# Occurances of integrals of the form $Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx$ (and perturbation techniques) [closed]

I am writing a review on perturbation techniques (actually hyperasymptotic techniques) for integrals of the form $$Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx,$$ where the interest is in the small-$$\lambda$$ limit, in physics. While I have understood the mathematical details of several of these methods, I am still at a loss for the extent of their utility in 'real' physics. It is this that requires more background familiarity with physics problems, rather than just picking up a single book/paper and learning a mathematical technique.

At the moment, I have the following ideas:

• Partition functions can take the form $$Z(\beta) = \int g(E)e^{-\beta E} dE$$ for a continuous energy spectrul (or approximated as such) and $$g(E)$$ would be some degeneracy factor for the energy levels. Here you get an asymptotic series in the limit of T-> 0.
• As an infinite dimensional generalisation, path integrals, as $$\hbar \rightarrow 0$$
• The exponential generally comes in where one has a Markovian process. The Greens function often has an exponential-like form, so when convolving it with a forcing term it will give something of the above form. I can't think of a particular example for this though, and its restricted to linear ODEs (or linear in some approximation, in which case one is generaly not interested in going beyond superasymptotics anyway)

In any case, I would be interested to get an idea for just how useful hyperasymptotic methods for integrals fo the above form are, and what kind of important perturbation problems do not fall under this category (the first instance od perturbation theory that I came across, in quantum mechanics, in which we just 'match powers of the perturbative potential strength lambda', seems to be a case)

• For clarity, you're interested in an asymptotic analysis at low $\lambda$? – Emilio Pisanty Mar 7 '19 at 20:25
• @EmilioPisanty indeed! Specifically, hyperasymptoyic techniques – Meep Mar 8 '19 at 14:09
• I'm closing this question as too broad, since Laplace's method & the method of steepest descent are used in virtually all subfields of physics. – Qmechanic Mar 9 '19 at 16:55