$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} \newcommand{\sech}{\text{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\Ln}{\text{Ln}} \newcommand{\p}{\partial} \newcommand{\intd}[1]{\int\hspace{-0.7em}\int\limits_{\hspace{-0.7em}{#1}}} $

Lecture 33, 11/14/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

Chapter 7: Fourier Series and Transform

  1. Average of a function:
    Average of $f(x)$ on $(a,b) = \frac1{b-a}\int_a^b f(x) dx$.
  2. Fourier Series: $-\pi< x< \pi$

    $\begin{array}{rcl} f(x) & \sim & \frac12 a_0 + a_1 \cos x + a_2 \cos 2x + a_3 \cos 3x + \cdots \\ && \mm + b_1 \sin x + b_2 \sin 2x + b_3 \sin 3x + \cdots \end{array}$

    $\mm\mm $
    $a_n = \frac1\pi\int_{-\pi}^\pi f(x) \cos nx dx $


    $\mm\mm $
    $b_n = \frac1\pi\int_{-\pi}^\pi f(x) \sin nx dx $



    The example we did during the lecture: $f(x) = \left\{ \begin{array}{ll} 0, & -\pi < x \le 0 \\ 1, & 0 < x <\pi \end{array}\right.$
    $f(x) \sim \frac12 +\frac2\pi\Big( \frac{\sin x}1 + \frac{\sin 3x}3 + \frac{\sin 5x}5+\cdots \Big)$ 
      N = 100;
  x = -pi:.01:pi;
  f = ones(1,length(x))*0.5;
  for i = 1:2:N
    f = f + (2/(pi*i)) * sin(i*x);
  end
  plot(x,f)