# Transseries as germs of surreal functions

## TL;DR

This paper demonstrates that Ecalle's transseries and related formal series can be viewed as functions on surreal numbers, establishing a new connection between transseries and surreal analysis, and introducing omega-series as a minimal subfield with rich properties.

## Contribution

It introduces omega-series as the smallest subfield of surreal numbers containing reals, omega, and closed under exp, log, and infinite sums, and interprets transseries as surreal functions.

## Key findings

- Omega-series form a proper class of surreal functions.
- Surreal numbers can be interpreted as a large field of transseries.
- Omega-series are surreal analytic and closed under composition and differentiation.

## Abstract

We show that \'Ecalle's transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.