Order of the width of a peak in perturbation methods

Question

My question relates to this excerpt from page 63 of the lecture notes on perturbation methods by W.R. Young.

How does the author find the 'width of the peak' to be $s^{-3/4}$ in this example? I have never come across this expression before and have no idea how the 'width' of a peak is defined mathematically here. Some help would be very much appreciated.

Answer 1

Here is another explanation of the "saddle point method". The idea is that the integral is approximated by a Gaussian whose width is a function of the second order coefficient in a Taylor expansion of the exponent (eqn. 126).