Author | Beals, Richard. author |
---|---|
Title | Advanced Mathematical Analysis [electronic resource] : Periodic Functions and Distributions, Complex Analysis, Laplace Transform and Applications / by Richard Beals |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1973 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9886-8 |
Descript | XII, 234 p. online resource |
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. Laplaceโs 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