1 Introduction. - 1. 1 An Overview. - 1. 2 Topics by Chapter. - 1. 3 Applying Mathematics. - References. - 2 Fourier Series. - 2. 1 Introduction. - 2. 2 Inner Products and Fourier Expansions. - 2. 3 Convergence of Fourier Series. - 2. 4 Pointwise and Uniform Convergence of Fourier Series. - 2. 5 Gibb's Phenomenon and Summation Methods. - 2. 6 Summation Methods. - 2. 7 Fourier Series Properties. - 2. 8 Periodic Solutions of Differential Equations. - 2. 9 Impedance Methods and Periodic Solutions. - 2. 10 Power Spectrum and Parseval's Theorem. - References. - 3 Elementary Boundary Value Problems. - 3. 1 Introduction. - 3. 2 The One-Dimensional Diffusion Equation. - 3. 3 The Wave Equation. - 3. 4 The Potential Equation. - 3. 5 Discrete Models of Boundary Value Problems. - 3. 6 Separation of Variables. - 3. 7 Half-Range Expansions and Symmetries. - 3. 8 Some Matters of Detail. - References. - 4 Sturm-Liouville Theory and Boundary Value Problems. - 4. 1 Further Boundary Value Problems. - 4. 2 Selfadjoint Eigenvalue Problems. - 4. 3 Sturm-Liouville Problems. - 4. 4 Power Series and Singular Sturm-Liouville Problems. - 4. 5 Cylindrical Problems and Bessel's Equation. - 4. 6 Multidimensional Problems and Forced Systems. - 4. 7 Finite Differences and Numerical Methods. - 4. 8 Variational Models and Finite Element Methods. - 4. 9 Computational Finite Element Methods. - References. - 5 Functions of a Complex Variable. - 5. 1 Complex Variables and Analytic Functions. - 5. 2 Domains of Definition of Complex Functions. - 5. 3 Integrals and Cauchy's Theorem. - 5. 4 Cauchy's Integral Formula Taylor Series and Residues. - 5. 5 Complex Variables and Fluid Flows. - 5. 6 Conformal Mappings and the Principle of the Argument. - References. - 6 Laplace Transforms. - 6. 1 Introduction. - 6. 2 Definitions of the Laplace Transform. - 6. 3 Mechanical Properties of LaplaceTransforms. - 6. 4 Elementary Transforms and Fourier Series Calculations. - 6. 5 Elementary Applications to Differential Equations. - 6. 6 Convolutions Impulse Responses and Weighting Patterns. - 6. 7 Vector Differential Equations. - 6. 8 Impedance Methods. - References. - 7. Fourier Transforms. - 7. 1 Introduction. - 7. 2 Basic Fourier Transforms. - 7. 3 Formal Properties of Fourier Transforms. - 7. 4 Convolutions and Parseval's Theorem. - 7. 5 Comments on the Inversion Theorem. - 7. 6 Fourier Inversion by Contour Integration. - 7. 7 The Laplace Transform Inversion Integral. - 7. 8 An Introduction to Generalized Functions. - 7. 9 Fourier Transforms Differential Equations and Circuits. - 7. 10 Transform Solutions of Boundary Value Problems. - 7. 11 Band-limited Functions and Communications. - References. - 8 Discrete Variable Transforms. - 8. 1 Some Discrete Variable Models. - 8. 2 Z-Transforms. - 8. 3 Z-Transform Properties. - 8. 4 z-Transform Inversion Integral. - 8. 5 Discrete Fourier Transforms. - 8. 6 Discrete Fourier Transform Properties. - 8. 7 Some Applications of Discrete Transform Methods. - 8. 8 Finite and Fast Fourier Transforms. - 8. 9 Finite Fourier Properties. - 8. 10 Fast Finite Transform Algorithm. - 8. 11 Computing The 1-1. 1. - References. - 9 Additional Topics. - 9. 1 Local Waveform Analysis. - 9. 2 Uncertainty Principle. - 9. 3 Short-Time Fourier Transforms. - 9. 4 Function Shifts and Scalings. - 9. 5 Orthonormal Shifts. - 9. 6 Multi-Resolution Analysis and Wavelets. - 9. 7 On Wavelet Applications. - 9. 8 Two-Sided Transforms. - 9. 9 Walsh Functions. - 9. 10 Geometrically Based Transforms. - References. - A Linear Algebra Overview. - A. 1 Vector spaces. - A. 2 Linear Mappings. - A. 3 Inner Products. - A. 4 Linear Functionals and Dual Spaces. - A. 5 Canonical Forms. - References. - B Software Resources. - B. 1 Computational andVisualization Software. - B. 2 MATLAB Data Structures. - B. 3 MATLAB Operators and Syntax. - B. 4 MATLAB Programming Structures. - B. 5 MATLAB Programs and Scripts. - B. 6 Common Idioms. - B. 7 Graphics. - B. 8 Toolboxes and Enhancemants. - References. - C Transform Tables. - C. 1 Laplace Transforms. - C. 2 Fourier Transforms. - C. 3 Z Transforms. - C. 4 Discrete Fourier Transforms. Language: English