Two-dimensional stationary-phase formula ($\frac{2\pi a(q)e^{i\lambda \varphi (q)+i\pi \mathrm{sgn}(H)/4}}{\lambda \sqrt{|\det H|}}$)

For a smooth real phase with an isolated nondegenerate interior critical point $q$, smooth amplitude $a$, and no competing boundary contribution, the leading contribution is $2\pi {\lambda }^{-1}a(q)|\det H{|}^{-1/2}{e}^{i\lambda \varphi (q)+i\pi \mathrm{sgn}(H)/4}$, where $H$ is the Hessian matrix. Orthogonal diagonalization gives two one-dimensional Fresnel integrals. A degenerate critical point or a stationary boundary needs its own local analysis.