Discrepancies from Asymptotic Series and Their Relation to Complex Classical Trajectories

There exist functions $F\left(x\right)\mathrm{}$ whose asymptotic expansions, when computed at fixed $x$ and optimal order, seem to converge to a numerical value which deviates significantly from $F\left(x\right)\mathrm{}$ itself. Several examples are given, showing that this phenomenon is not exceptional, and should occur in quantum theory. In particular, the semiclassical expansion, at large quantum numbers, of levels of the quartic oscillator $V=x^4$ presents such discrepancies, which we explain quantitatively as contributions from classical trajectories for which space and time coordinates become complex.