$\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 34, 11/16/2022. 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. Fourier coefficients: more examples.
  2. Dirichlet Conditions (convergence):
    If $f(x)$ is extended to be a $2\pi$ periodic function with finitly many jump discontinuities, with \[ \int_{-\pi}^\pi |f(x)| dx <\infty \] then
    $\mm$ (a) the Fourier series converges to $f(x)$ at any continuous point, and
    $\mm$ (b) the Fourier series converges to midpoint at any jump.