An Introduction to Fourier Analysis

Russell L. Herman

• Publisher: CRC Press
• Year: 2016
• ISBN:
  9781498773720
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

This book helps students explore Fourier analysis and its related topics, helping them appreciate why it pervades many fields of mathematics, science, and engineering.

This introductory textbook was written with mathematics, science, and engineering students with a background in calculus and basic linear algebra in mind. It can be used as a textbook for undergraduate courses in Fourier analysis or applied mathematics, which cover Fourier series, orthogonal functions, Fourier and Laplace transforms, and an introduction to complex variables. These topics are tied together by the application of the spectral analysis of analog and discrete signals, and provide an introduction to the discrete Fourier transform. A number of examples and exercises are provided including implementations of Maple, MATLAB, and Python for computing series expansions and transforms.

After reading this book, students will be familiar with:

• Convergence and summation of infinite series

• Representation of functions by infinite series

• Trigonometric and Generalized Fourier series

• Legendre, Bessel, gamma, and delta functions

• Complex numbers and functions

• Analytic functions and integration in the complex plane

• Fourier and Laplace transforms.

• The relationship between analog and digital signals

Dr. Russell L. Herman is a professor of Mathematics and Professor of Physics at the University of North Carolina Wilmington. A recipient of several teaching awards, he has taught introductory through graduate courses in several areas including applied mathematics, partial differential equations, mathematical physics, quantum theory, optics, cosmology, and general relativity. His research interests include topics in nonlinear wave equations, soliton perturbation theory, fluid dynamics, relativity, chaos and dynamical systems.

Chapter 1

Review of Sequences and Infinite Series

“Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” Sherlock Holmes (by Sir Arthur Conan Doyle, 1859–1930)

The material in this chapter is a review of material covered in a standard course in calculus with some additional notions from advanced calculus. It is provided as a review before encountering the notion of Fourier series and their convergence as seen in the next chapter.

IN THIS CHAPTER WE WILL REVIEW and extend some of the concepts and definitions related to infinite series that you might have seen previously in your calculus class¹.. Working with infinite series can be a little tricky and we need to understand some of the basics before moving on to the study of series of trigonometric functions.

For example, one can show that the infinite series

S=1−12+13−14+15−⋅⋅⋅

converges to ln 2. However, the terms can be rearranged to give

1+(13−12+15)+(17−14+19)+(111−16+113)+⋅⋅⋅=32ln2.

In fact, other rearrangements can be made to give any desired sum!

As we will see,

ln(1+x)=x−x2+x3−….

So, inserting x = 1 yields the first result - at least formall...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Dedication
• Table of Contents
• Introduction
• 1 Review of Sequences and Infinite Series
• 2 Fourier Trigonometric Series
• 3 Generalized Fourier Series and Function Spaces
• 4 Complex Analysis
• 5 Fourier and Laplace Transforms
• 6 From Continuous to Discrete Signals
• 7 Signal Analysis
• Bibliography
• Index