Watson's lemma

Watson's lemma is the precise, term-by-term bookkeeping that turns Laplace's method into a full asymptotic series. Where Laplace's method gives you just the leading estimate of a peaked integral, Watson's lemma tells you the entire expansion in powers of the large parameter — and does so by a tidy recipe: expand the slowly varying part of the integrand near the relevant endpoint and integrate the expansion term by term.

The setting is a Laplace-type integral, the integral from 0 to infinity (or to some upper limit) of e^(-x t) f(t) dt, studied as x tends to infinity. Because the exponential e^(-x t) is sharply concentrated near t = 0 for large x, only the behaviour of f near t = 0 matters. If f(t) has a small-t expansion f(t) ~ a_0 t^(b_0) + a_1 t^(b_1) + ... (with exponents b_k greater than -1 so each piece is integrable), then integrating each power against e^(-x t) using the gamma-function formula gives the asymptotic series term by term: the integral ~ sum of a_k Gamma(b_k + 1) / x^(b_k + 1). Each successive term contributes a higher inverse power of x.

This lemma is the standard tool for reading off the large-argument asymptotics of special functions defined by integrals — the gamma function, error function, Bessel functions, the exponential integral, and many others — and it is the rigorous backbone behind 'just expand and integrate'. The honest caveat: the resulting series is typically asymptotic, not convergent, and it requires the integrand's heavy weight to sit at the endpoint t = 0; if the dominant contribution comes from an interior maximum instead, you are in Laplace's-method or steepest-descent territory, not Watson's lemma directly.