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A Complex Analysis Problem Book

A Complex Analysis Problem Book This is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications and aspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises we wish to show to the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space) are given. The book also includes a part where relevant facts from topology, functional analysis and Lebesgue integration are reviewed. ; This introduction to aspects of analytic function theory not usually touched on in a first course is a compilation of exercises in the theory that includes completed and detailed solutions. It also discusses topics that are relevant to electrical engineering. ; Prologue.- I Complex numbers.- 1 Complex numbers: algebra.- 2 Complex numbers: geometry.- 3 Complex numbers and analysis.- 4. Remarks and generalizations: quaternions, etc.- II Functions of a complex variable.- 5 C-differentiable functions.- 6 Cauchy's theorem.- 7 First applications.- 8 Laurent expansions and applications.- 9 Computations of definite integrals.- 10 Harmonic functions.- 11 Conformal mappings.-III Complements.- 12 Some useful theorems.- 13 Some topology.- References.- Index. ; From the reviews: “This volume contains a collection of exercises in the theory of analytic functions. The author also provides detailed solutions to all proposed problems. … The book under review is mainly addressed to upper undergraduate students from mathematics and electrical engineering.” (Vicenţiu D. Rădulescu, Zentralblatt MATH, Vol. 1226, 2012) “This is a substantial book (more than 500 pages long) which starts with a sketch of the construction of the field of complex numbers … and proceeds to much more advanced material. … this book is more than just a collection of exercises and solutions … . anyone teaching such a course would undoubtedly find this book a useful supplement, as would graduate students studying for their qualifier examinations, who could use this book to brush up on both theory and technique.” (Mark Hunacek, The Mathematical Association of America, December, 2011) ; Prof. Daniel Alpay is a faculty member of the department of mathematics at Ben-Gurion University, Beer-Sheva, Israel. He is the incumbent of the Earl Katz Family chair in algebraic system theory. He has a double formation of electrical engineer (Telecom Paris, graduated 1978) and mathematician (PhD, Weizmann Institute, 1986). His research includes operator theory, stochastic analysis, and the theory of linear systems. Daniel Alpay is one of the initiators and responsible of the dual track electrical-engineering mathematics at Ben-Gurion University. Together with co-authors, he has written two books and close to 190 research papers, and edited ten books of research papers. ; This is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications and aspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises show the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space) are given. The book also includes a part where relevant facts from topology, functional analysis and Lebesgue integration are reviewed. ; Connections with electrical engineering and the theory of linear systems A variety of non trivial and interesting examples are given as exercises (for instance the Bohr phenomenon, integral representations of certain analytic functions, Blaschke products, the Schur algorithm) Examples using positive definite functions and reproducing kernel spaces are given in the exercises ; NL http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Complex Analysis Problem Book

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Publisher
Springer Basel
Copyright
Copyright � Springer Basel AG
DOI
10.1007/978-3-0348-0078-5
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Abstract

This is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications and aspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises we wish to show to the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space) are given. The book also includes a part where relevant facts from topology, functional analysis and Lebesgue integration are reviewed. ; This introduction to aspects of analytic function theory not usually touched on in a first course is a compilation of exercises in the theory that includes completed and detailed solutions. It also discusses topics that are relevant to electrical engineering. ; Prologue.- I Complex numbers.- 1 Complex numbers: algebra.- 2 Complex numbers: geometry.- 3 Complex numbers and analysis.- 4. Remarks and generalizations: quaternions, etc.- II Functions of a complex variable.- 5 C-differentiable functions.- 6 Cauchy's theorem.- 7 First applications.- 8 Laurent expansions and applications.- 9 Computations of definite integrals.- 10 Harmonic functions.- 11 Conformal mappings.-III Complements.- 12 Some useful theorems.- 13 Some topology.- References.- Index. ; From the reviews: “This volume contains a collection of exercises in the theory of analytic functions. The author also provides detailed solutions to all proposed problems. … The book under review is mainly addressed to upper undergraduate students from mathematics and electrical engineering.” (Vicenţiu D. Rădulescu, Zentralblatt MATH, Vol. 1226, 2012) “This is a substantial book (more than 500 pages long) which starts with a sketch of the construction of the field of complex numbers … and proceeds to much more advanced material. … this book is more than just a collection of exercises and solutions … . anyone teaching such a course would undoubtedly find this book a useful supplement, as would graduate students studying for their qualifier examinations, who could use this book to brush up on both theory and technique.” (Mark Hunacek, The Mathematical Association of America, December, 2011) ; Prof. Daniel Alpay is a faculty member of the department of mathematics at Ben-Gurion University, Beer-Sheva, Israel. He is the incumbent of the Earl Katz Family chair in algebraic system theory. He has a double formation of electrical engineer (Telecom Paris, graduated 1978) and mathematician (PhD, Weizmann Institute, 1986). His research includes operator theory, stochastic analysis, and the theory of linear systems. Daniel Alpay is one of the initiators and responsible of the dual track electrical-engineering mathematics at Ben-Gurion University. Together with co-authors, he has written two books and close to 190 research papers, and edited ten books of research papers. ; This is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications and aspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises show the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space) are given. The book also includes a part where relevant facts from topology, functional analysis and Lebesgue integration are reviewed. ; Connections with electrical engineering and the theory of linear systems A variety of non trivial and interesting examples are given as exercises (for instance the Bohr phenomenon, integral representations of certain analytic functions, Blaschke products, the Schur algorithm) Examples using positive definite functions and reproducing kernel spaces are given in the exercises ; NL

Published: Aug 20, 2011

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