A Complex Analysis Problem Book

This second edition presents a collection of exercises on the theory of analytic functions, including completed and detailed solutions. It introduces students to various applications and aspects of the theory of analytic functions not always touched on in a first course, while also addressing topics of interest to electrical engineering students (e.g., the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). It provides examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space), and also includes a section reviewing essential aspects of topology, functional analysis and Lebesgue integration. Benefits of the 2nd edition Rational functions are now covered in a separate chapter. Further, the section on conformal mappings has been expanded. Front Matter....Pages i-xviii Front Matter....Pages 9-9 Complex Numbers: Algebra....Pages 11-64 Complex Numbers: Geometry....Pages 65-91 Complex Numbers and Analysis....Pages 93-148 Front Matter....Pages 149-149 Cauchy–Riemann Equations and \(\mathbb{C}\) -differentiable Functions....Pages 151-207 Cauchy’s Theorem....Pages 209-292 Morera, Liouville, Schwarz, et les autres: First Applications....Pages 293-335 Laurent Expansions, Residues, Singularities and Applications....Pages 337-379 Computations of Definite Integrals Using the Residue Theorem....Pages 381-413 Front Matter....Pages 415-415 Harmonic Functions....Pages 417-442 Conformal Mappings....Pages 443-456 A Taste of Linear System Theory and Signal Processing....Pages 457-471 Rational Functions....Pages 473-490 Special Functions and Transforms....Pages 491-509 Front Matter....Pages 511-511 Some Useful Theorems....Pages 513-527 Some Topology....Pages 529-543 Some Functional Analysis Essentials....Pages 545-562 A Brief Survey of Integration....Pages 563-578 Back Matter....Pages 579-596