Nonstandard asymptotic analysis. (English) Zbl 0633.41001

Lecture Notes in Mathematics, 1249. Berlin etc.: Springer-Verlag. IX, 187 p.; DM 28.50 (1987).
This book is a good addition to the existing literature on nonstandard analysis. It presents nonstandard methods of asymptotic reasoning along with new results obtained by using them. This also provides a nonstandard alternative to the classical theory of asymptotic expansions. The new results presented here deal with asymptotic expressions concerning the remainders of convergent and divergent Taylor series and asymptotic series. These expressions are further used to study the following well known problems: (1) How to use divergent series in approximations? (2) To present a mathematical description of the phenomenon: The configuration of Taylor polynomials of some familiar functions is strikingly regular.
The book is divided into three parts. The first part treats in a nonstandard way the well known examples of asymptotics such as the exponential integral, Stirling’s formula, the divergent expansions of the Bessel functions, the Taylor polynomials of the exponential function. In the second part the author embeds the results obtained on these examples into a general theory. In the last part the author develops in a linear way the foundations of a general theory of nonstandard asymptotics.
The book is very well written and is presented with an axiomatic form of nonstandard analysis, namely, the internal set theory. It is very valuable to those who are conversant with the basic principles of nonstandard analysis and also to those interested in classical analysis.
Reviewer: Y.Sitaraman


41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
26E35 Nonstandard analysis
03H05 Nonstandard models in mathematics
26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations