Chapter 7
Asymptotic Expansions

The special functions of Chapter 2 were defined by integrals, and the transforms of Chapters 5 and 6 produce more of them. Most such integrals cannot be evaluated in closed form, and a convergent series for them is often useless in practice: the series for \(\operatorname {erf}(x)\) converges for every \(x\), yet at \(x=5\) its terms grow to about \(10^{9}\) before they begin to fall, and any finite-precision computation of it is destroyed by cancellation long before it converges.

What is wanted in that regime is different in kind. An asymptotic expansion is a series that need not converge at all, but whose first few terms approximate the function extremely well when the argument is large. This chapter develops the idea, and its central tool — Watson’s lemma — turns out to be the gamma function once again.