Hyperasymptotics and the Linear Boundary Layer Problem: Why Asymptotic Series Diverge

The simplest problem with boundary layers, $\epsilon^{2} u_{xx} - u = - f(x)$, is used to illustrate (i) why the perturbation series in powers of $\epsilon$ is asymptotic but divergent, (ii) why the optimally truncated expansion is ``superasymptotic'' in the sense that that error is proportional to $\exp(- \mbox{[constant]} / \epsilon)$, and (iii) how to obtain an improved "hyperasymptotic" approximation.

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