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.
Author(s): Daniel Alpay (auth.)
Edition: 1
Publisher: Birkhäuser Basel
Year: 2011
Language: English
Pages: 526
Tags: Functions of a Complex Variable
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
