An Introduction to Fourier Analysis.
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| Format: | eBook |
| Language: | English |
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Milton :
CRC Press,
2016.
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| Online Access: | Connect to the full text of this electronic book |
Table of Contents:
- Cover; Half Title; Title; Copyright; Dedication; Table of Contents; Introduction; 1 Review of Sequences and Infinite Series; 1.1 Sequences of Real Numbers; 1.2 Convergence of Sequences; 1.3 Limit Theorems; 1.4 Infinite Series; 1.5 Geometric Series; 1.6 Convergence Tests; 1.7 Sequences of Functions; 1.8 Infinite Series of Functions; 1.9 Special Series Expansions; 1.10 Power Series; 1.11 Binomial Series; 1.12 The Order of Sequences and Functions; Problems; 2 Fourier Trigonometric Series; 2.1 Introduction to Fourier Series; 2.2 Fourier Trigonometric Series.
- 2.3 Fourier Series over Other Intervals2.3.1 Fourier Series on [0, L]; 2.3.2 Parseval's Identity; 2.3.3 Fourier Series on [a, b]; 2.4 Sine and Cosine Series; 2.5 The Gibbs Phenomenon; 2.6 Multiple Fourier Series; 2.7 Appendix: Convergence of Trigonometric Fourier Series; 2.8 Plotting Fourier Series; 2.8.1 MATLAB and GNU Octave Files; 2.8.2 Python Scripts; Problems; 3 Generalized Fourier Series and Function Spaces; 3.1 Finite Dimensional Vector Spaces; 3.2 Function Spaces; 3.3 Classical Orthogonal Polynomials; 3.4 Fourier-Legendre Series; 3.4.1 Properties of Legendre Polynomials.
- 3.4.2 The Generating Function for Legendre Polynomials3.4.3 The Differential Equation for Legendre Polynomials; 3.4.4 Fourier-Legendre Series Examples; 3.5 Gamma Function; 3.6 Fourier-Bessel Series; 3.7 Appendix: The Least Squares Approximation; Problems; 4 Complex Analysis; 4.1 Complex Numbers; 4.2 Complex Valued Functions; 4.2.1 Complex Domain Coloring; 4.3 Complex Differentiation; 4.4 Complex Integration; 4.4.1 Complex Path Integrals; 4.4.2 Cauchy's Theorem; 4.4.3 Analytic Functions and Cauchy's Integral Formula; 4.4.4 Laurent Series; 4.4.5 Singularities and The Residue Theorem.
- 4.4.6 Infinite Integrals4.4.7 Integration over Multivalued Functions; 4.4.8 Appendix: Jordan's Lemma; Problems; 5 Fourier and Laplace Transforms; 5.1 Introduction; 5.2 Complex Exponential Fourier Series; 5.3 Exponential Fourier Transform; 5.4 The Dirac Delta Function; 5.5 Properties of the Fourier Transform; 5.5.1 Fourier Transform Examples; 5.6 The Convolution Operation for Fourier Transforms; 5.6.1 Convolution Theorem for Fourier Transforms; 5.6.2 Application to Signal Analysis; 5.6.3 Plancherel's Formula; 5.7 The Laplace Transform; 5.7.1 Properties and Examples of Laplace Transforms.
- 5.8 Applications of Laplace Transforms5.8.1 Series Summation Using Laplace Transforms; 5.8.2 Solution of ODEs Using Laplace Transforms; 5.8.3 Step and Impulse Functions; 5.9 The Convolution Theorem for Laplace Transforms; 5.10 The Inverse Laplace Transform; 5.11 Transforms and Partial Differential Equations; 5.11.1 Fourier Transform and the Heat Equation; 5.11.2 Laplace's Equation on the Half-Plane; 5.11.3 Heat Equation on Infinite Interval, Revisited; 5.11.4 Nonhomogeneous Heat Equation; 5.12 Computing Fourier and Laplace Transforms; 5.12.1 Using MATLAB for Transforms.