2010 Mathematics Subject Classification: 30-01This 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 we wish to show to 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. "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. "--Publisher's Web-Site Complex numbers : Algebra Complex numbers : Geometry Complex numbers and analysis Cauchy-riemann equations and C-differentiable functions Cauchy's theorem Morera, Liouville, Schwarz, et les autres : first applications Laurent expansions, residues, singularities and applications Computations of definite integrals using the residue theorem Harmonic functions Conformal mappings A taste of linear system theory and signal processing Some useful theorems Some topology Some functional analysis essentials A brief survey of integration. Prologue.- I Complex numbers.- 1 Complex numbers: algebra.- 2 Complex numbers: geometry.- 3 Complex numbers and analysis.- 4. Remarks and generalizations: quaternions, etc.- II Functions of a complex variable.- 5 C-differentiable functions.- 6 Cauchy's theorem.- 7 First applications.- 8 Laurent expansions and applications.- 9 Computations of definite integrals.- 10 Harmonic functions.- 11 Conformal mappings.-III Complements.- 12 Some useful theorems.- 13 Some topology.- References.- Index. This introduction to aspects of analytic function theory not usually touched on in a first course is a compilation of exercises in the theory that includes completed and detailed solutions. It also discusses topics that are relevant to electrical engineering. I. Complex Numbers -- Ii. Functions Of A Complex Variable -- Iii. Applications And More Advanced Topics -- Iv. Appendix. Daniel Alpay. Includes Bibliographical References (p. 513-522) And Index.
