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 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.

Authors: Alpay, Daniel
Features a new chapter on rational functions
Elaborates on connections with electrical engineering and the theory of linear systems
Uses a variety of non-trivial and interesting examples as the basis for exercises (e.g. the Bohr phenomenon, integral representations of certain analytic functions, Blaschke products, and the Schur algorithm)
Some of the exercises use the notions of positive matrix, positive definite function and reproducing kernel Hilbert space

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.

by Daniel Alpay (Author)
Provides a wide range of advanced exercises
Hints at current state of research
Offers an overview of what a beginning graduate student should know