An Introduction to Fourier Analysis

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.

這本書幫助學生探索傅立葉分析及其相關主題，幫助他們理解為什麼它在許多數學、科學和工程領域中普遍存在。

這本入門教材是針對具有微積分和基礎線性代數背景的數學、科學和工程學生而寫的。它可以作為傅立葉分析或應用數學的本科課程教材使用，涵蓋傅立葉級數、正交函數、傅立葉和拉普拉斯變換以及複變數的介紹。這些主題通過模擬和離散信號的頻譜分析應用相互關聯，並介紹了離散傅立葉變換。書中提供了許多例子和練習，包括使用Maple、MATLAB和Python進行計算級數展開和變換的實現。

閱讀本書後，學生將熟悉以下內容：
• 無窮級數的收斂和求和
• 函數的無窮級數表示
• 三角和廣義傅立葉級數
• 勒讓德、貝塞爾、伽馬和德爾塔函數
• 複數和函數
• 複平面上的解析函數和積分
• 傅立葉和拉普拉斯變換
• 模擬和數字信號之間的關係

Russell L. Herman博士是北卡羅來納威爾明頓大學的數學和物理學教授。他曾獲得多個教學獎項，並在應用數學、偏微分方程、數學物理學、量子理論、光學、宇宙學和廣義相對論等多個領域教授過入門到研究生課程。他的研究興趣包括非線性波動方程、孤立子擾動理論、流體動力學、相對論、混沌和動力系統。