Let $\varphi \in C^\infty (\mathbb R^n ;\mathbb R)$ such that
1) $\varphi(0)=0$
2) $\varphi(x)>0$ on $\mathbb R^n\setminus 0$
3) $\text{Hess}_{\varphi}(0)>0 $
and let $B_1(0)$ be the unitary ball around $0$. Then the classical Laplace method gives that for $h>0$ small
$\int_{B_1(0)} e^{-\varphi(x)/h} dx \sim h^{n/2}\sum_{k=0}^\infty h^k b_k$
where $(b_k)$ is a sequence in $\mathbb R$.
My question is: if I have also a function $g\in C^\infty (\mathbb R^n ;\mathbb R)$ such that $g(0)=0$, is it still true that there is a sequence $(c_k) \in \mathbb R$ such that
$\int_{B_1(0)} e^{-(\varphi(x) + \sqrt h g(x))/h} dx \sim h^{n/2}\sum_{k=0}^\infty h^k c_k$
or do there appear also half powers of $h$ in the expansion?