ABSTRACT
With the foregoing conditions, the nature of the asymptotic approximation to I(x) for large x depends on the sign of L -p . When 1 < p the contribution from the endpoint a dominates, when L > p the contribution from b dominates, and when L = p the contributions from a and b are equally important. The commonest case in physical applications is L < p, and we begin with this. 13.2 Theorem 13.1 In addition to the conditions of $1 3.1, assume that 1 < p, the first of (13.02) is twice differentiable, and the second of (1 3.02) is differentiable.' Then
I ) - 2 ) 2 1 - (J g:;# (x -+ m). To prove this result we take a new integration variable v = p(t)-p(a). In con-
sequence of Condition (i), the relationship between t and v is one to one. Denote
Then
As in 57.3, Condition (ii) implies that
Moreover in the present case this relation can be differentiated. We now express
where
according as v lies inside or outside the interval (0, B). The first term on the right-hand side of (13.06) is evaluable by means of Lemma
12.1 and yields the required approximation (1 3.04). Next, Lemma 12.2 shows that
E1(x)=0(x-l) ( x + a ) . For the remaining error term, it is readily verified by reference to the given con-
ditions that the function 4(v), defined by (13.07) and (1 3.05), satisfies the conditions of Lemma 12.3 with a = Alp. Therefore
~2 ( 4 = 0 (x - I/") (x + a ) . Since A/p < 1, the estimate O(x-') for E~(x) may be absorbed in the estimate
consequence of Conditions (ii) and (iv) of $1 3.1. Equation (1 3.03) yields the following integral for the error term:
where p and f (v) are defined by (13.05). The given conditions show that this integral converges absolutely and uniformly throughout its range; accordingly the RiemannLebesgue lemma immediately yields the desired result ~ ( x ) = x-lo(]).
