ABSTRACT
From Theorem 3.1 (with z replaced by l/z) it is seen that if the series C a,z-' converges for all sufficiently large Izl, then it is the asymptotic expansion of its sum, defined in the usual way, without restriction on ph z. Naturally, however, greater interest attaches to asymptotic expansions that diverge. An example has been provided by (1.05); this is a consequence of (1.08). 7.2 Theorem 7.1 A necessary and sufficient condition thatf(z) possesses an asymptotic expansion of theform (7.03) is thatfor each nonnegative integer n
as z -+ co in R, uniformly with respect to ph z.
