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alpay d complex analysis problem book 2011

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Textbook in PDF format 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. The audience we have in mind consists of undergraduate students from mathematics and electrical engineering, with an eye on advanced students from both tracks. Analytic functions are the bread and butter of mathematicians. For engineers, analytic functions appear everywhere, in particular in the theory of linear dynamical systems, signal processing, circuit theory, sampling theorems, optimal control, to name a few. For instance, a motivation for an engineering student would be to know that transfer functions of discrete-time shift-invariant dissipative linear systems are functions analytic in the open unit disk, or in a half-plane, and bounded by one in modulus there (the celebrated Schur functions); see for instance the book [5]. Unfortunately, most, if not all, electrical engineering students do not know what a transfer function is when they begin studying the theory of analytic functions. For the convenience of engineering students we give, in the second part of this prologue, a short discussion of time-invariant linear bounded systems, and their connections to complex variables. The book consists of four parts. The first two parts, respectively entitled Complex numbers and Function of a complex variable, form the bulk of the book. Most of the exercises presented in these two parts have been given in the past years by the author in classes on Introduction to the theory of functions of a complex variable for second year electrical engineering students, and Theory of functions of a complex variable for mathematics students, at the department of mathematics at Ben-Gurion University. The exercises rely only on classical real analysis, but sometimes we use measure theory (mainly via the dominated convergence theorem) to avoid lengthy arguments. Study of some special Hilbert spaces of analytic functions is also scattered in the text, and requires some elementary functional analysis. When studying a function analytic in a domain (for instance in the open unit disk), we will usually (but not always) assume that it is analytic in a neighborhood of this domain, to avoid problems with boundary values. The student will in particular meet in this second part, in simplified forms, Bohr’s inequality and the Herglotz integral representation of a function analytic in the open unit disk, and with a real positive part there. The third part, entitled Applications and more advanced topics, contains more advanced material, which was taught by the author to graduate students and also to undergraduate students from the double major program mathematics and electrical engineering at Ben-Gurion University. Topics include harmonic functions, conformal mappings and a brief introduction to the theory of linear systems. We hope to come back to these topics, and discuss other advanced topics such as elliptic functions, various aspects of several complex variables and Riemann surfaces in a sequel to the present book. For the convenience of the reader, we give in the first three parts of the book a number of reminders of known facts from complex analysis, mostly without proofs, in the text. The solutions of most of the exercises are presented at the end of the chapter where they are given. The fourth part, entitled Advanced prerequisites, contains material from real analysis, topology, functional analysis and measure theory, which are needed to solve the exercises (and, in fact, to fully understand a first course on complex variables). Since we mention in the text a number of Hilbert spaces of analytic functions, we also have taken the liberty of mentioning the definition of a reproducing kernel Hilbert space. Complex Numbers Complex Numbers: Algebra Complex Numbers: Geometry Complex Numbers and Analysis Functions of a Complex Variable Cauchy-Riemann Equations and C-differentiable Functions Cauchy’s Theorem Morera, Liouville, Schwarz, et les autres: First Applications Laurent Expansions, Residues, Singularities and Applications Computations of Definite Integrals Using the Residue Theorem Applications and More Advanced Topics Harmonic Functions Conformal Mappings A Taste of Linear System Theory and Signal Processing Appendix Some Useful Theorems Some Topology Some Functional Analysis Essentials A Brief Survey of Integration

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alpay d complex analysis problem book 2011

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