Hyperasymptotics

Hyperasymptotics is a framework in asymptotic analysis that provides more accurate approximations to functions and integrals by going beyond standard asymptotic expansions. It systematically analyzes the remainder after an optimal truncation of an asymptotic series and expresses that remainder as a new, often smaller, asymptotic expansion. This process can be iterated, yielding a nested sequence of expansions sometimes referred to as hyperasymptotic expansions or transseries.

Origin and development: The approach arose in the late 20th century in the work of researchers such as Jonathan S. Howls and colleagues, as part of the broader development of resurgence theory and the understanding of Stokes phenomena within asymptotics. It is related to, but distinct from, Borel resummation and ordinary asymptotics, and it provides a practical way to capture terms that standard methods miss.

Methodology: The procedure begins with an asymptotic expansion for a quantity as a parameter becomes large. One truncates at an optimal order to minimize the leading remainder. The remainder is then represented by an integral or a dominant late-term series, often exhibiting factorial-over-power growth, which is itself approximated by a secondary asymptotic expansion. This secondary expansion captures exponentially small corrections associated with the nearest singularities or saddles. The process can be repeated to obtain higher levels of refinement, yielding a hierarchy of corrections valid across Stokes lines.

Applications and scope: Hyperasymptotics has been applied to integrals, differential equations, wave problems, and a range of special functions such as Bessel, Airy, and gamma functions. It is valued for producing highly accurate approximations in regimes where conventional asymptotics fail or exhibit rapid changes.

See also: asymptotic expansion, Stokes phenomenon, resurgence, transseries, terminant.