Articles liés à Function Theoy Of One Complex Variable
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Synopsis
Complex analysis is one of the most beautiful subjects that we learn as graduate students. Part of the joy comes from being able to arrive quickly at some "real theorems". The fundamental techniques of complex variables are also used to solve real problems in neighboring subjects, such as number theory or PDEs. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors. All the material usually treated in such a course is covered here, but following somewhat different principles. To begin with, the authors emphasize how this subject is a natural outgrowth of multivariable real analysis. Complex function theory has long been a flourishing independent field. However, an efficient path into the subject is to observe how its rudiments arise directly from familiar ideas in calculus. The authors pursue this point of view by comparing and contrasting complex analysis with its real variable counterpart. Explanations of certain topics in complex analysis can sometimes become complicated by the intermingling of the analysis and the topology. Here, the authors have collected the primary topological issues in a separate chapter, leaving the way open for a more direct and less ambiguous approach to the analytic material. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat $Hp$ spaces and Painleve's theorem on smoothness to the boundary for conformal maps. A large number of exercises are included. Some are simply drills to hone the students' skills, but many others are further developments of the ideas in the main text. The exercises are also used to explore the striking interconnectedness of the topics that constitute complex analysis.
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Quatrième de couverture
A new approach that treats complex analysis in a broad context
This book presents a new approach to one of mathematics′ oldest fields. It departs from the tradition of teaching complex analysis as a self–contained subject and, instead, treats the subject as a natural development from calculus. It also shows how complex analysis is used in other areas, exploring connections with calculus, algebra, geometry, topology, and other parts of analysis.
The authors provide the ideal framework for a first–year graduate course in complex analysis building upon ideas the student is already familiar with and simplifying the transition to advanced topics. The book is also for those using complex numbers and functions in applied fields, including engineering, physics, and other areas.
Function Theory of One Complex Variable
- Compares and contrasts complex variable theory with real variable theory
- Clarifies analytical ideas belonging to complex analysis by separating them from topological issues
- Discusses some of the current research in the field, including a number of interesting topics not discussed in other textbooks
- Features many examples as well as 75 illustrations
- Provides especially thorough exercise sets
Présentation de l'éditeur
This book offers an excellent modern introduction to complex analysis, one of the core components of mathematics. It establishes the basic analytical ideas before moving on to more topological subjects, easing the transition from familiar ideas of calculus to the more advanced ones distinctive to complex analysis.
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