Definition 2

The field of (H) functions is called the (H) extension of the Hardy field $\mathbf{R}(x)$ of rational functions with real coefficients. Any function belonging to this extension is called an (H) function.

Following this definition, if $f$ is an (H) function, not identically zero on a neighbourhood of $+\infty$, then $e^f$ and $\log |f|$ are also (H) functions. More generally, if $g$ is a second (H) function, and $u_1, u_2, \ldots, u_n$ a definition sequence for $g$, and if $f(x)$ tends to $+\infty$ with $x$, one sees by induction on $n$ that the composite functions $u_1 \circ f, u_2 \circ f, \ldots, u_n \circ f$ and $g \circ f$ are (H) functions.