Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 31C / b / Solution

The unique minimum is at zero with phase $t^2$ and amplitude $\cos 0=1$. The interior Laplace method givesFor an additional check, extending to the real line gives the exact Gaussian Fourier integral $\sqrt{\pi /\lambda }{e}^{-1/(4\lambda )}$. The omitted tails are exponentially small, so the displayed leading term follows directly.