Example 2

A function may be comparable to all the functions of a scale and yet not admit a principal part with respect to this scale. For example, for $x$ real tending to $+\infty$, $\sqrt{x}$ has no principal part with respect to the scale $x^n$ where $n$ is a rational integer; $\log x$ has no principal part with respect to the scale of $x^\alpha$ ($\alpha$ arbitrary real); $\exp(\sqrt{\log x})$ and $x' = e^{x \log x}$ have no principal part with respect to the scale of the $x^\alpha (\log x)^\beta$, nor with respect to the scale of the $\exp(p(x))$ ($p$ a polynomial with no constant term).

The concept of principal part admits extensive generalization. Suppose that a function $f \in \mathscr{H}(\mathfrak{F}, V)$ has a principal part $a_1 g_1$ with respect to a scale $\mathscr{E}$; the relation $f \sim a_1 g_1$ is equivalent to $f - a_1 g_1 \ll g_1$ (V, p. 216, def. 4); to study the function $f$ more closely one is thus led to consider the function $f - a_1 g_1$. If this function has a principal part $a_2 g_2$ with respect to $\mathscr{E}$ one must have $g_2 \ll g_1$ and $f - a_1 g_1 - a_2 g_2 \ll g_2$.

More generally, suppose that the scale $\mathscr{E}$ is written parametrically in the form $(g_\alpha)$ where $\alpha$ runs through a set of indices $A$ endowed with a totally ordered structure isomorphic to the opposite of the order structure of $\mathscr{E}$: the relation $\alpha < \beta$ is thus equivalent to $g_\beta \ll g_\alpha$. In these circumstances: