We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn's book {\em Asymptotic Methods in Analysis}. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.

asymptotic analysis, method of stationary phase, saddle point method, uniform asymptotic expansions, special functions
Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60), Asymptotic representations in the complex domain (msc 30E15), Elementary classical functions (msc 33Bxx), Hypergeometric functions (msc 33Cxx), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60)
Other (theme 6)
Special issue in memory of N.G. (Dick) de Bruijn (1918–2012)
DOI: 10.1016/j.indag.2013.08.001

Temme, N.M. (2013). Uniform Asymptotic Methods for Integrals. Indagationes Mathematicae, 24(4), 739–765. doi:10.1016/j.indag.2013.08.001