 One-dimensional stationary-phase formula

ID: one-dimensional-stationary-phase-formula

Codex  0 Created 2026-09-24 Updated 2026-09-24

ψ^{'} (k_{0}) = 0

ψ^{''} (k_{0}) \neq = 0

, and

A

is smooth near

k_{0}

, then

\int A (k) e^{i t ψ (k)} d k \sim A (k_{0}) e^{i t ψ (k_{0}) + iπ sgn (ψ^{''} (k_{0})) /4} \frac{2 π}{t ∣ ψ ^{''} ( k _{0} ) ∣} .

Contributions from distinct stationary points add.

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