Laplace's method

Summary

Laplace's method approximates integrals of the form $\int_a^b e^{M f(x)}\, dx$ when the parameter $M$ is large. For large $M$ almost all of the integral comes from a small neighbourhood of the point where $f$ is largest. There $f$ is close to a downward parabola. Replacing $f$ by this parabola turns the integral into a Gaussian integral, which gives the leading term of an asymptotic equivalence. Stirling's formula is the classical application.