Asymptotic expansions of exp-log functions

Extended admissible functions and Gaussian limiting distributions

“…For example, if f is an exp-log function (that is, it is built by finitely many compositions of rational functions as well as e z and log z), then we can apply the results and implementations for multiseries inversion and substitution by Salvy and Shackell [SS99] (cf. also Richardson et al [RSSVdH96]) in order to compute automatically an asymptotic expression for r n and, consequently, for the mean and variance if they exist. (The problem is that not every exp-log function has an inverse that is asymptotically equal to an exp-log functionl; see Shackell [Sha93].…”

Symbolic Asymptotics: Multiseries of Inverse Functions

“…We now give more precise definitions for these notions, very similar to those in Richardson et al (1996). Definition 1.1.…”

“…with U = log log x 2 − 1 8 8 log log x + 1 + 1 8 , defined as the inverse function of 2y(x) + y(x) 1/2 composed with log log x. Multiseries were introduced in Van der Hoeven (1997) and Richardson et al (1996). Other names for these kinds of asymptotic expansions or very similar ones are hyperasymptotics (Berry and Howls, 1990), exponential asymptotics (Meyer, 1980), asymptotics beyond all orders (Costin and Kruskal, 1996).…”

“…Multiseries seem to have some advantages over nested expansions especially in the way in which results are presented, although we would claim that nested expansions also have advantages; in particular they are canonical and they often make it easier to develop and prove algorithms. In Richardson et al (1996) an algorithm was given to compute multiseries for the class of exp-log functions. The present paper can be viewed as the natural next step.…”

Relax, but Don’t be Too Lazy

“…, b n ∈ R + * ), then we are essentially handling power series in several variables, so we can gain on the complexity. For applications, see Richardson et al (1996) and van der Hoeven (1997a).…”