a theory of analyzable functions are present in
the works of Euler, Cauchy, Stokes, Hardy, Borel, and others. The theory took
a great leap forward in the early 1980s with the work of J. Ecalle; similar tech-
niques and concepts emerged at essentially the same time in analysis, logic, applied
mathematics and surreal number theory and developed rapidly through the 1990s.
The links between the various approaches soon became apparent and this body
of ideas is now recognized as a field of its own with numerous nontrivial applications.
The contributions to this volume are a sequel of the International Workshop on
Analyzable Functions and Applications, Edinburgh, June 17-21, 2002, in which
experts from many areas of mathematics participated.
The meeting was possible through generous funding by UK's Engineering and
Physical Sciences Research Council.
We thank the International Centre for Mathematical Sciences and in particular
Mrs. Tracey Dart for their great help and cooperation.