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基本説明
A collection of exercises in the theory of analytic functions, with completed and detailed solutions. A variety of non trivial and interesting examples are given as exercises (for instance the Bohr phenomenon, integral representations of certain analytic functions, Blaschke products, the Schur algorithm).
Full Description
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 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.
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.
