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264 Approximation of Integrals

To verify the second inequality, note that
∞ ∞

2
2
2
2
(1 + 1/z ) exp(−x /2) dx ≥ (1 + 1/x )exp(−x /2) dx.
z z
Since
d 1 2 2 2
exp(−x /2) = (1 + 1/x )exp(−x /2),
dx x
∞ 1

2 2 2
(1 + 1/x )exp(−x /2) dx = exp(−z /2),
z z
which proves the result for z > 0.
The result for z < 0 follows from the fact that, for z < 0, (z) = 1 − (−z).

9.3 Asymptotic Expansions
In many cases, exact calculation of integrals is not possible and approximations are needed.
Here we consider aproximations as some parameter reaches a limiting value. In general
discussions that parameter is denoted by n and we consider approximations as n →∞;
different notation for the parameter and different limiting values are often used in specific
examples. Note that, in this context, n is not necessarily an integer.
Let f denote a real-valued function of a real variable n. The series
∞
− j
a j n
j=0
is said to be an asymptotic expansion of f if, for any m = 1, 2,...,
m
− j
f (n) = a j n + R m+1 (n)
j=0
where R m+1 (x) = O(n −(m+1) )as n →∞; that is, where n m+1 R m+1 (n) remains bounded as
n →∞. This is often written
∞
− j
f (n) ∼ a j n as n →∞.
j=0
It is important to note that this does not imply that

∞
− j
f (n) = a j n ,
j=0
which would require additional conditions on the convergence of the series. An asymptotic
expansion represents a sequence of approximations to f (n) with the property that the order
−1
of the remainder term, as a power of n ,is higher than that of the terms included in the
approximation. It is also worth noting that, for a given value of n, the approximation based
on m terms in the expansion may be more accurate than the approximation based on m + 1
terms in the approximation.
In many cases, an entire asymptotic expansion is not needed; that is, we do not need to
be able to compute an approximation to f (n) with error of order O(n −m ) for any value of