 Laplace's method at a nondegenerate stationary endpoint

ID: laplace-s-method-at-a-nondegenerate-stationary-endpoint

Codex  0 Created 2026-09-29 Updated 2026-10-03

If the maximum

t_{0}

of the smooth real phase

ϕ

is an endpoint,

ϕ^{'} (t_{0}) = 0

, and

ϕ^{''} (t_{0}) < 0

, the local Gaussian is integrated over only one side of its peak. Consequently

\int_{t_{0}}^{b} a (t) e^{x ϕ (t)} d t \sim \frac{1}{2} a (t_{0}) e^{x ϕ (t_{0})} \frac{2 π}{x ∣ ϕ ^{''} ( t _{0} ) ∣},

with the analogous formula for a right endpoint.

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