§ 2. Asymptotic expansions

1. Scales of comparison

Let E be a set filtered by a filter with base $\mathfrak{F}$, and K a non-discrete valued field (most often $K = \mathbf{R}$ or $K = \mathbf{C}$). In the set of functions in $\mathscr{H}(\mathfrak{F}, K)$ not equivalent to 0 modulo $R_\infty$ (that is, those for which in every set in $\mathfrak{F}$ there is at least one point where the function does not vanish), the relation " $f \ll g$ or $f = g$" is an order relation.

In other words, if $f$ and $g$ are functions in $\mathscr{E}$ then one (and only one) of the relations $f \ll g$, $g \ll f$, $f = g$ always holds. It follows that on $\mathscr{E}$ the relation $f \asymp g$ (and a fortiori $|f| \sim a\,|g|$, where $a$ is a number $> 0$) implies $f = g$.

Every subset of a comparison scale is clearly also a comparison scale.

2. Principal parts and asymptotic expansions

Let $\mathscr{E}$ be a comparison scale formed by functions with values in a non-discrete valued field K. Let V be a normed space over K, and let $\mathbf{f}$ be a function in $\mathscr{H}(\mathfrak{F}, V)$; if there are a function $g \in \mathscr{E}$ and an element $\mathbf{a} \neq 0$ in V such that $\mathbf{f} \sim \mathbf{a}g$, one says that $\mathbf{a}g$ is a principal part of $\mathbf{f}$ relative to the scale $\mathscr{E}$. From def. 1 of V, p. 10, $\mathbf{f}$ can have only one principal part relative to $\mathscr{E}$, for if $g_1,\ g_2$ are two functions in $\mathscr{E}$ and $\mathbf{a}_1,\ \mathbf{a}_2$ are two $\neq 0$ elements of V, then the relation $\mathbf{a}_1 g_1 \sim \mathbf{a}_2 g_2$ implies $|g_1| \asymp |g_2|$ and consequently $g_1 = g_2$, whence $(\mathbf{a}_2 - \mathbf{a}_1)g_1 \ll g_1$, and since $g_1$ is not identically zero on any set in $\mathscr{E}$ this implies $\mathbf{a}_2 = \mathbf{a}_1$.

If $\mathbf{f}$ has a principal part relative to a comparison scale $\mathscr{E}$ it has the same principal part relative to every comparison scale $\mathscr{E}' \supset \mathscr{E}$.

To express the fact that $\sum_{\lambda \leq \alpha} a_\lambda g_\lambda$ is an asymptotic expansion of $f$ to precision $g_\alpha$ one most often restricts oneself to writing

if there are several functions in the proof) following the notation of V, p. 219 and 220.

Of two asymptotic expansions (of two functions, distinct or not) relative to the same scale $\mathscr{E}$, one says that the one with precision of greater index is the more precise.

If $\sum_{\lambda \leq \alpha} a_\lambda g_\lambda$ is an asymptotic expansion of $f$ to precision $g_\alpha$, then, for every $\beta < \alpha$, $\sum_{\lambda \leq \beta} a_\lambda g_\lambda$ is an asymptotic expansion of $f$ to precision $g_\beta$ (V, p. 215, prop. 5): one says that it is obtained by reducing the precision of the given expansion $\sum_{\lambda \leq \alpha} a_\lambda g_\lambda$ of $f$ to $g_\beta$.

If $\sum_{\lambda \leq \alpha} a_\lambda g_\lambda$ and $\sum_{\lambda \leq \alpha} b_\lambda g_\lambda$ are asymptotic expansions to the same precision $g_\alpha$ of two functions $f_1, f_2$, then $\sum_{\lambda \leq \alpha} (a_\lambda + b_\lambda) g_\lambda$ is an asymptotic expansion of $f_1 + f_2$ to precision $g_\alpha$ (V, p. 215, prop. 5); and for every scalar $c$, $\sum_{\lambda \leq \alpha} a_\lambda c g_\lambda$ is an asymptotic expansion of $f_1 c$ to precision $g_\alpha$. It follows that if a function admits an asymptotic expansion to precision $g_\alpha$ then this expansion is unique: it is sufficient to see that the function 0 does not admit an asymptotic expansion with precision $g_\alpha$ having coefficients $\neq 0$. For, if $0 = \sum_{\lambda \leq \alpha} a_\lambda g_\lambda + r_\alpha$, and if $\gamma$ were the least of the indices $\lambda \leq \alpha$ such that $a_\lambda \neq 0$, one would have $a_\gamma g_\gamma = - \sum_{\gamma < \lambda \leq \alpha} a_\lambda g_\lambda - r_\alpha \ll g_\gamma$, which is absurd.

