By Michael Berry (auth.), Harvey Segur, Saleh Tanveer, Herbert Levine (eds.)

An asymptotic growth is a sequence that offers a chain of more and more actual approximations to a functionality in a selected restrict. The formal definition, given through Poincare (1886, Acta Math. 8:295), is as follows. Given a function,

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Let us show how it works in the present simple case. (28), (29)). (32)) is therefore strongly divergent. In order to proceed we follow Borel's strategy(12) and define a new series: B(x) = ~ an tl (2n)! (29): (36) I I B(x) is therefore a convergent series for x < 1 and it defines an analytic function there. (36) shows that the closest singularity to zero is located at x = - 1. 1/2 (37) In order to go backward from B(x) to (z)+ln(-z), we define (38) We then identify 'P(z) and (z)+ln(-z) which satisfy the same differential equation and have the same asymptotic series on the real negative axis.

Lower case letters denote negative values of ~, and upper case letters denote positive~. 1 < 2- 19. 2 translating letters into decimal ranges. 2 should be lower case, because ~(k,O) < 0 for all k, as proved in [2]. 209 x 10-13 (obtained by DEND) has been swamped by STIFF's output error of magnitude 6 x 10-8 • Not only do the true values of t::. 389 x 1O-3942}, but STIFF also finds it harder to control the increasing frequencies of spurious solutions (fo~ example STIFF's output errors have risen to magnitude 10-6 when k = -15); so there is little point in running STIFF for k < - 10.

This is not because higher approximations give higher powers of 8 but because they are suppressed by powers of n (see below). 13): r .. 4) and depends only on the limiting behavior of the most diverging term of en near i1t/2. The use of gO is sufficient because the contributions of the higher order contributions have exactly the same form but with n in the argument of the r function replaced by a larger integer. Therefore as mentionned above these contributions are suppressed by powers of n compared to the one that has been retained and do not contribute to the result in the limit n~oo.

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