Asymptotic analysis, part 1
H.A. Lauwerier · Data Archiving and Networked Services (DANS) · 1974
This tract is an enlarged and improved version of tract 13 by the same author.It contains the expanded lecture notes of courses on asymptotic methods in applied analysis given for students in mathematical physics at the University of Amsterdam.In particular we consider the asymptotic behaviour of real and complex functions which are explicitly given by an integral expression.Asymptotics is an art that can be mastered only by studying a great number of special cases rather than using general theorems.The author therefore has attempted to illustrate the techniques of asymptotic analysis by a great variety of worked-out examples.Much attention is given to the asymptotic expansion of functions which can be expressed as• a Laplace integral or more generally as an integral to which the saddle point technique can be applied.A number of special functions such as Bessel functions and confluent hypergeometric functions are treated in this way.A few topics are included here which are not usually found in the text-books.We mention in particular the chapters on factorial series, the Euler transformation, the asymptotic behaviour of Cauchy integrals and asymptotics in the theory of probability.Also much original material is included.The text is aimed at students in analysis and in mathematical physics.The style may be reminescent of the lecture room.In general the systematic treatment anf the theorems are preceded or followed by examples and special cases.Some theorems could easily be generalized and conditions may be relaxed somewhat.However, for the sake of simplicity and clearness the theorems are sometimes stated in a simple and special form.Obvious generalizations are left to the reader.H.A.L.