Asymptotic Analysis and Singular Perturbation Theory

Chapter 2
Asymptotic Expansions
In this chapter, we define the order notation and asymptotic expansions. For additional
discussion, see [4], and [17].
2.1 Order notation
The O and o order notation provides a precise mathematical formulation of ideas
that correspond — roughly — to the ‘same order of magnitude’ and ‘smaller order
of magnitude.’ We state the definitions for the asymptotic behavior of a real-valued
function f(x) as x → 0, where x is a real parameter. With obvious modifications,
similar definitions apply to asymptotic behavior in the limits x → 0 + , x → x0,
x → ∞, to complex or integer parameters, and other cases. Also, by replacing | · |
with a norm, we can define similiar concepts for functions taking values in a normed
linear space.
Definition 2.1 Let f, g : R \ 0 → R be real functions. We say f = O(g) as x → 0
if there are constants C and r > 0 such that
|f(x)| ≤ C|g(x)| whenever 0 < |x| < r.
We say f = o(g) as x → 0 if for every δ > 0 there is an r > 0 such that
|f(x)| ≤ δ|g(x)| whenever 0 < |x| < r.
If g = 0, then f = O(g) as x → 0 if and only if f/g is bounded in a (punctured)
neighborhood of 0, and f = o(g) if and only if f/g → 0 as x → 0.
We also write f ≪ g, or f is ‘much less than’ g, if f = o(g), and f ∼ g, or f is
asymptotic to g, if f/g → 1.
Example 2.2 A few simple examples are:
(a) sin 1/x = O(1) as x → 0
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