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The following theorem is due to Laplace:

Theorem: Suppose that $h$ is a real-valued $C^2$-function defined on the interval $(a,b) \subset \mathbb{R}$. If $h$ has a unique maximum at $c$ with $a<c<b$ so that $h'(c)=0$ and $h''(0)<0$, then, $$ \int _{a}^{b} e^{\lambda h(t)} \mathrm{d}{t} \sim e^{\lambda h(c)} \left( \frac{-2\pi}{\lambda h''(c)} \right)^{\frac{1}{2}} $$ as $\lambda \to \infty$.

Question: Is there any similar result for the case when $\lambda \to 0$?

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    The key observation here is that when $\lambda \rightarrow \infty$, the behaviour of the integral is dominated by $e^{\lambda h(c)}$ with a particular "bandwidth". When $\lambda \rightarrow 0$, the contrary happens and it would seem most useful to have bounds on $h$. – Olivier Apr 16 '17 at 14:08
  • I asked about one specific case here. – Antonio Vargas Apr 17 '17 at 11:17

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