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Uniform asymptotic expansions of integrals: A selection of problems. (English) Zbl 0849.41024
Summary: On the occasion of the conference we mention examples of Stieltjes’ work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several “standard” problems and examples, in which known special functions (error functions, Airy functions, Bessel functions, etc.) are needed to construct uniform approximations. Finally, we discuss the recent interest and new insights in the Stokes phenomenon. An extensive bibliography on uniform asymptotic methods for integrals is given, together with references to recent papers on the Stokes phenomenon for integrals and related topics.
MSC:
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33B20Incomplete beta and gamma functions
33C10Bessel and Airy functions, cylinder functions, 0 F 1
33C45Orthogonal polynomials and functions of hypergeometric type
11B73Bell and Stirling numbers
30E15Asymptotic representations in the complex domain