Fourier Series and Integral TransformsThis volume provides the reader with a basic understanding of Fourier series, Fourier transforms and Laplace transforms. The book is an expanded and polished version of the authors' notes for a one semester course, for students of mathematics, electrical engineering, physics and computer science. Prerequisites for readers of this book are a basic course in both calculus and linear algebra. Otherwise the material is self-contained with numerous exercises and various examples of applications. |
Contents
Notation and Terminology | 1 |
2 Calculus Notation | 2 |
3 Useful Trigonometric Formulae | 4 |
Background Inner Product Spaces | 5 |
2 The Norm | 10 |
3 Orthogonal and Orthonormal Systems | 15 |
4 Orthogonal Projections and Approximation in the Mean | 19 |
5 Infinite Orthonormal Systems | 24 |
The Fourier Transform | 93 |
2 Examples | 98 |
3 Properties and Formulae | 102 |
4 The Inverse Fourier Transform and Plancherels Identity | 108 |
6 Applications of the Residue Theorem | 119 |
7 Applications to Partial Differential Equations | 125 |
8 Applications to Signal Processing | 130 |
The Laplace Transform | 140 |
Fourier Series | 32 |
2 Evenness Oddness and Additional Examples | 40 |
3 Complex Fourier Series | 42 |
4 Pointwise Convergence and Dirichlets Theorem | 46 |
5 Uniform Convergence | 56 |
6 Parsevals Identity | 63 |
7 The Gibbs Phenomenon | 68 |
8 Sine and Cosine Series | 72 |
9 Differentiation and Integration of Fourier Series | 76 |
10 Fourier Series on Other Intervals | 81 |
11 Applications to Partial Differential Equations | 85 |
2 More Formulae and Examples | 143 |
3 Applications to Ordinary Differential Equations | 149 |
4 The Heaviside and DiracDelta Functions | 155 |
5 Convolution | 162 |
6 More Examples and Applications | 168 |
7 The Inverse Transform Formula | 173 |
8 Applications of the Inverse Transform | 175 |
The Residue Theorem and Related Results | 182 |
Leibnizs Rule and Fubinis Theorem | 186 |
Index | 188 |
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Common terms and phrases
absolutely integrable b₁ b₂ sin nx Bessel's inequality Cauchy-Schwarz inequality complex Fourier series constant continuous function converges uniformly convolution define the function definition denote the Fourier differential equations Dirichlet's Theorem E G(R e-st Example exists f(x+ finite number follows Fourier coefficients Fubini's Theorem function ƒ ƒ converges ƒ is continuous heat equation infinite orthonormal system inner product space interval inverse Fourier transform inverse Laplace transform Let f Let f(x Let ƒ linear space method n=1 denote natural number number of points nx dx obtain odd function orthonormal system Parseval's identity partial sums piecewise continuous piecewise continuous function Plancherel's identity points of discontinuity polynomial properties Proposition real number Residue Theorem scalars sequence series converges series of ƒ sin(m sinnx Sm(x solution solve Theorem Theorem transform of f uniform convergence vector
References to this book
Handbook of Mathematics for Engineers and Scientists Andrei D. Polyanin,Alexander V. Manzhirov No preview available - 2006 |
Numerische Verbrennungssimulation: Effiziente numerische Simulation ... Peter Gerlinger No preview available - 2005 |