$\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 36, 11/21/2022. 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. Even and Odd functions
    $\fcolorbox{white}{yellow}{$f(x)$ is even if $f(-x)=f(x)$, i.e., symmetric along $y$-axis.}$
    $\fcolorbox{white}{yellow}{$f(x)$ is odd if $f(-x)=-f(x)$, i.e., symmetric with respect to origin.}$
    $\mm$
    $\int_{-l}^l f(x)dx = \left\{\begin{array}{ll} 0 & \text{ if $f(x)$ is odd}\\ 2\int_{0}^l f(x)dx & \text{ if $f(x)$ is even} \end{array}\right.$


    In particular,
    $\mm$
    $\fcolorbox{white}{yellow}{Sine Series:}$ If $f(x)$ is odd, then $a_n=0$, $b_n = \frac2l \int_0^l f(x)\sin \frac{n\pi x}l dx$

    $\mm$
    $\fcolorbox{white}{yellow}{Cosine Series:}$ If $f(x)$ is even, then $b_n=0$, $a_n = \frac2l \int_0^l f(x)\cos \frac{n\pi x}l dx$


    Product:
    $\mm$ (odd)$\cdot$(even) $=$ (odd); $\mm$ (even)$\cdot$(even) $=$ (even); $\mm$ (odd)$\cdot$(odd) $=$ (even);