# Transseries for beginners

@article{Edgar2008TransseriesFB, title={Transseries for beginners}, author={G. A. Edgar}, journal={arXiv: Rings and Algebras}, year={2008} }

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

#### 44 Citations

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 interacting… Expand

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

- Physics
- 2014

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 method… Expand

L O ] 1 3 D ec 2 01 7 On Numbers , Germs , and Transseries

- 2018

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 interacting… Expand

FRACTIONAL ITERATION OF SERIES AND TRANSSERIES

- Mathematics
- 2010

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 by… Expand

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. A… Expand

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 shown… Expand

Trans-Series Asymptotics of Solutions to the Degenerate Painlev\'{e} III Equation: A Case Study

- Mathematics, Physics
- 2020

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 linear… Expand

Asymptotics, ambiguities and resurgence

- Mathematics
- 2017

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 or… Expand

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