Remark 2

When $g(x)$ is an (H) function equivalent to a constant $k \neq 0$ one has $g(x) = k + g_1(x)$, with $g_1 \ll 1$; the function $u_1(x) = u(x) - k$ is the inverse of the function $x - g_1(x + k)$, and one is brought back to the case treated in prop. 5 of V, p. 256.

To have an asymptotic expansion for the function $u$ it thus suffices to have such an expansion for the function $u_n$: if $g$ admits an asymptotic expansion relative to the scale under consideration one is thus led (by the definition of (H) functions) to the problems examined in V, p. 223 to 227.

The most general case, as follows, reduces to the case treated in prop. 5 of V, p. 256: the function $y = u(x)$ is assumed to satisfy the relation

where $\varphi$ is an (H) function, $\psi$ is an (H) function such that $\psi \gg 1$ and such that the inverse function $\theta$ of $\psi$ is also an (H) function, and $g$ is an (H) function such that $g \ll \psi$. Let now $v(x)$ be the inverse function of $x - g(\theta(x))$; one has $u = \theta \circ v \circ \varphi$, and $g(\theta(x)) \ll x$; if one knows an asymptotic expansion for $v$ thanks to prop. 5 of V, p. 256, one can then deduce an asymptotic expansion for $u$ as in V, p. 223 to 227.