To say that a function $f$ admits an asymptotic expansion to precision $g_\alpha$, all of whose coefficients are zero, is equivalent to saying that $f \ll g_\alpha$. If $f$ admits an asymptotic expansion $\sum_{\lambda \leq \alpha} a_\lambda g_\lambda$ to precision $g_\alpha$ whose coefficients are not all zero, and if $\gamma$ is the smallest of the indices $\lambda$ such that $a_\lambda \neq 0$, then $a_\gamma g_\gamma$ is the principal part of $f$ relative to the scale $\mathscr{E}$, for one has $f - a_\gamma g_\gamma = \sum_{\gamma < \lambda \leq \alpha} a_\lambda g_\lambda + r_\alpha \ll g_\gamma$; similarly, if $\mu \leq \alpha$ is an index such that $a_\mu \neq 0$, then $a_\mu g_\mu$ is the principal part of $f - \sum_{\lambda < \mu} a_\lambda g_\lambda$.

The most important asymptotic expansions in applications are those relative to the scale of the $x^{-n}$ (resp. of the $z^{-n}$), where $n$ is a positive or negative integer, as $x$ tends to $+\infty$ or to $-\infty$ (resp. when the complex number $z$ tends to $\infty$), or relative to the scale of the $(x - c)^n$ (resp. $(z - c)^n$) when the real number $x$ tends to $c$ from the right or left (resp. when the complex number $z$ tends to $c$). We saw in I, p. 21 that every vector function of a real variable $x$ which is $k$ times differentiable at a point $c \in \mathbf{R}$ admits a Taylor expansion of order $k$ at this point, that is, an asymptotic expansion to precision $(x - c)^k$ with respect to the scale of the $(x - c)^n$.

3. Sums and products of asymptotic expansions

If $f_1, f_2$ admit asymptotic expansions to precision $g_\alpha$ and $g_\beta$ respectively, relative to a comparison scale $\mathscr{E}$, one deduces expansions to precision $g_{\min(\alpha, \beta)}$ by limiting the two expansions to this precision; we have seen in V, p. 222 how one obtains an asymptotic expansion for $f_1 + f_2$ to precision $g_{\min(\alpha, \beta)}$.

Let $V_1, V_2$ and $V$ be three normed spaces over the field $K$, and let $(\mathbf{x}, \mathbf{y}) \mapsto [\mathbf{x}. \mathbf{y}]$ be a continuous bilinear map from $V_1 \times V_2$ into $V$; we suppose further, for the rest of this section, that the scale $\mathscr{E}$ is such that the product of any two functions in $\mathscr{E}$ is again in $\mathscr{E}$ (which is true for all the comparison scales given as examples (in V, p. 220)).

