# §8.20 Asymptotic Expansions of

## §8.20(i) Large

As

and

again denoting an arbitrary small positive constant. Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii).

For an exponentially-improved asymptotic expansion of see §2.11(iii).

## §8.20(ii) Large

For and let and define ,

so that is a polynomial in of degree when . In particular,

8.20.5

Then as

uniformly for .

For further information, including extensions to complex values of and , see Temme (1994b, §4) and Dunster (1996b, 1997).