Laplace's method
/ luh-PLAHSS /
Many integrals that resist exact evaluation contain a large exponential weight that is overwhelmingly concentrated at one spot, like a spotlight that brightens a single point of a stage and leaves the rest in near darkness. Laplace's method exploits exactly this: when an integrand is sharply peaked, almost all of the integral comes from a tiny region around the peak, and you can approximate the whole thing by studying that little neighbourhood.
The mechanism: to estimate the integral of e^(M phi(t)) over an interval as the large parameter M tends to infinity, find where the smooth function phi(t) reaches its maximum, say at t = t0 with phi'(t0) = 0 and phi''(t0) < 0. Near that point phi is well approximated by its parabola phi(t0) + (1/2) phi''(t0) (t - t0)^2, turning the integrand into a Gaussian bump. Doing the resulting Gaussian integral gives the leading estimate: the integral is approximately e^(M phi(t0)) times sqrt(2 pi / (M |phi''(t0)|)). The taller and the larger M, the more peaked the bump and the better the approximation, with relative error of order 1/M.
This single idea is the engine behind a remarkable amount of applied mathematics: it yields Stirling's formula for n! by writing the factorial as an integral, it underlies the saddle-point method in the complex plane, and it is the 'Laplace approximation' used throughout statistics and machine learning to approximate posterior integrals by a Gaussian centred at the most probable value. The honest caveat: it needs a genuine interior maximum where phi'' is nonzero; if the maximum sits at an endpoint, or phi'' vanishes there (a flatter peak), the leading formula changes and you must redo the local analysis.
Writing n! = integral from 0 to infinity of t^n e^(-t) dt and applying Laplace's method around the peak at t = n recovers Stirling: n! ~ sqrt(2 pi n) (n/e)^n.
Approximating the sharp peak of t^n e^(-t) by a Gaussian turns an intractable integral into Stirling's formula.
It is the maximum of the exponent phi, not of the whole integrand, that matters at leading order; a slowly varying prefactor in front of e^(M phi) only contributes its value at the peak, not its shape.