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298 Approximation of Integrals

are useful in statistics. Many imporant properties of special functions have been catalogued in Erd´elyi
(1953a,b).
The basic theory of asymptotic expansions is outlined in de Bruijn (1956) and Erdelyi (1956).
Asymptotic expansions of integrals, including Watson’s lemma and related results, are discussed in
Andrews et al. (1999, Appendix C), Bleistein and Handelsman (1986), Temme (1996, Chapter 2),
and Wong (1989). The form of Watson’s lemma and Laplace’s method presented here is based on
Breitung (1994, Chapter 4). Evans and Swartz (2000) discuss the problem of approximating integrals
that are useful in statistics; see also Barndorff-Nielsen and Cox (1989).
The uniform asymptotic approximation presented in Section 9.6 is known as Temme’s method
(Temme 1982); see Jensen (1995, Chapter 3) for further discussion of this result and some general-
izations. The approximation of sums and the Euler-Maclaurin summation formula is discussed in de
Bruijn (1956, Chapter 3); see also Andrews et al. (1999, Appendix D). Exercise 9.15 is discussed
further in Barndorff-Nielsen and Cox (1989).