Let $N \in \mathbb{N}$ with $N \geq 1$, let $U \subset \mathbb{R}^N$ be open, let $\phi \in C^\infty(U;\mathbb{R})$, and let $a \in C_c^\infty(U;\mathbb{C})$. Suppose that $y_0 \in \operatorname{supp} a$ is the unique point of $\operatorname{supp} a$ at which $\nabla \phi$ vanishes, and suppose that the Hessian matrix $H := J(\nabla\phi)_{y_0} \in \mathbb{R}^{N \times N}$ is invertible. Let $\operatorname{sgn}(H)$ denote the number of positive eigenvalues of $H$ minus the number of negative eigenvalues of $H$, counted with multiplicity. Then there exist complex-valued linear differential functionals $L_j:C_c^\infty(U;\mathbb{C}) \to \mathbb{C}$, one for each integer $j \geq 0$, of differential order at most $2j$ and evaluated at $y_0$, such that $L_0a = a(y_0)$ and, as $\lambda \to +\infty$, the oscillatory integral admits the full asymptotic expansion

\begin{align*} \int_U e^{i\lambda\phi(y)}a(y)\,d\mathcal{L}^N(y) \sim e^{i\lambda\phi(y_0)}e^{i\pi\operatorname{sgn}(H)/4}\left(\frac{2\pi}{\lambda}\right)^{N/2}|\det H|^{-1/2}\sum_{j=0}^{\infty}\lambda^{-j}L_j a. \end{align*}

Equivalently, for every integer $M \geq 0$, there are constants $C_M > 0$ and $\lambda_M \geq 1$, depending on $M$, $\phi$, and $a$, such that for all $\lambda \geq \lambda_M$,

\begin{align*} \left|\int_U e^{i\lambda\phi(y)}a(y)\,d\mathcal{L}^N(y) - e^{i\lambda\phi(y_0)}e^{i\pi\operatorname{sgn}(H)/4}\left(\frac{2\pi}{\lambda}\right)^{N/2}|\det H|^{-1/2}\sum_{j=0}^{M-1}\lambda^{-j}L_j a\right| \leq C_M\lambda^{-N/2-M}, \end{align*}

with the convention that the sum is $0$ when $M = 0$.