Integration in the surreal numbers In the appendix to ONAG (2nd edition), Conway points that the definition of integration (using Riemann sums as left and right options) gives the "wrong" answer : $\int_0^\omega \exp(t)\thinspace dt=\exp(\omega)$ (instead of $\exp(\omega)-1$). I wonder if this was not due to lack of some options (presumably right ones), and if a better definition could not be sought by using Kurzweil-Henstock integration (but I was not able to concoct one, of course, else I would not ask here). Has the idea already been   tried?
 A: In a comment on Timothy Chow's recent answer that cited a paper a by O. Costin, H. Friedman and myself concerned with integration on the surreals, I noted that the cited paper has now been superseded by a revised and expanded version of portions that paper, and that Costin and I expect to post the revised version on arXiv within the next few weeks. I'm pleased to note that the revision (Integration on the Surreals: https://arxiv.org/abs/2208.14331) has now appeared. A separate paper by Costin and Friedman concerned with a different portion of the original paper will appear on another occasion.
A: In a recent article in the Notices of the AMS, Philip Ehrlich briefly describes some progress in this area. Below is a relevant excerpt from the article.

Conway originally expressed doubt that “reasonable” global definitions of exponentiation, logarithm, sine, and cosine could be defined on $\mathbf{No}$. Through the collective efforts of Kruskal, Norton, Gonshor, van den Dries, Ehrlich, and Kaplan, however, this doubt has been put to rest. Van den Dries and Ehrlich (2001) showed that $\mathbf{No}$ together with the Kruskal–Gonshor exponential function $\exp$ defined thereon has the same elementary properties of the ordered field of real numbers with real exponentiation, and Ehrlich and Kaplan have further shown that $\mathbf{No}$ has canonical sine and cosine functions which in turn lead to a canonical exponential function on $\mathbf{No}$’s surcomplex counterpart ${\mathbf{No}}[i]$ that extends $\exp$.

Additional rudiments of analysis on the surreals have also been developed by Alling, Fornasiero, Rubinstein–Salzedo and Swaminathan, and Costin, Ehrlich and Friedman. Costin and Ehrlich, in particular, have developed a theory of integration (and differentiation) that extends the range of analysis from the reals to the surreals for a large subclass of resurgent functions that arise in applied analysis. The resurgent functions, which generalize the analytic functions, were introduced by Écalle in the early 1980s in connection with work related to Hilbert’s 16th problem. Unlike nonstandard analysis, which provides an infinitesimalist approach to integration on the extended reals ($\mathbb{R}\cup \{\pm \infty \}$), surreal integration deals with integrals whose bounds and values need not be extended reals at all. For example, in the surreal theory (setting $e^x=\exp x$) we have
$$\begin{equation*} \int _{0}^{\omega }e^{x}dx=e^{\omega }-1=\omega ^{\omega }-1. \end{equation*}$$
This work makes contributions towards realizing some of the analytic goals expressed by Kruskal and Norton in their unsuccessful early attempts to establish a theory of surreal integration as described by Conway.

In particular, the theory of integration mentioned above is developed in Integration on the Surreals: a Conjecture of Conway, Kruskal and Norton,  arXiv:1505.02478, 2015.
See also Ehrlich's answer to a related MO question.
