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. Front Matter....Pages i-xvii Front Matter....Pages 9-9 Complex Numbers: Algebra....Pages 11-60 Complex Numbers: Geometry....Pages 61-86 Complex Numbers and Analysis....Pages 87-140 Front Matter....Pages 141-141 Cauchy-Riemann Equations and $$\mathbb{C}$$ -differentiable Functions....Pages 143-191 Cauchy’s Theorem....Pages 193-273 Morera, Liouville, Schwarz, et les autres: First Applications....Pages 275-316 Laurent Expansions, Residues, Singularities and Applications....Pages 317-364 Computations of Definite Integrals Using the Residue Theorem....Pages 365-392 Front Matter....Pages 393-393 Harmonic Functions....Pages 395-419 Conformal Mappings....Pages 421-430 A Taste of Linear System Theory and Signal Processing....Pages 431-446 Front Matter....Pages 447-447 Some Useful Theorems....Pages 449-461 Some Topology....Pages 463-477 Some Functional Analysis Essentials....Pages 479-494 A Brief Survey of Integration....Pages 495-511 Back Matter....Pages 513-526 "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. "--Publisher's Web-Site 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. I. Complex Numbers -- Ii. Functions Of A Complex Variable -- Iii. Applications And More Advanced Topics -- Iv. Appendix. Daniel Alpay. Includes Bibliographical References (p. 513-522) And Index.