Integration in the surreal numbers

Question

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?

Answer 1

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.

Answer 2

Edit (8/21/24). The paper Integration on the Surreals by Ovidiu Costin and myself referred to below has now appeared online in Advances in Mathematics. It is a substantially revised and expanded version of the below referenced preprint that appeared on mathatXiv. A pdf file of the just-said version is available at https://doi.org/10.1016/j.aim.2024.109823.

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.

Answer 3

Here is another answer by me, in which I will give an explicit Mathematica code for evaluating integrals over surreal numbers. It is based on the same approach as my prevuous answer, but uses a more general formula.

It should be noted that my approach does not satisfy the property of linearity against an infinite coefficient. In other words, moving an infinite factor from under the integral changes the value of the expression: $\alpha\int f\ne \int (\alpha f)$. The property of linearity against infinite factors has been included in other approaches to surreal integration problem, but I consider it unnatural.

In the code below, for simplicity, in input w corresponds to $\omega$ and W to $\omega_1$. The input accepts any surreal numbers composed of these values as integration limits and any surreal-valued function of integration variable x as the expression under integral. In output, derivation is denoted as ꝺ.

f[x_] = w^W;
a = 0;
b = w;
f[x_] = f[x] /. W -> W[w];
Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
Fin[t_] := (PowerExpand[f[t]] /. Log[w] -> 0 /. w -> 0) + 
     Limit[Evaluate[LaplaceTransform[ D[f[t], w], t, x]] + 
        Evaluate[LaplaceTransform[ D[f[t], w], t, -x]], x -> 0]/
      2 /. \[Infinity] -> 0 /. -\[Infinity] -> 0 // 
  FullSimplify  (*Finding the finite part so to integrate separately*)
int = \[Pi]  W  Integrate[D[f[t], w], {t, a, b}] + 
    Integrate[Fin[t], {t, a, b}] // FullSimplify;
int = If[a == b, \[Pi]  D[f[t], w], int, int] /. 
     Derivative[1][W][w] -> ꝺ[W] /. W[w_] :> W // Normal;
Print[Inactivate[
  Integrate[
   f[x] /. W[w] -> W /. w -> \[Omega] /. 
    W -> Subscript[\[Omega], 1], {x, 
    a /. w -> \[Omega] /. W -> Subscript[\[Omega], 1], 
    b /. w -> \[Omega] /. W -> Subscript[\[Omega], 1]}], 
  Integrate], "=", 
 int /. w -> \[Omega] /. W -> Subscript[\[Omega], 1]]

The code correctly gives all the results from the other my answer, but also can find more complicated results, such as:

• $\int _0^{\omega }e^{c x \omega }dx=\frac{c \omega ^3+\omega +\pi \left(e^{c \omega ^2} \left(c \omega ^2-1\right)+1\right) \omega _1}{c \omega ^2}$

• $\int _0^{\omega }e^{c x^2 \omega }dx=\omega +\frac{1}{2} e^{c \omega ^3} \pi \omega _1-\frac{\pi ^{3/2} \text{erfi}\left(\sqrt{c} \omega ^{3/2}\right) \omega _1}{4 \sqrt{c} \omega ^{3/2}}$

• $\int _0^{\omega }\log (c x \omega )dx=\omega (\log (c)+\log (\omega )-1)+\pi \omega _1$

• $\int _0^{\omega }\omega ^xdx=\frac{\frac{\pi \left((\omega \log (\omega )-1) \omega ^{\omega }+1\right) \omega _1}{\omega }+1}{\log ^2(\omega )}$

• $\int _0^{\omega }\omega ^{\omega }dx=\pi (\log (\omega )+1) \omega _1 \omega ^{\omega +1}+\omega$

• $\int _0^{\omega }p \omega ^{q \omega }dx=p \omega \left(\pi q (\log (\omega )+1) \omega _1 \omega ^{q \omega }+1\right)$

• $\int _0^{\omega }\omega ^{\omega _1}dx=\pi \omega _1 \left(\omega \partial \left(\omega _1\right) \log (\omega )+\omega _1\right) \omega ^{\omega _1}$