Now let $f_1, f_2$ be two functions in $\mathscr{H}(\mathfrak{F}, V_1)$ and $\mathscr{H}(\mathfrak{F}, V_2)$ respectively, having asymptotic expansions $f_1 = \sum_{\lambda \leq \alpha} a_\lambda g_\lambda + r_\alpha$ and $f_2 = \sum_{\mu \leq \beta} b_\mu g_\mu + r_\beta$ to precision $g_\alpha$ and $g_\beta$ respectively, with respect to the scale $\mathscr{E}$. Suppose further that neither the $a_\lambda$ nor the $b_\mu$ are all zero, and let $a_\gamma g_\gamma$ and $b_\delta g_\delta$ be the principal parts of $f_1$ and $f_2$. By hypothesis, one can write $g_\gamma g_\beta = g_\rho$ and $g_\delta g_\alpha = g_\sigma$; let us show that the sum $\sum [\mathbf{a}_\lambda . \mathbf{b}_\mu] g_\lambda g_\mu$ taken over all pairs $(\lambda, \mu)$ such that $g_{\min(\rho, \sigma)} \ll g_\lambda g_\mu$, is an asymptotic expansion of $[f_1.f_2]$ to precision $g_{\min(\rho, \sigma)}$. Now the difference between $[f_1.f_2]$ and this sum is the sum of a finite number of terms, each of which is either of the form $[\mathbf{a}_\lambda . \mathbf{b}_\mu] g_\lambda g_\mu$ with $g_\lambda g_\mu \ll g_{\min(\rho, \sigma)}$, or of the form $[\mathbf{a}_\lambda . \mathbf{r}_\beta] g_\lambda$ where $\lambda \geq \gamma$, or of the form $[\mathbf{r}_\alpha . \mathbf{b}_\mu] g_\mu$ where $\mu \geq \delta$; but since $[\mathbf{x}. \mathbf{y}]$ is continuous, one has from (V, p. 213, prop. 3 and V, p. 215, prop. 6) that $[\mathbf{a}_\lambda . \mathbf{r}_\beta] g_\lambda \preccurlyeq \mathbf{r}_\beta g_\lambda \ll g_\beta g_\gamma = g_\rho$ for $\lambda \geq \gamma$, and similarly $[\mathbf{r}_\alpha . \mathbf{b}_\mu] g_\mu \preccurlyeq \mathbf{r}_\alpha g_\mu \ll g_\alpha g_\delta = g_\sigma$ for $\mu \geq \delta$, whence the proposition (V, p. 215, prop. 5).

If all the $\mathbf{a}_\lambda$ are zero one has $[f_1.f_2] \ll g_\alpha g_\delta$: in other words, one has an asymptotic expansion of $[f_1.f_2]$ with zero terms, to precision $g_\alpha g_\delta$; similarly if all the $\mathbf{a}_\lambda$ and $\mathbf{b}_\mu$ are zero one has an asymptotic expansion of $[f_1.f_2]$ with zero terms to precision $g_\alpha g_\beta$.

We shall apply the preceding result principally to the case where V is a normed algebra over K and the bilinear function $[\mathbf{x}. \mathbf{y}]$ is the product xy in this algebra; the most important cases are those where V is equal to $\mathbf{R}$ or to $\mathbf{C}$.

In particular, if $f_i$ ($1 \leq i \leq n$) are n functions in $\mathscr{H}(\mathfrak{F}, K)$ each of which admits an asymptotic expansion with respect to $\mathscr{E}$ one can obtain an asymptotic expansion with respect to $\mathscr{E}$ for every polynomial $\sum_{(v_i)} \mathbf{a}_{v_1 v_2} v_n f_1^{v_1} \ldots f_n^{v_n}$ in the $f_i$ with coefficients in a normed space V; furthermore, the preceding rules allow one to determine the precision of the expansion obtained, if one knows those of the expansions of the functions $f_i$.

4. Composition of asymptotic expansions

Let $f$ be a function in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$ (resp. $\mathscr{H}(\mathfrak{F}, \mathbf{C})$), admitting an asymptotic expansion to precision $g_\alpha$ with respect to a scale $\mathscr{E}$, and having limit 0 along the filter with base $\widetilde{\mathfrak{F}}$. On the other hand let $\mathbf{h}$ be a function with values in a normed space V over $\mathbf{R}$ (resp. $\mathbf{C}$), defined on a neighbourhood of the point 0 in $\mathbf{R}$ (resp. $\mathbf{C}$), and n times differentiable on this neighbourhood; then

on this neighbourhood (I, p. 21), whence, on a suitable set in $\widetilde{\mathfrak{F}}$

We have seen, in n 3, how to form an asymptotic expansion for $c_0 + c_1 f + \cdots + c_n f^n$ to a precision $g_\rho$ determined by the precision of the expansion of $f$; moreover, suppose that the coefficients of the asymptotic expansion of $f$ are not all zero, and that $a_\gamma g_\gamma$ is the principal part of $f$, and let $g_\sigma = g_\gamma^n$; if $\sigma < \rho$ one will have an expansion of $\mathbf{h} \circ f$ to precision $g_\sigma$ on limiting the expansion of $\sum_{k=0}^n c_k f^k$ to this precision; if, on the contrary, $\rho \leq \sigma$, then the expansion of $\sum_{k=0}^n c_k f^k$ is also an expansion of $\mathbf{h} \circ f$ to precision $g_\rho$.

