Course Materials 

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The main text for this course is a book written by Dr. Herman: An Introduction to Fourier Analysis. Pages with color.

We will also use resources from the web.

• Gauss and the FFT

• The Vibrating String Controversy

  • Unpublished resolution

• The Fourier Transforms and its Applications - Stanford U, YouTube
• See Fourier Series and Gibbs Phenomena by Donny Lee
• Mobius Transformations Revealed - complex analysis
• A Crash Course in Contour Integration - complex analysis
• Fourier Transform Film, I will derive
• Last Class Example: PDF
  • Files SigAnal.mlx, firstbird.wav, LIGO Data NumL.txt, NumH.txt

Reviews for Exams

• Review #1, Sample Exam

• Review #2, Sample Exam

• Review #3, Sample Exam

• Final Review, Sample Final

• Project Files

Sequences and Infinite Series

• Sequences and Series in Maple

• Overview of sequences and series - See additional material, problems and videos

• SOSMath - Sequences, Series, Geometric Series, Power Series, 

• Wolfram Alpha - Series Examples

• Hyperbolic functions

• How Small is Small? - Talk by Dr. Herman

Fourier Series

• Fourier Series and Waves

• SOS Math - Fourier Series

• Fourier Series and Convergence

• Products of Trigonometric Functions

• Fourier Series - Eric Weinstein's World of Math

• Computer Computation

  • Maple Worksheets - fseries.mws, PlotFS.mws, EvenOdd.mws

  • Wolframalpha.com - Fourier series examples

    • sum_(k=1)^10(4/pi sin((2 k-1)*x)/(2 k-1))

    • sum_(k=1)^10(4/pi sin((2 k-1)*x)/(2 k-1))

    • sum_(n=1)^10(cos(2 n pi x)/(n pi)^2- sin(2 n pi x)/(n pi))

  • SageMath http://www.sagemath.org/, Ex:  http://sage.brandoncurtis.com/fourier.html

  • MATLAB plotfs.m (symbolic), plotfs0.m (not symbolic), FScoeff.m

    • GNU Octave - MATLAB clone

  • Python fsplot.py,

    • MIT Binder
    • https://www.educative.io/answers/how-to-implement-fourier-series-in-python

  • desmos https://www.desmos.com/calculator/oowvp6lolf

  • Software for Computing

• The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet ...- Google Books Result

• Note on Gibbs Phenomenon, 1925

• The Overshoot (12:28)

Complex Variables

• Notes on Complex Analysis

  • Complex Analysis - George Cain

  • Complex Analysis Notes - Harold V. McIntosh

  • Other books

• A Short History of Complex Numbers

• Short Course on Complex Numbers

• Visualizing Complex Functions

  • Complex Function Plots - Maple

  • ComplexGrid.m, plotcomplex.m, domcol2.m - Matlab

  • Complex Function Grapher

  • Complex Function Viewer

  • Domain Coloring

  • Domain Coloring for Complex Functions - Interactive

• Hyperbolic Function Identities

• Residue Theorem and examples

• The Residue Theorem and its Applications

• Definite integrals using the residue theorem - Orloff

• Short Practical Guide Computing Integrals With Closed Contours

Dirac Delta Function

• MathWorld

• K. Bryan's The Delta Function

• Dirac Delta Function Applications

• A Brief History - March 6

• Who invented the delta function?

Fourier Transforms

• Table of Fourier Transform Pairs - (Let j = -i)

• Differing Conventions

• Maple File for Fourier Transforms

• Finite Wave Train - Maple

• Fourier transform in Maple, Other transforms

• Fourier Transform, Fourier Series, Discrete Fourier Transform - Eric Weinstein's World of Math

• Wikipedia Entry

• Wolframalpha Integral Transforms

• MATLAB Files

  • Student Access - https://uncw.edu/itsd/working/matlab.html

  • MATLAB function (complex conjugate definition)

  • Fourier transform and convolution: ftrans.m, ftrans2.m, myconv.m

  • Laplace transform for ODEs: lapsolve.m, shoimpulse.m, odeprob.m

  • Laplace - symsum and convolution: sqwave.m, lconv.m

Laplace Transforms

• Laplace Transform Table - MS Word, PDF

• Practice Laplace Transform Problems, PDF

• Mathworld

• Tables

  • Collections of Worksheets! http://www.equationsheet.com/ 

  • http://www.vibrationdata.com/Laplace.htm 

  • eFunda: Laplace Transform Table

  • http://cnyack.homestead.com/files/alaplace/laptre.htm 

  • http://www.sosmath.com/tables/laplace/laplace.html 

  • Integral Transforms https://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm

• The Inverse Laplace Transform

• More Examples - Inverse Laplace Transform

• Bromwich Integral Example - video

• Key hole Example

• Using Laplace Transforms to Sum Series, a paper, examples

Signal Analysis

• An Introduction to the Sampling Theorem

• MATLAB DSP

• STFT Video

• Window functions

Discrete Transforms

• The Fast Fourier Transform

• Comparison of Fourier Transform, Fourier Series and DFT - Maple File

• Sampling Effects - Maple File

• Signals Talk

• Software

  • Spectrum Lab 

  • Cool Edit

  • Gold Wave

  • Audacity

MATLAB

• Basic Matlab

• Matlab TiddlyWiki - Herman

• Octave - Open Source Software that recognizes many Matlab functions

• Introduction to Computer Programming with MATLAB

• Creating a GUI in Matlab 5

MAPLE

• Basic Maple

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