This expansion gives the asymptotic behavior of the integral I0, i.e. its behavior for large positive values of cd. Results such as (1.6), in which we know only that the remainder terra possesses at most a certain suitable order of magnitude, belong to the first part of asymptotics, which I call "pure asymptotics", or the asymptotics of Poincare*, as it was this mathematician who found for asymptotic developments of a particular kind a rigorous mathematical basis. It is true that for the evaluation of the integral Io for a certain value of ω , say as -10, with a prescribed degree of accuracy, the above result is not absolutely reliable, due to the fact that we do not know a numerical upper bound for the absolute value of the remainder term. It is for this reason that some mathematicians are not interested in expansions of this kind;they are only satisfied if they have at their disposal a sharp upper bound for the absolute value of the remainder term. I do not agree with that severe point of view.
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