Graduate Texts in Mathematics

One Basis concepts.- x1. Sets and functions.- x2. Real and complex numbers.- x3. Sequences of real and complex numbers.- x4. Series.- x5. Metric spaces.- x6. Compact sets.- x7. Vector spaces.- Two Continuous functions.- x1. Continuity, uniform continuity, and compactness.- x2. Integration of complex-valued functions.- x3. Differentiation of complex-valued functions.- x4. Sequences and series of functions.- x5. Differential equations and the exponential function.- x6. Trigonometric functions and the logarithm.- x7. Functions of two variables.- x8. Some infinitely differentiable functions.- Three Periodic functions and periodic distributions.- x1. Continuous periodic functions.- x2. Smooth periodic functions.- x3. Translation, convolution, and approximation.- x4. The Weierstrass approximation theorems.- x5. Periodic distributions.- x6. Determining the periodic distributions.- x7. Convolution of distributions.- x8. Summary of operations on periodic distributions.- Four Hilbert spaces and Fourier series.- x1. An inner product in ?, and the space ?2.- x2. Hilbert space.- x3. Hilbert spaces of sequences.- x4. Orthonormal bases.- x5. Orthogonal expansions.- x6. Fourier series.- Five Applications of Fourier series.- x1. Fourier series of smooth periodic functions and periodic distributions.- x2. Fourier series, convolutions, and approximation.- x3. The heat equation: distribution solutions.- x4. The heat equation: classical solutions; derivation.- x5. The wave equation.- x6. Laplace's equation and the Dirichlet problem.- Six Complex analysis.- x1. Complex differentiation.- x2. Complex integration.- x3. The Cauchy integral formula.- x4. The local behavior of a holomorphic function.- x5. Isolated singularities.- x6. Rational functions; Laurent expansions; residues.- x7. Holomorphic functions in the unit disc.- Seven The Laplace transform.- x1. Introduction.- x2. The space ?.- x3. The space ??.- x4. Characterization of distributions of type ??.- x5. Laplace transforms of functions.- x6. Laplace transforms of distributions.- x7. Differential equations.- Notes and bibliography.- Notation index.