Definition 2

One says that a function $f \in \mathscr{H}(\mathfrak{F}, V)$ admits an asymptotic expansion to precision $g_\alpha$ (relative to the scale $\mathscr{E}$*) if there exists a family $(a_\lambda)_{\lambda \leq \alpha}$ of elements of $V$, all but a finite number of them being $0$, such that $f - \sum_{\lambda \leq \alpha} a_\lambda g_\lambda \ll g_\alpha$. One says that $\sum_{\lambda \leq \alpha} a_\lambda g_\lambda$ is an asymptotic expansion of $f$ to precision $g_\alpha$, that the $a_\lambda g_\lambda$ ($\lambda \leq \alpha$) are its terms, the $a_\lambda$ its coefficients, and the function $r_\alpha = f - \sum_{\lambda \leq \alpha} a_\lambda g_\lambda$ the remainder of this expansion.

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 $(x - c)^n$.