• $\int _0^{\omega }\omega _1^{\omega }dx=\pi \omega \left(\frac{\omega \partial\left(\omega _1\right)}{\omega _1}+\log \left(\omega _1\right)\right) \omega _1^{\omega +1}+\omega$

• $\int _0^{\omega }\omega _1^xdx=\frac{\omega _1^{\omega }-1}{\log \left(\omega _1\right)}+\frac{\partial\left(\omega _1\right) \left(\pi \left(\omega \log \left(\omega _1\right)-1\right) \omega _1^{\omega +1}+\pi \omega _1+\omega \right)}{\log ^2\left(\omega _1\right) \omega _1}$

• $\int _0^{\omega _1}\omega ^xdx=\frac{\omega _1 \left(\pi \left(\log (\omega ) \omega _1-1\right) \omega ^{\omega _1}+\pi +1\right)}{\omega \log ^2(\omega )}$

• $\int _0^{\omega }\log \left(\omega _1\right)dx=\omega \left(\log \left(\omega _1\right)+\pi \partial\left(\omega _1\right) \omega _1\right)$

etc.

Answer 4

The following rules define a natural extension of integration to Hardy fields, and when Hardy field structure is introduced in surreals, works for surreals as well. It does not employ any specific properties of surreal numbers beyond the H-field structure.

But first, some notes on notation.

• We assume the canonical embedding of Hardy field into surreals, so $\omega$ is equivalent of the germ of the function $f(x)=x$ at infinity, $\infty$ and a defined derivation function $\partial$ such that $\partial(\omega)=1$.

• We denote the numerosity of the interval $[0,1)$ as $\omega_1$. Although this is not necessary, we consider it equal to the first uncountable ordinal, which we have freedom to do due to the automorphisms of surreals.

• In the context of $a,b\in\mathbb{R}$, we use notation $\int_a^b$ to denote an integral where the limits of integration, $a$ and $b$ are half-included, in other words, $\int_a^b=\frac12\left(\int_{(a,b)}+\int_{ [a,b ]}\right)$.

• We equate the germ of the real-valued function $f(x)$ at infinity if the germ belongs to the Hardy field and is a sum of a purely infinite and a real number (no infinitesimal terms in Conway normal form), with a (generally, divergent) expression $f(a)+\int_a^\infty f'(t)dt=f(\omega),$ where $a$ is an arbitrary real number.

Provided that, the following fundamental rule emerges:

$$\int_S u dt=\pi N(S) \partial(u),$$ where $S\subset\mathbb{R}$ is the integration domain, $N(S)$ is the numerosity expressed as a surreal number and $u$ is a surreal number with no finite part (the finite part can be separated from any surreal number using Conway normal form). If the value under integral contains finite and non-finite (infinite and infinitesimal) parts, the integration can be done separately.

Consequences:

• $\int_a^b u dt=\pi \omega_1 (b-a)\partial(u)$.

• $\int_a^b u f(t) dt =\pi \omega_1 \partial(u)\int_a^b f(t)dt,$

The both of the above formulas can be used to generalize integration to arbitrary $a,b\in \text{No}$. In the case of the later formula we employ Newton-Leibnitz formula, which we postulate to still hold in surreal domain.

• $\int_S \ln \omega \, dt=\frac{1}{\omega }\int_S \omega \, dt$

Examples:

• $\int _0^1\omega dt=\pi \omega _1$

• $\int _0^1\exp \omega \, dt=\pi \omega _1 \exp \omega $

• $\int_0^\omega \ln \omega \,dt=\int_0^\infty \ln \omega \,dt=\frac12\int_{\mathbb{R}} \ln \omega \,dt= \pi \omega _1$

• $\int_0^1 \ln \omega_1\, dt = \frac1{\omega_1}\int_0^1\omega_1\,dt= \pi \partial(\omega_1)$

• $\frac1\pi\int_0^{\omega_1}\omega\,dt=\omega_1^2$

(numerosity of the Long Line or $1\times 1$ square with half-included perimeter)

• $\int_{\mathbb Z}\ln \omega=2\pi$