Let be a point set with a limit point . As in
The symbol can also apply to the whole set , and not just as .
In (2.1.5) can be replaced by any fixed ray in the sector , or by the whole of the sector . (Here and elsewhere in this chapter is an arbitrary small positive constant.) But (2.1.5) does not hold as in (for example, set and let .)
If converges for all sufficiently small , then for each nonnegative integer
The symbols and can be used generically. For example,
it being understood that these equalities are not reversible. (In other words here really means .)
Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. For example, suppose is continuous and as in , where () is a constant. Then
Differentiation requires extra conditions. For example, if is analytic for all sufficiently large in a sector and as in , being real, then as in any closed sector properly interior to and with the same vertex (Ritt’s theorem). This result also holds with both ’s replaced by ’s.
Let be a formal power series (convergent or divergent) and for each positive integer ,
as in an unbounded set in or . Then is a Poincaré asymptotic expansion, or simply asymptotic expansion, of as in . Symbolically,
Condition (2.1.13) is equivalent to
for each . If converges for all sufficiently large , then it is automatically the asymptotic expansion of its sum as in .
If is a finite limit point of , then
means that for each , the difference between and the th partial sum on the right-hand side is as in .
Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series. These include addition, subtraction, multiplication, and division. Substitution, logarithms, and powers are also permissible; compare Olver (1997b, pp. 19–22). Differentiation, however, requires the kind of extra conditions needed for the symbol (§2.1(ii)). For reversion see §2.2.
Some asymptotic approximations are expressed in terms of two or more Poincaré asymptotic expansions. In those cases it is usually necessary to interpret each infinite series separately in the manner described above; that is, it is not always possible to reinterpret the asymptotic approximation as a single asymptotic expansion. For an example see (2.8.15).
Asymptotic expansions of the forms (2.1.14), (2.1.16) are unique. But for any given set of coefficients , and suitably restricted there is an infinity of analytic functions such that (2.1.14) and (2.1.16) apply. For (2.1.14) can be the positive real axis or any unbounded sector in of finite angle. As an example, in the sector () each of the functions , and (principal value) has the null asymptotic expansion
If the set in §2.1(iii) is a closed sector , then by definition the asymptotic property (2.1.13) holds uniformly with respect to as . The asymptotic property may also hold uniformly with respect to parameters. Suppose is a parameter (or set of parameters) ranging over a point set (or sets) , and for each nonnegative integer
is bounded as in , uniformly for . (The coefficients may now depend on .) Then
as in , uniformly with respect to .
Similarly for finite limit point in place of .
Let , , be a sequence of functions defined in such that for each
where is a finite, or infinite, limit point of . Then is an asymptotic sequence or scale. Suppose also that and satisfy
for . Then is a generalized asymptotic expansion of with respect to the scale . Symbolically,
Care is needed in understanding and manipulating generalized asymptotic expansions. Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. It can even happen that a generalized asymptotic expansion converges, but its sum is not the function being represented asymptotically; for an example see §18.15(iii).