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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 $

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

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

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

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

• $\int_{\mathbb Z}\ln \omega=2\pi$

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)

• $\int_{\mathbb Z}\ln \omega=2\pi$

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,♾️ 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 $

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

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

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

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

• $\int_{\mathbb Z}\ln \omega=2\pi$

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 $

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

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

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

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

• $\int_{\mathbb Z}\ln \omega=2\pi$

The following rules is 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, 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 $

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

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

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

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

• $\int_{\mathbb Z}\ln \omega=2\pi$

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,♾️ 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 $

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

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

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

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

• $\int_{\mathbb Z}\ln \omega=2\pi$