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2 Asymptotic ApproximationsAreas

§2.1 Definitions and Elementary Properties

Contents

§2.1(i) Asymptotic and Order Symbols

Let X be a point set with a limit point c. As xc in X

2.1.1 f(x)ϕ(x) f(x)/ϕ(x)1.
2.1.2 f(x)=o(ϕ(x)) f(x)/ϕ(x)0.
2.1.3 f(x)=O(ϕ(x)) |f(x)/ϕ(x)| is bounded.

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

Examples

2.1.4 tanhxx,
x0 in .
2.1.5 -x=o(1),
x+ in .
2.1.6 sin(πx+x-1)=O(x-1),
x± in .
2.1.7 x=O(1),
x.

In (2.1.5) can be replaced by any fixed ray in the sector |phx|<12π, or by the whole of the sector |phx|12π-δ. (Here and elsewhere in this chapter δ is an arbitrary small positive constant.) But (2.1.5) does not hold as x in |phx|<12π (for example, set x=1+t and let t±.)

If s=0aszs converges for all sufficiently small |z|, then for each nonnegative integer n

2.1.8 s=naszs=O(zn),
z0 in .

Example

The symbols o and O can be used generically. For example,

2.1.10 o(ϕ) =O(ϕ),
o(ϕ)+o(ϕ) =o(ϕ),

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 f(x) is continuous and f(x)xν as x+ in , where ν () is a constant. Then

2.1.11 xf(t)t -xν+1ν+1,
ν<-1,
2.1.12 f(x)x {a constant,ν<-1,lnx,ν=-1,xν+1/(ν+1),ν>-1.

Differentiation requires extra conditions. For example, if f(z) is analytic for all sufficiently large |z| in a sector S and f(z)=O(zν) as z in S, ν being real, then f(z)=O(zν-1) as z in any closed sector properly interior to S and with the same vertex (Ritt’s theorem). This result also holds with both O’s replaced by o’s.

§2.1(iii) Asymptotic Expansions

Let asx-s be a formal power series (convergent or divergent) and for each positive integer n,

2.1.13 f(x)=s=0n-1asx-s+O(x-n)

as x in an unbounded set X in or . Then asx-s is a Poincaré asymptotic expansion, or simply asymptotic expansion, of f(x) as x in X. Symbolically,

2.1.14 f(x)a0+a1x-1+a2x-2+,
x in X.

Condition (2.1.13) is equivalent to

2.1.15 xn(f(x)-s=0n-1asx-s)an,
x in X,

for each n=0,1,2,. If asx-s converges for all sufficiently large |x|, then it is automatically the asymptotic expansion of its sum as x in .

If c is a finite limit point of X, then

2.1.16 f(x)a0+a1(x-c)+a2(x-c)2+,
xc in X,

means that for each n, the difference between f(x) and the nth partial sum on the right-hand side is O((x-c)n) as xc in X.

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 O 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 a0,a1,a2,, and suitably restricted X there is an infinity of analytic functions f(x) such that (2.1.14) and (2.1.16) apply. For (2.1.14) X can be the positive real axis or any unbounded sector in of finite angle. As an example, in the sector |phz|12π-δ (<12π) each of the functions 0,-z, and -z (principal value) has the null asymptotic expansion

2.1.17 0+0z-1+0z-2+,
z.

§2.1(iv) Uniform Asymptotic Expansions

If the set X in §2.1(iii) is a closed sector αphxβ, then by definition the asymptotic property (2.1.13) holds uniformly with respect to phx[α,β] as |x|. The asymptotic property may also hold uniformly with respect to parameters. Suppose u is a parameter (or set of parameters) ranging over a point set (or sets) U, and for each nonnegative integer n

|xn(f(u,x)-s=0n-1as(u)x-s)|

is bounded as x in X, uniformly for uU. (The coefficients as(u) may now depend on u.) Then

2.1.18 f(u,x)s=0as(u)x-s

as x in X, uniformly with respect to uU.

Similarly for finite limit point c in place of .

§2.1(v) Generalized Asymptotic Expansions

Let ϕs(x), s=0,1,2,, be a sequence of functions defined in X such that for each s

2.1.19 ϕs+1(x)=o(ϕs(x)),
xc in X,

where c is a finite, or infinite, limit point of X. Then {ϕs(x)} is an asymptotic sequence or scale. Suppose also that f(x) and fs(x) satisfy

2.1.20 f(x)=s=0n-1fs(x)+O(ϕn(x)),
xc in X,

for n=0,1,2,. Then fs(x) is a generalized asymptotic expansion of f(x) with respect to the scale {ϕs(x)}. Symbolically,

2.1.21 f(x)s=0fs(x); {ϕs(x)},
xc in X.

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).