Laplace's method with multiple global maxima

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 30D / a / Solution

{c}

If the maximum $t_0$ of the smooth real phase $\phi$ is an endpoint, $\phi'(t_0)=0$, and $\phi''(t_0)<0$, the local Gaussian is integrated over only one side of its peak. Consequently
$$
\int_{t_0}^{b}a(t)e^{x\phi(t)}\,dt
\sim\frac12a(t_0)e^{x\phi(t_0)}
\sqrt{\frac{2\pi}{x|\phi''(t_0)|}},
$$
with the analogous formula for a right endpoint.

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 Incoming links: Laplace's method at a nondegenerate stationary endpoint