Advanced Mathematical Analysis

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. Rational functions; 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