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14 1. Mathematical preliminaries

Now, with respect to an asymptotic sequence (that is, using the chosen sequence),
we may write down a set of terms, such as (1.34); this is called an asymptotic expansion.
We now give a formal definition of an asymptotic expansion (which is usually credited
to Henri Poincaré (1854–1912)).

Definition (asymptotic expansion)

The series of terms written as

where the are constants, is an asymptotic expansion of f(x), with respect to the
asymptotic sequence if, for every

If this expansion exists, it is unique in that the coefficients, are completely
determined.

There are some comments that we should add in order to make clear what this defi-
nition says and implies—and what it does not.
First, given only a function and a limit of interest (i.e. f (x) and the asymp-
totic expansion is not unique; it is unique (if it exists—we shall comment on this
shortly) only if the asymptotic sequence is also prescribed. To see that this is the case,
let us consider our function sin(3x) again; we will demonstrate that this can be repre-
sented, as in any number of different ways, by choosing different asymptotic
sequences (although, presumably, we would wish to use the sequence which is the
simplest). So, for example,

indeed, this last example, is a familiar identity for sin(3x). (Another simple example of
this non-uniqueness is discussed in Q1.10.) So, given a function and the limit, we need
to select an appropriate asymptotic sequence—appropriate because, for some choices,
the asymptotic expansion does not exist.