Advanced Mathematical Analysis - R. Beals

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Sommaire:
One Basis concepts.- ?1. Sets and functions.- ?2. Real and complex numbers.- ?3. Sequences of real and complex numbers.- ?4. Series.- ?5. Metric spaces.- ?6. Compact sets.- ?7. Vector spaces.- Two Continuous functions.- ?1. Continuity, uniform continuity, and compactness.- ?2. Integration of complex-valued functions.- ?3. Differentiation of complex-valued functions.- ?4. Sequences and series of functions.- ?5. Differential equations and the exponential function.- ?6. Trigonometric functions and the logarithm.- ?7. Functions of two variables.- ?8. Some infinitely differentiable functions.- Three Periodic functions and periodic distributions.- ?1. Continuous periodic functions.- ?2. Smooth periodic functions.- ?3. Translation, convolution, and approximation.- ?4. The Weierstrass approximation theorems.- ?5. Periodic distributions.- ?6. Determining the periodic distributions.- ?7. Convolution of distributions.- ?8. Summary of operations on periodic distributions.- Four Hilbert spaces and Fourier series.- ?1. An inner product in ?, and the space ?2.- ?2. Hilbert space.- ?3. Hilbert spaces of sequences.- ?4. Orthonormal bases.- ?5. Orthogonal expansions.- ?6. Fourier series.- Five Applications of Fourier series.- ?1. Fourier series of smooth periodic functions and periodic distributions.- ?2. Fourier series, convolutions, and approximation.- ?3. The heat equation: distribution solutions.- ?4. The heat equation: classical solutions; derivation.- ?5. The wave equation.- ?6. Laplace's equation and the Dirichlet problem.- Six Complex analysis.- ?1. Complex differentiation.- ?2. Complex integration.- ?3. The Cauchy integral formula.- ?4. The local behavior of a holomorphic function.- ?5. Isolated singularities.- ?6. Rationalfunctions; Laurent expansions; residues.- ?7. Holomorphic functions in the unit disc.- Seven The Laplace transform.- ?1. Introduction.- ?2. The space ?.- ?3. The space ??.- ?4. Characterization of distributions of type ??.- ?5. Laplace transforms of functions.- ?6. Laplace transforms of distributions.- ?7. Differential equations.- Notes and bibliography.- Notation index.