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.

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

Autorentext

Prof. Daniel Alpay is a faculty member of the department of mathematics at Ben-Gurion University, Beer-Sheva, Israel. He is the incumbent of the Earl Katz Family chair in algebraic system theory. He has a double formation of electrical engineer (Telecom Paris, graduated 1978) and mathematician (PhD, Weizmann Institute, 1986). His research includes operator theory, stochastic analysis, and the theory of linear systems. Daniel Alpay is one of the initiators and responsible of the dual track electrical-engineering mathematics at Ben-Gurion University. He is the author of "An Advanced Complex Analysis Problem Book" (Birkhäuser, 2015). Together with co-authors, he has written seven books and close to 240 research papers, and edited fifteen books of research papers, and in particular the Springer Reference Work on Operator Theory.

Inhalt
Part I Complex Numbers.- Complex Numbers: Algebra.- Complex Numbers: Geometry.- Complex Numbers and Analysis.- Part II Functions of a Complex Variable.- CauchyRiemann Equations and C-differentiable Functions.- Cauchy'sTheorem.- Morera, Liouville, Schwarz, et les autres: First Applications.- Laurent Expansions, Residues, Singularities and Applications.- Computations of Definite Integrals Using the Residue Theorem.- Part III Applications and More Advanced Topics.- Harmonic Functions.- Conformal Mappings.- A Taste of Linear System Theory and Signal Processing.- Rational Functions.- Special Functions and Transforms.- Part IV Appendix.- Some Topology.- Some Functional Analysis Essentials.- A Brief Survey of Integration.