Proposition 3

If $\mathscr{K}_0$ is a Hardy field there exists a Hardy field $\mathscr{K}$ containing $\mathscr{K}_0$ and such that, for every function $z \in \mathscr{K}$, not identically zero on a neighbourhood of $+\infty$, both $e^z$ and $\log |z|$ belong to $\mathscr{K}$.

Denote by $\mathscr{K}$ the set of functions $f \in \mathscr{H}(\mathfrak{F}, \mathbf{R})$ having the following properties: for each function $f \in \mathscr{K}$ there is a finite number of Hardy fields $\mathscr{K}_1, \mathscr{K}_2, \ldots, \mathscr{K}_n$ (the number $n$ and the fields $\mathscr{K}_i$ depending on $f$) such that $f \in \mathscr{K}_n$ and, for $0 \leq i \leq n-1$, one has $\mathscr{K}_{i+1} = \mathscr{K}_i(u_{i+1})$ where $u_{i+1}$ is equal either to $e^{z_i}$ or to $\log |z_i|$ with $z_i$ belonging to $\mathscr{K}_i$ and not vanishing identically on a neighbourhood of $+\infty$. One says that $u_1, u_2, \ldots, u_n$ form a definition sequence for the field $\mathscr{K}_n$ and of the function $f$; the same function can naturally admit several definition sequences.

By def. 1 of V, p. 247, every function $f \in \mathscr{K}$, not identically zero on a neighbourhood of $+\infty$, has constant sign and is differentiable on an interval $[x_0, +\infty[$; if $f \in \mathscr{K}_n$, then $1/f$ and $f'$ are equal to functions in $\mathscr{K}_n$, thus to functions in $\mathscr{K}$, on a neighbourhood of $+\infty$. To see that $\mathscr{K}$ is a Hardy field it is enough to prove that if $f$ and $g$ are two functions in $\mathscr{K}$ then $f - g$ and $fg$ are equal to functions in $\mathscr{K}$ on a neighbourhood of $+\infty$. Now let $u_1, u_2, \ldots, u_m$ be a definition sequence for $f$, and $v_1, v_2, \ldots, v_n$ a definition sequence for $g$. The sequence $u_1, u_2, \ldots, u_m, v_1, v_2, \ldots, v_n$ obtained by concatenating the sequences $(u_i)$ and $(v_i)$ is again a definition sequence of a Hardy field $\mathscr{K}_{m+n}$ and this field contains $f$ and $g$, so $f - g$ and $fg$ are equal to functions in $\mathscr{K}_{m+n}$ on a neighbourhood of $+\infty$.

One says that the Hardy field $\mathscr{K}$ defined in the proof of prop. 3 is the (H) extension of the Hardy field $\mathscr{K}_0$.

If $\mathscr{K}'$ is another Hardy field possessing the properties stated in prop. 3, it follows from the construction of $\mathscr{K}$ that $\mathscr{K}/\mathbf{R}_\sim$ is contained in $\mathscr{K}'/\mathbf{R}_\sim$. By abuse of language, one says that the (H) extension of the Hardy field $\mathscr{K}_0$ is the smallest Hardy field $\mathscr{K}$ having these properties.