R U S S E L L L. H E R M A N

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1 R U S S E L L L. H E R M A N A N I N T R O D U C T I O N T O F O U R I E R A N D C O M P L E X A N A LY S I S W I T H A P P L I C AT I O N S T O T H E S P E C T R A L A N A LY S I S O F S I G N A L S R. L. H E R M A N - V E R S I O N D AT E : J A N U A R Y 1 3,

2 Copyright by Russell L. Herman published by r. l. herman This text has been reformatted from the original using a modification of the Tufte-book documentclass in LATEX. See tufte-latex.googlecode.com. an introduction to fourier and complex analysis with applications to the spectral analysis of signals by Russell Herman is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. These notes have resided at since Spring Sixth printing, January 2016

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4 Contents Introduction vii 1 Review of Sequences and Infinite Series Sequences of Real Numbers Convergence of Sequences Limit Theorems Infinite Series Geometric Series Convergence Tests Sequences of Functions Infinite Series of Functions Special Series Expansions Power Series Binomial Series The Order of Sequences and Functions Problems Fourier Trigonometric Series Introduction to Fourier Series Fourier Trigonometric Series Fourier Series over Other Intervals Fourier Series on [a, b] Sine and Cosine Series The Gibbs Phenomenon Multiple Fourier Series Problems Generalized Fourier Series and Function Spaces Finite Dimensional Vector Spaces Function Spaces Classical Orthogonal Polynomials Fourier-Legendre Series Properties of Legendre Polynomials The Generating Function for Legendre Polynomials The Differential Equation for Legendre Polynomials Fourier-Legendre Series Examples

5 iv 3.5 Gamma Function Fourier-Bessel Series Appendix: The Least Squares Approximation Appendix: Convergence of Trigonometric Fourier Series Problems Complex Analysis Complex Numbers Complex Valued Functions Complex Domain Coloring Complex Differentiation Complex Integration Complex Path Integrals Cauchy s Theorem Analytic Functions and Cauchy s Integral Formula Laurent Series Singularities and The Residue Theorem Infinite Integrals Integration over Multivalued Functions Appendix: Jordan s Lemma Problems Fourier and Laplace Transforms Introduction Complex Exponential Fourier Series Exponential Fourier Transform The Dirac Delta Function Properties of the Fourier Transform Fourier Transform Examples The Convolution Operation Convolution Theorem for Fourier Transforms Application to Signal Analysis Parseval s Equality The Laplace Transform Properties and Examples of Laplace Transforms Applications of Laplace Transforms Series Summation Using Laplace Transforms Solution of ODEs Using Laplace Transforms Step and Impulse Functions The Convolution Theorem The Inverse Laplace Transform Transforms and Partial Differential Equations Fourier Transform and the Heat Equation Laplace s Equation on the Half Plane Heat Equation on Infinite Interval, Revisited Nonhomogeneous Heat Equation

6 v Problems From Analog to Discrete Signals Analog to Periodic Signals The Dirac Comb Function Discrete Signals Summary The Discrete (Trigonometric) Fourier Transform Discrete Trigonometric Series Discrete Orthogonality The Discrete Fourier Coefficients The Discrete Exponential Transform FFT: The Fast Fourier Transform Applications MATLAB Implementation MATLAB for the Discrete Fourier Transform Matrix Operations for MATLAB MATLAB Implementation of FFT Problems Signal Analysis Introduction Periodogram Examples Effects of Sampling Effect of Finite Record Length Aliasing The Shannon Sampling Theorem Nonstationary Signals Simple examples The Spectrogram Short-Time Fourier Transform Harmonic Analysis Problems Bibliography 317 Index 319

7 vi Dedicated to those students who have endured the various editions of an introduction to fourier and complex analysis with applications to the spectral analysis of signals.