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A course in analysis. Volume III: Measure and integration theory, complex-valued functions of a complex variable. (English) Zbl 1381.28002

Hackensack, NJ: World Scientific (ISBN 978-981-3221-59-8/hbk). xxvi, 757 p. (2018).
The third volume of the ambitious course in analysis written by N. Jacob and K. P. Evans (began with [Zbl. 1327.26012] and continued with [Zbl. 06605224]) is dedicated to two main themes, measure and integration theory and complex-valued functions of a complex variable (also known as the theory of functions or function theory), keeping the high standard of the previous two books of the series. Like its predecessors, this volume is a strong candidate for becoming one of the basic references in the mentioned fields of analysis, both for the width of the covered material and for the quality and clarity of the presentation, nice and illustrative examples arising everywhere among theoretical results. Like in the previous two volumes of the series, the reader can feel the rich experience of the authors in teaching mathematics, in particular analysis, stressed, for instance, by the motivations of the various introduced concepts and the fine balance between the hard theoretical results and applications. This over 750-pages long book is divided into two parts dedicated to the main themes mentioned above, containing also some appendices that include a list of contributors to analysis that continues its counterparts from the first two volumes as well as detailed solutions to the 275 carefully chosen problems and the standard comprehensive lists of symbols and indexed subjects, respectively. The first part of the book, where Lebesque’s measure and integration theory is presented in great detail, is divided into 12 chapters dedicated to the following themes: \(\sigma\)-fields and measures, pre-measures and Carathéodory’s theorem, Lebesgue-Borel measure and Hausdorff measures, measurable mappings, Lebesque integration with respect to a measure, Radon-Nikodym theorem and transformation theorem, almost everywhere statements, convergence theorems and their applications, integration on product spaces and applications, convolutions of functions and measures, differentiation revisited and to some additional selected topics such as critical points and values, Lusin’s theorem, weak convergence, characterizations of compactness and the Kolmogorov-Riesz theorem. The second part consists of a 17 chapters long presentation of the theory and properties of complex-valued functions of a complex variable, connections to geometry and topology as well as complex-valued mappings being also discussed. After introducing the complex numbers as a complete field and talking about the above mentioned related issues, the authors offer a nice introduction to the complex-valued functions of a complex variable, with some important examples mentioned separately, followed by the complex differentiation and some topological investigations. In the next chapters one can read about complex integration, more precisely about line integrals and the Cauchy integral statements. Power series, holomorphy and differential equations follow, with a whole chapter dedicated to the Residue Theorem. Before introducing the Riemann mapping theorem the \(\Gamma\) and \(\zeta\) functions are presented, as well as Dirichlet series and elliptic integrals and functions, respectively. The chapter closes with some facts about power series in several variables. The appendices, about one hundred pages shorter than the ones from the first volume, contain various basic results on topics like point set topology, measure and set theory, Möbius transformations and Bernoulli numbers, that are included here for reader’s convenience as some of them are needed within the main parts of the book.
All in one, I highly recommend this book to anyone teaching or studying measure, integration and/or function theory and am looking forward to reading the next volume of this series that was announced to be dedicated to Fourier analysis, ordinary differential equations and to calculus of variations.

MSC:

28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
30-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable
28Axx Classical measure theory
30Dxx Entire and meromorphic functions of one complex variable, and related topics
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