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AuthorBeals, Richard. author
TitleAdvanced Mathematical Analysis [electronic resource] : Periodic Functions and Distributions, Complex Analysis, Laplace Transform and Applications / by Richard Beals
ImprintNew York, NY : Springer New York : Imprint: Springer, 1973
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Descript XII, 234 p. online resource


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 engiยญ 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. Matheยญ 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 repreยญ 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 legitiยญ 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 encompassยญ ing different examples


One Basis concepts -- ยง1. Sets and functions -- ยง2. Real and complex numbers -- ยง3. Sequences of real and complex numbers -- ยง4. Series -- ยง5. Metric spaces -- ยง6. Compact sets -- ยง7. Vector spaces -- Two Continuous functions -- ยง1. Continuity, uniform continuity, and compactness -- ยง2. Integration of complex-valued functions -- ยง3. Differentiation of complex-valued functions -- ยง4. Sequences and series of functions -- ยง5. Differential equations and the exponential function -- ยง6. Trigonometric functions and the logarithm -- ยง7. Functions of two variables -- ยง8. Some infinitely differentiable functions -- Three Periodic functions and periodic distributions -- ยง1. Continuous periodic functions -- ยง2. Smooth periodic functions -- ยง3. Translation, convolution, and approximation -- ยง4. The Weierstrass approximation theorems -- ยง5. Periodic distributions -- ยง6. Determining the periodic distributions -- ยง7. Convolution of distributions -- ยง8. Summary of operations on periodic distributions -- Four Hilbert spaces and Fourier series -- ยง1. An inner product in ?, and the space ?2 -- ยง2. Hilbert space -- ยง3. Hilbert spaces of sequences -- ยง4. Orthonormal bases -- ยง5. Orthogonal expansions -- ยง6. Fourier series -- Five Applications of Fourier series -- ยง1. Fourier series of smooth periodic functions and periodic distributions -- ยง2. Fourier series, convolutions, and approximation -- ยง3. The heat equation: distribution solutions -- ยง4. The heat equation: classical solutions; derivation -- ยง5. The wave equation -- ยง6. Laplaceโ{128}{153}s equation and the Dirichlet problem -- Six Complex analysis -- ยง1. Complex differentiation -- ยง2. Complex integration -- ยง3. The Cauchy integral formula -- ยง4. The local behavior of a holomorphic function -- ยง5. Isolated singularities -- ยง6. Rational functions; Laurent expansions; residues -- ยง7. Holomorphic functions in the unit disc -- Seven The Laplace transform -- ยง1. Introduction -- ยง2. The space ? -- ยง3. The space ?? -- ยง4. Characterization of distributions of type ?? -- ยง5. Laplace transforms of functions -- ยง6. Laplace transforms of distributions -- ยง7. Differential equations -- Notes and bibliography -- Notation index

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