Let $f$ and $g$ two real smooth functions, $x_0 \in \mathbb{R}$ such that $f'(x_0) = 0$ and $f''(x_0) > 0$. Given $a, b \in \mathbb{R}$, Laplace's method gives an equivalent of the form, as $t$ goes to $+\infty$,
$$ I_t := \int_a^b g(x) \exp{(-tf(x))} dx \sim \sqrt{\frac{2\pi}{t|f''(x_0)|}} g(x_0) \exp{(-tf(x_0))} $$
I am looking for an extension of this result that gives a more general asymptotic expansion of the form
$I_t \exp{(tf(x_0))} = \sum_{k=1}^N a_k t^{-k/2} + o(t^{-N/2})$
where the coefficients $a_k$ depend on the coefficients of the Taylor expansions of $f$ and $g$. Thank you for your help.