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Could anyone give a reference for the error bound in Laplace's approximation? Namely, reference for the proof of the following statement:

Let $f>0$ and $\phi$ be smooth functions on $[a,b]$ and $\phi$ achieves a unique maximum at $c \in (a,b)$ then

$$\int_a^b f(t) e^{x \phi(t)} \text{d} t = \sqrt{\frac{2\pi}{x |\phi''(c)|}} f(c) e^{x\phi(c)}\left(1 + O\left(\frac{1}{x}\right) \right) \qquad \text{as} \ x \rightarrow +\infty$$

P.S. wiki page has proof that the relative error goes to $0$ but it is not very readable for me.

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    The proof on the Wikipedia page looks pretty much like the standard proof. Which part of it is hard to understand? – 2017-02-12
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    For example the last estimate $(c)=|\int_{-D_y}^{D_y} A -A_0 d y| \leq \int_{-D_y}^{D_y} e^{- \pi y^2} |\frac{h'(\xi)}{h(0)} s y| dy$ (I guess he uses MVT but I don't quite see how to get rid of the exponential terms). Also how does he choose the point $D_y$ (and what is the subscript $y$ for, what kind of dependence is that because $y$ is the integration variable?) Thank you. – 2017-02-13

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