# Transseries for beginners

```@article{Edgar2008TransseriesFB,
title={Transseries for beginners},
author={G. A. Edgar},
journal={arXiv: Rings and Algebras},
year={2008}
}```
• G. A. Edgar
• Published 2008
• Mathematics
• arXiv: Rings and Algebras
From the simplest point of view, transseries are a new kind of expansion for real-valued functions. But transseries constitute much more than that--they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries is a large ordered field, extending the real number field, and endowed with additional operations such as exponential, logarithm, derivative, integral, composition. Over the course of the last 20 years or so, transseries have emerged in several areas of… Expand
On numbers, germs, and transseries
• Mathematics
• 2017
Germs of real-valued functions, surreal numbers, and transseries are three ways to enrich the real continuum by infinitesimal and infinite quantities. Each of these comes with naturally interactingExpand
Toward a Model Theory for Transseries
• Mathematics, Computer Science
• Notre Dame J. Formal Log.
• 2013
An overview of the algebraic and model-theoretic aspects of this differential field of transseries is given, and efforts to understand its first-order theory are reported on. Expand
An Introduction to Resurgence, Trans-Series and Alien Calculus
In these notes we give an overview of different topics in resurgence theory from a physics point of view, but with particular mathematical flavour. After a short review of the standard Borel methodExpand
L O ] 1 3 D ec 2 01 7 On Numbers , Germs , and Transseries
Germs of real-valued functions, surreal numbers, and transseries are three ways to enrich the real continuum by infinitesimal and infinite quantities. Each of these comes with naturally interactingExpand
FRACTIONAL ITERATION OF SERIES AND TRANSSERIES
We investigate compositional iteration of fractional order for transseries. For any large positive transseries \$T\$ of exponentiality 0, there is a family \$T^{[s]}\$ indexed by real numbers \$s\$Expand
A Primer on Resurgent Transseries and Their Asymptotics
• Physics, Mathematics
• 2018
The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed byExpand
Integration on the Surreals: a Conjecture of Conway, Kruskal and Norton
• Mathematics
• 2015
In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field No of surreal numbers containing the reals and the ordinals, as well as a vast array of less familiar numbers. AExpand
Resurgent Transseries and the Holomorphic Anomaly
• Mathematics, Physics
• 2013
The gauge theoretic large N expansion yields an asymptotic series which requires a nonperturbative completion to be well defined. Recently, within the context of random matrix models, it was shownExpand
Trans-Series Asymptotics of Solutions to the Degenerate Painlev\'{e} III Equation: A Case Study
A one-parameter family of trans-series asymptotics of solutions to the Degenerate Painleve III Equation (DP3E) are parametrised in terms of the monodromy data of an associated two-by-two linearExpand
Asymptotics, ambiguities and resurgence
The appearance of resurgent functions in the context of the perturbative study of observables in physics is now well established. Whether these arise from the related study of non-linear systems orExpand

#### References

SHOWING 1-10 OF 75 REFERENCES
Logarithmic-exponential series
• Mathematics, Computer Science
• Ann. Pure Appl. Log.
• 2001
Evidence is given for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields, and its basic properties are established, including the existence of compositional inverses. Expand
Topological construction of transseries and introduction to generalized Borel summability
Transseries in the sense of \'Ecalle are constructed using a topological approach. A general contractive mapping principle is formulated and proved, showing the closure of transseries under a wideExpand
Global reconstruction of analytic functions from local expansions and a new general method of converting sums into integrals
• Mathematics
• 2006
A new summation method is introduced to convert a relatively wide family of Taylor series and infinite sums into integrals. Global behavior such as analytic continuation, position of singularities,Expand
FRACTIONAL ITERATION OF SERIES AND TRANSSERIES
We investigate compositional iteration of fractional order for transseries. For any large positive transseries \$T\$ of exponentiality 0, there is a family \$T^{[s]}\$ indexed by real numbers \$s\$Expand
Nested expansions and hardy fields
This short survey of some recent results is based on talks given at the Universities of Kent and Leipzig based on talk given by Wolfgang Lassner and the University ofLeipzig. Expand
Power series in one variable
Abstract This paper is concerned with the ring A of all complex formal power series and the group G of substitution-invertible formal series. The two main questions of interest will be these. How canExpand
Asymptotic differential algebra
• Mathematics
• 2005
We believe there is room for a subject named as in the title of this paper. Motivating examples are Hardy fields and fields of transseries. Assuming no previous knowledge of these notions, weExpand
Exponentiation in power series fields
• Mathematics
• 1996
We prove that for no nontrivial ordered abelian group G, the ordered power series fleld R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its orderedExpand
Logarithmic-Exponential Power Series
• Mathematics
• 1997
We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zetaExpand
Elementary Inversion of the Laplace Transform
at least for those s for which the integral converges. In practice when one uses the Laplace transform to, for example, solve a differential equation, one has to at some point invert the LaplaceExpand