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Features connections with electrical engineering and the theory of linear systems
Uses a variety of non-trivial and interesting examples as exercises (for instance the Bohr phenomenon, integral representations of certain analytic functions, Blaschke products, and the Schur algorithm)
Exercises include those using positive definite functions and reproducing kernel spaces
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.
Content Level »Upper undergraduate
Keywords »Cauchy formula - analytic function - complex variables - conformal mapping - holomorphic function