A course in mathematical analysis. Volume III: Complex analysis, measure and integration (Q2871220)

This last volume of the series of three volumes of ``A course in mathematical analysis'' covers complex analysis and the theory of measure and integration. The book offers an expanded version of the author's own undergraduate courses taught at the University of Cambridge. The spirit of this volume is clearly developed. The author formulates concepts clearly and explains the relationship between them. The subject matter is important and interesting.NEWLINENEWLINEIn this volume, the author carefully develops the classical theory of functions of a complex variable. He establishes several basic properties of the complex plane, including a proof of the Jordan curve theorem. A central place is given to the Lebesgue measure, which is used as a model for other measure spaces. Another basic result in this volume is the Radon-Nikodym theorem, in relationship with the differentiation of measures.NEWLINENEWLINEThe content of this volume is divided into two parts, as follows. Part I is devoted to \textit{Complex analysis} and contains the following seven chapters: Holomorphic functions and analytic functions; The topology of the complex plane; Complex integration; Zeros and singularities; The calculus of residues; Conformal transformations; Applications. Part II deals with \textit{Measure and integration} and contains the following eight chapters: Lebesgue measure on \({\mathbb R}\); Measurable spaces and measurable functions; Integration; Constructing measures; Signed measures and complex measures; Measures on metric spaces; Differentiation; Applications. Over 250 exercises and numerous examples and applications challenge the reader to learn through practice and illustrate the abstract results in this volume.NEWLINENEWLINEThis is a very beautiful book, coming at the right moment. It is an intuitively motivated, original, and insightful presentation of many topics in classical complex analysis and related areas. The volume is a remarkable mix of classical results and research topics at the forefront of actual research. It is full of cross fertilizations of different theories and will be useful to undergraduate and graduate students, as well as to many researchers in other fields (functional analysis, partial differential equations, mathematical physics).