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This question is a generalized version of another question. My old question asked about bounds for the Stirling series, while this question concerns Laplace's method in general.

Laplace's method for the leading order term is described well on the relevant Wikipedia page. This method can be pushed to yield a full asymptotic series. One reference where this is described is Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory by Bender and Orszag. A proof is given in volume 2B of Simon's A Comprehensive Course in Analysis. An example of such an expansion is given in the question linked above.

Are any rigorous quantitative estimates known on the relative error of such an expansion, truncated at some arbitrary term? This seems like a fairly obvious question to ask about, yet I couldn't find any results after searching for a while.

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    I am interested in an answer to this question as well. One thing which may cause some trouble is that two different functions may have the same asymptotic expansion, so the truncation error can probably not just be "read" from the asymptotic series itself. It may require additional information about the function which is being expanded. – 2017-01-31
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    I would also add that any proof of the Laplace method expansion can be converted to an explicit error bound by resisting the urge to use O notation throughout the proof (since the O notation bounds are themselves proved by explicit estimates). – 2017-01-31

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