= Two-dimensional stationary-phase formula
{title2=$\frac{2\pi a(q)e^{i\lambda\phi(q)+i\pi\operatorname{sgn}(H)/4}}{\lambda\sqrt{|\det H|}}$}
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 $2\pi\lambda^{-1}a(q)|\det H|^{-1/2}e^{i\lambda\phi(q)+i\pi\operatorname{sgn}(H)/4}$, 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.
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