Once upon a time students of mathematics and students of science or engineering took the same courses in mathematical analysis beyond calculus. Now it is common to separate" advanced mathematics for science and engi- neering" from what might be called "advanced mathematical analysis for mathematicians." It seems to me both useful and timely to attempt a reconciliation. The separation between kinds of courses has unhealthy effects. Mathe- matics students reverse the historical development of analysis, learning the unifying abstractions first and the examples later (if ever). Science students learn the examples as taught generations ago, missing modern insights. A choice between encountering Fourier series as a minor instance of the repre- sentation theory of Banach algebras, and encountering Fourier series in isolation and developed in an ad hoc manner, is no choice at all. It is easy to recognize these problems, but less easy to counter the legiti- mate pressures which have led to a separation. Modern mathematics has broadened our perspectives by abstraction and bold generalization, while developing techniques which can treat classical theories in a definitive way. On the other hand, the applier of mathematics has continued to need a variety of definite tools and has not had the time to acquire the broadest and most definitive grasp-to learn necessary and sufficient conditions when simple sufficient conditions will serve, or to learn the general framework encompass- ing different examples.
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.
