Page 282 - Elements of Distribution Theory

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052184472Xc09 CUNY148/Severini June 2, 2005 12:8

268 Approximation of Integrals

or,
1 − 1 1 1 1
z
3
¯
(z) = φ(z) z
= φ(z) − + O
1
1 + O z z 3 z 5
z 4
as z →∞.
Note that this approach may be continued indefinitely, leading to the result that
1 1 3 k (2k)! 1 1

¯
(z) = φ(z) − + + ··· + (−1) + O
k
z z 3 z 5 2 k! z 2k+1 z 2k+2
as z →∞, for any k = 0, 1, 2,....
9.4 Watson’s Lemma

Note that an integral of the form
∞

α
t exp(−nt) dt,
0
where α> −1 and n > 0, may be integrated exactly, yielding

∞ (α + 1)
α
t exp(−nt) dt = α+1 .
0 n
Now consider an integral of the form
∞

h(t)exp(−nt) dt
0
where h has an series representation of the form
∞
j
h(t) = a j t .
j=0
Then, assuming that summation and integration may be interchanged,
∞ ∞ ∞ ∞

( j + 1)
j
h(t)exp(−nt) dt = a j t exp(−nt) dt = a j .
0 j=0 0 j=0 n j+1
Note that the terms in this series have increasing powers of 1/n; hence, if n is large,
the value of the integral may be approximated by the first few terms in the series. Watson’s
lemma is a formal statement of this result.
Theorem 9.12 (Watson’s lemma). Let h denote a real-valued continuous function on
[0, ∞) satisfying the following conditions:
(i) h(t) = O(exp(bt)) as t →∞ for some constant b
(ii) there exist constants c 0 , c 1 ,..., c m+1 ,a 0 , a 1 ,..., a m+1 ,

−1 < a 0 < a 1 < ··· < a m+1
such that
m

ˆ a j
h m (t) = c j t
j=0