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Laplace's method allows to get an asympotic approximation, for $N\to \infty$, of integrals of the form

$$ \int e ^{N f(x)}dx $$

I plead guilty of abusing several times (eg: exhibit 1 and exhibit 2) of the method, by using a function in the exponent that actually depends also on $N$, say $f(x,N)$.

That's obviously illegal, and, in general, will produce nonsense. But in the above examples it worked nicely for me.

I guess there should be some justification, or some known recipe for extending Laplace's method for these scenarios - perhaps imposing some restriction on $f(x,N)$ - probably that it varies "slowly" (or "asymptotically almost constant") with $N$.

Can someone devise such extension, or provide some reference?

leonbloy
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1 Answers1

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The power of the Laplace method is that it really is much more than a single theorem, and the statements on the wiki page page are specializations of the general method.

In his book Asymptotic Methods in Analysis, N. G. de Bruijn works out several examples in full detail which aren't of the usual $\int f e^{Ng}$ form. See, in particular, chapter 6.

From the preface of the book:

The reader will not find anything like a general theory in this book. Many asymptotic methods are very flexible, and in such cases it is not possible to formulate a single theorem covering all applications. Any attempt at generalization would actually result in a restriction.

I've fully worked out (relatively tame) examples on Math.SE here and here too.

Antonio Vargas
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