# §2.1 Definitions and Elementary Properties

## §2.1(i) Asymptotic and Order Symbols

Let be a point set with a limit point . As in

2.1.1
2.1.2
2.1.3

The symbol can also apply to the whole set , and not just as .

### ¶ Examples

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

### ¶ Example

The symbols and can be used generically. For example,

2.1.10
,
,

it being understood that these equalities are not reversible. (In other words here really means .)

## §2.1(ii) Integration and Differentiation

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.

## §2.1(iii) Asymptotic Expansions

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,

2.1.14 in .

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.

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

2.1.17.

## §2.1(iv) Uniform Asymptotic Expansions

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 .

## §2.1(v) Generalized Asymptotic Expansions

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,

As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. For an example see §14.15(i).

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).