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I am looking for a rigorous, textbook-style proof of the Laplace approximation of general exponential integrals of the form $$ \int e^{\lambda h(\lambda,t)}g(\lambda,t)\,dt \quad\text{as}\quad \lambda\to\infty. \tag{*} $$

Every treatment I have found assumes that $h(t)$ does not depend on $\lambda$. Frustratingly, Miller's "Applied Asymptotic Analysis" introduces the subject using the general integral $(*)$, but only proves results for $h(t)$.

(There are many more resources and lecture notes online, e.g. these notes, but they only treat $h(t)$ as well.)

JohnA
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  • You may have a look at F. W. J. Olver's book Asymptotics and Special Functions. Especially, Chapter 9, Sections 2 and 4. – Gary Apr 15 '23 at 08:03
  • Related https://math.stackexchange.com/questions/2234358/asymptotics-justification-for-abusing-laplaces-method – leonbloy Apr 17 '23 at 02:14

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