ABSTRACT
Under appropriate smoothness and boundedness conditions, the integral!(..\) = I: h(..\t)j(t) dt bas the asymptotic expansion
gral!(..\) = I: ei~t f(t) dt (which is a special case of Special Case 1), has the asymptotic expansion
gral!(..\) = I: e-~t f(t) dt (which is a special case of Special Case 1), bas the asymptotic expansion
Watson's Lemma (Bleistein and Handelsman [2], page 103, or Wong [5], page 20): If /(t) is locally absolutely integrable on (O,oo), as t-+ oo, /(t) = O(e0 ') for some real number a, and, as t -+ 0+, /(t) "' E:=0 c,.t•m, where Re(am)
43. Asymptotic Expansions
Theorem (Bleistein and Handelsman [2], page 120): Let h(t) and /(t) be sufficiently smooth functions on the infinite interval {0, oo) having the asymptotic forms
with some conditions on the range of the parameters appearing in the expansion. Let the Mellin transforms of h and I be denoted by M[h; z] and M[/; z] (see the Notes). If some technical conditions are satisfied, then
represents a finite asymptotic expansion as ~ -+ oo with respect to the asymptotic sequence p-oi(log~)nrm}. The expression in {43.2) represents a sum of the residues over all of the poles in a specific region of the complex plane.
