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Laplace's method

If a smooth real phase

ϕ

has a unique nondegenerate interior maximum at

t_{0}

, then

\int a (t) e^{x ϕ (t)} d t \sim a (t_{0}) e^{x ϕ (t_{0})} \frac{2 π}{x ∣ ϕ ^{''} ( t _{0} ) ∣} .

For a maximum at an endpoint

b

with

ϕ^{'} (b) > 0

, the leading factor is

a (b) e^{x ϕ (b)} / (x ϕ^{'} (b))

.

Table of contents 65 1
- Laplace asymptotic for the derivative of the Gamma function Laplace's method 28

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