If all the terms of the asymptotic expansion of $f$ are zero, and if $g_\alpha \ll 1$, then $f \ll g_\alpha$ and so $f^k \ll g_\alpha^k \ll g_\alpha$ for every integer $k > 0$; if $c_m$ is the first coefficient of index $> 0$ which is not zero (assuming that the $c_k$ for indices $k > 0$ are not all 0), then $c_0$ is an asymptotic expansion of $h \circ f$ to precision $g_\alpha^m$.

In the remainder of this section we shall restrict ourselves to the case where the functions in $\mathscr{E}$ have real values and are strictly positive on a set in $\mathfrak{F}$, and we shall consider asymptotic expansions only of functions in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$. Suppose first that for every function $g \in \mathscr{E}$ and every real number $\nu$, $g^\nu$ again belongs to $\mathscr{E}$: this condition is fulfilled, for example, by the scale of the $x^\alpha$ or by that of the $x^\alpha |\log x|^\beta$ ($\alpha$ and $\beta$ being arbitrary real numbers) on a neighbourhood of $+\infty$ or a neighbourhood of 0 in $\mathbf{R}$. This property implies that the quotient of two functions in $\mathscr{E}$ again belongs to $\mathscr{E}$. This being so, from an asymptotic expansion relative to $\mathscr{E}$ of a function $f \in \mathscr{H}(\mathfrak{F}, \mathbf{R})$, to precision $g_\alpha$, one can derive an expansion for $|f|^\nu$ for every real number $\nu$. Let us restrict ourselves to the case where the coefficients of the expansion of $f$ are not all zero, and let $a_\gamma g_\gamma$ be the principal part of $f$; one can write $|f|^\nu = |a_\gamma|^\nu g_\gamma^\nu (1 + h)^\nu$, with

With the same hypotheses on $f$ one can write

5. Asymptotic expansions with variable coefficients

One can generalize the concept of principal part, and that of an asymptotic expansion, in the following way. Let $\mathscr{E}$ be a comparison scale formed of real (resp. complex) functions such that, for each of them, there is a set in $\mathfrak{F}$ on which the function does not vanish at any point. Further, let $C$ be a set of functions in $\mathscr{H}(\mathfrak{F}, V)$ satisfying the following three conditions:

(CO₁) For every function $a \in C$ one has $a \preccurlyeq 1$. (CO₂) The relation $a \ll 1$ for a function $a \in C$ implies $a = 0$. (CO₃) $C$ is a vector space over $\mathbf{R}$ (resp. $\mathbf{C}$).

Now let $f$ be any function in $\mathscr{H}(\mathfrak{F}, V)$; if there exist a function $g \in \mathscr{E}$ and a nonzero function $a \in C$ such that $f - ag \ll g$ one will say that $ag$ is a principal part of $f$, relative to the comparison scale $\mathscr{E}$ and to the domain of coefficients $C$. If such a principal part exists, it is unique: for suppose that there are two principal parts $a_1 g_1$ and $a_2 g_2$; one cannot have $g_1 \ll g_2$ since from (CO₁) one could deduce that $a_1 g_1 \ll g_2$ and $f - a_1 g_1 \ll g_1 \ll g_2$, so $f \ll g_2$; but one would also have $a_2 g_2 \ll g_2$ and consequently $a_2 \ll 1$ contradicting the hypothesis $a_2 \neq 0$ and (CO₂). So one must have $g_1 = g_2$; from the relations $f - a_1 g_1 \ll g_1, f - a_2 g_1 \ll g_1$ one deduces that $(a_2 - a_1)g_1 \ll g_1$ whence $a_2 - a_1 \ll 1$, and consequently $a_2 = a_1$, by (CO₂) and (CO₃).