Measured limits and multiseries

Abstract

We introduce the notion of a measured limit and show how it can give meaning to asymptotic expansions in which the coefficients are variable and may even have poles. We give an algorithm for computing the measured multiseries of a large class of elementary functions.

Backward Error Analysis for Perturbation Methods

“…We thus see an example of the use of f (x 0 ) instead of just A, as discussed in section 2, to approximately double the number of correct terms in the approximation. This analysis can be implemented in Maple as follows: Note that we had to use the MultiSeries package [31] to expand the series in equation ( 79), for understanding how accurate z 2 was. z 2 is slightly more lacunary than the two-variable expansion in [5], because we have a zero coefficient for W 2 .…”

Measured Multiseries and Integration

“…Trigonometric functions whose arguments tend to infinity have long been used in expansions. [16] offered a theoretical basis for these which gave them precise meaning in terms of the asymptotics of functions, and extended the methods of [11] to such cases. By using a result from [21], it was shown that these can be applied to a large class of elementary functions, although restrictions are needed on trigonomentric functions appearing inside exponentials or other trigonometric functions.…”

“…This work provided a theoretical underpining for the Maple c implementation which Salvy was developing at that time. If H denotes any Hardy field in which multiseries can be calculated, the algorithm of [16] can easily be extended to an algebra H G , generated by H and a finite number of sines or cosines of its elements. Our concern here is to give an algorithm for computing measured multiseries when integration is added to the signature of H G .…”

“…This paper makes use of a number of results from elsewhere. Firstly we need Theorem 2 of [16], which itself makes substantial use of Theorem 7 from [21]. The latter represents the culmination of a classical line of research and provides an important stepping stone in the continuing development of the asymptotics of elementary funtions.…”

“…The definitions and basic properties of measured limits, coefficients and measured multiseries from [16] are presented. Some slight differences arise in that [16] treated asymptotics at x = 0, whereas we make the convention here that limits are at +∞. The obvious change of variable transforms between these.…”