Stationary phase and oscillatory integrals

Let the amplitude be smooth with compact support and let the phase be real and smooth near it. Where the phase derivative is bounded away from zero, express the oscillatory exponential as its derivative divided by the large parameter times that phase derivative. Integration by parts moves the derivative onto the amplitude quotient. Compact support removes endpoint terms. Repeating this argument gives decay faster than every fixed inverse power when there are no stationary points on the support. The constants depend on derivative bounds and the lower bound for the phase derivative.

On a finite interval with nonzero amplitude at its endpoints, the boundary terms do not vanish. They often produce contributions of order the inverse parameter, even if no stationary point exists. This is why an interior stationary-phase remainder cannot be used unchanged for an arbitrary abruptly truncated integral. A cutoff that vanishes smoothly near artificial boundaries separates the interior calculation from endpoint effects. Absolute integration estimates alone cannot reveal this cancellation, since the integrand's magnitude need not decrease with the oscillation parameter.

Assume a smooth compactly supported amplitude and a real smooth phase with finitely many stationary points on the support, all nondegenerate and in its interior. Near each point, the phase is its stationary value plus a signed quadratic term at leading order. Rescaling by the inverse square root of the large parameter gives a Fresnel-type Gaussian. Positive phase curvature contributes a factor with argument plus one-quarter pi, and negative curvature contributes the conjugate factor. The amplitudes must be summed as complex numbers before taking a final magnitude.

Under these hypotheses, the sum of leading contributions has a remainder of order the parameter to minus three-halves. A vanishing amplitude can remove a leading contribution, so the next term may matter. If the second derivative also vanishes, the quadratic scaling fails: a cubic phase has a one-third-power scale and leads to Airy-type behavior in suitable parameterized problems. Coalescing critical points require a uniform treatment; adding separate formulas whose curvature denominators diverge does not cure their breakdown.

A Gaussian amplitude is not compactly supported, so its use needs an independent tail argument or direct evaluation. For amplitude $e^{-x^2}$ and phase $x^2/2$, combine the exponents into a complex Gaussian whose real quadratic coefficient stays positive. Its exact value is the square root of pi divided by the square root of $1-i\lambda/2$, using the branch continued from the positive value at zero parameter. This selects a definite phase and avoids an arbitrary sign in the square root.

For large positive parameter, the coefficient's argument approaches minus one-half pi, so the reciprocal square root approaches argument plus one-quarter pi. The exact magnitude and phase therefore recover the stationary-phase constant. This example also shows why a real-valued magnitude estimate cannot reconstruct interference: two equal-sized contributions with different phase factors can reinforce or cancel. Keep phase conventions consistent across critical points and across any contour continuation used to evaluate them.