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A Complex Analysis Problem Book [electronic resource] / by Daniel Alpay.

By: Alpay, Daniel [author.].
Contributor(s): SpringerLink (Online service).
Material type: materialTypeLabelBookPublisher: Basel : Springer Basel, 2011Description: X, 526p. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783034800785.Subject(s): Mathematics | Functions of complex variables | Mathematics | Functions of a Complex VariableDDC classification: 515.9 Online resources: Click here to access online
Contents:
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.
In: Springer eBooksSummary: 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.
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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 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.

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