Advanced Mathematical Analysis: Periodic Functions and Distributions, Complex Analysis, Laplace Transform and Applications

Cover......Page 1
Title Page......Page 3
Copyright......Page 4
Dedication......Page 5
Preface......Page 6
TABLE OF CONTENTS......Page 8
1. Sets and functions......Page 11
2. Real and complex numbers......Page 15
3. Sequences of real and complex numbers......Page 20
4. Series......Page 24
5. Metric spaces......Page 29
6. Compact sets......Page 33
7. Vector spaces......Page 37
1. Continuity, uniform continuity, and compactness......Page 44
2. Integration of complex-valued functions......Page 48
3. Differentiation of complex-valued functions......Page 52
4. Sequences and series of functions......Page 57
5. Differential equations and the exponential function......Page 61
6. Trigonometric functions and the logarithm......Page 67
7. Functions of two variables......Page 72
8. Some infinitely differentiable functions......Page 77
1. Continuous periodic functions......Page 79
2. Smooth periodic functions......Page 82
3. Translation, convolution, and approximation......Page 87
4. The Weierstrass approximation theorems......Page 91
5. Periodic distributions......Page 94
6. Determining the periodic distributions......Page 99
7. Convolution of distributions......Page 104
8. Summary of operations on periodic distributions......Page 109
1. An inner product in W, and the space 22......Page 113
2. Hilbert space......Page 119
3. Hilbert spaces of sequences......Page 123
4. Orthonormal bases......Page 126
5. Orthogonal expansions......Page 131
6. Fourier series......Page 135
1. Fourier series of smooth periodic functions and periodic distributions......Page 141
2. Fourier series, convolutions, and approximation......Page 144
3. The heat equation: distribution solutions......Page 147
4. The heat equation: classical solutions; derivation......Page 152
5. The wave equation......Page 155
6. Laplace\'s equation and the Dirichlet problem......Page 160
1. Complex differentiation......Page 165
2. Complex integration......Page 169
3. The Cauchy integral formula......Page 176
4. The local behavior of a holomorphic function......Page 181
5. Isolated singularities......Page 185
6. Rational functions; Laurent expansions; residues......Page 189
7. Holomorphic functions in the unit disc......Page 194
1. Introduction......Page 200
2. The space Phi......Page 203
3. The space Phi\'......Page 207
4. Characterization of distributions of type Phi\'......Page 211
5. Laplace transforms of functions......Page 215
6. Laplace transforms of distributions......Page 220
7. Differential equations......Page 223
Notes and bibliography......Page 233
Notation index......Page 235
Subject index......Page 237
Back Cover......Page 243