Section 6.4 Fourier Series introduction

Objective:

1. Definition of a Fourier series

2. Euler-Fourier formulas

3. Find the Fourier series of a linear piece-wise function

In previous section, we have for any integer [latex]n[/latex], [latex]\lambda_{n}=(\frac{n\pi}{L})^{2}[/latex] is an eigenvalue and [latex]X_{n}=c_{n}\text{sin}(\frac{n\pi}{L}x)[/latex] is an eigenfunction for the boundary value problem [latex]X''+\lambda X=0[/latex], [latex]X(0)=0=X(L)[/latex]. This means general solution is a linear combinations of [latex]X_{n}=c_{n}\text{sin}(\frac{n\pi}{L}x)[/latex]. We have the form [latex]\sum_{n=-\infty}^{\infty}c_{n}\text{sin}(\frac{n\pi}{L}x)[/latex]. We are interested in what exactly solution we should have and if the series is convergent. We introduce the Fourier series.

Definition: (a) We begin with a series of the form  \[\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\text{cos}(\frac{n\pi x}{L})+b_{n}\text{sin}(\frac{n\pi x}{L})\right). \] On the set of points where this series converges, it defines a function [latex]f(x)[/latex] whose value at each point [latex]x[/latex] is the sum of the series for that value of [latex]x[/latex]. The series is said to be the Fourier series of [latex]f(x)[/latex].  

(b) The coefficients [latex]a_{n}=\frac{1}{L}\int_{-L}^{L}f(x)\text{cos}(\frac{n\pi x}{L})dx[/latex], [latex]n=0,1,2,...,[/latex] and [latex]b_{n}=\frac{1}{L}\int_{-L}^{L}f(x)\text{sin}(\frac{n\pi x}{L})dx[/latex], [latex]n=1,2,3,...[/latex]. which are known as the Euler-Fourier formulas.

(c) A function is periodic} with period [latex]T \gt 0[/latex] if the domain of [latex]f(x)[/latex] contains [latex]x+T[/latex] whenever it contains [latex]x[/latex], and if [latex]f(x+T)=f(x)[/latex]. In particular, [latex]\text{sin}(n\pi x/L)[/latex] and [latex]\text{cos}(n\pi x/L)[/latex] are periodic with period [latex]T=2L/n[/latex]. Furthermore, [latex]f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\text{cos}(\frac{n\pi x}{L})+b_{n}\text{sin}(\frac{n\pi x}{L})\right)[/latex] has period [latex]2L[/latex]. 

Example 1: Sketch the graph for 3 periods. Find the Fourier series. [latex]f(x)=-x[/latex], [latex]-L \lt x \lt L[/latex], [latex]f(x+2L)=f(x)[/latex].

Exercise 1: Sketch the graph for 3 periods. Find the Fourier series. [latex]f(x)=x[/latex], [latex]-L \lt x \lt L[/latex], [latex]f(x+2L)=f(x)[/latex].

Example 2: Sketch the graph for 3 periods. Find the Fourier series.  \begin{align*} f(x) & =\begin{cases} 0 & -L\leq x \lt 0\\ 1 & 0\leq x \lt L \end{cases}, \end{align*}  [latex]-L \lt x \lt L[/latex], [latex]f(x+2L)=f(x)[/latex].

Exercise 2: Sketch the graph for 3 periods. Find the Fourier series.  \begin{align*} f(x) & =\begin{cases} 1 & -L\leq x \lt 0\\ 0 & 0\leq x \lt L \end{cases}, \end{align*} [latex]-L \lt x \lt L[/latex], [latex]f(x+2L)=f(x)[/latex].

Example 3: Sketch the graph for 3 periods. Find the Fourier series.  \begin{align*} f(x) & =\begin{cases} 0 & -\pi\leq x \lt 0\\ -x & 0\leq x \lt \pi \end{cases}, \end{align*}  [latex]-\pi \lt x \lt \pi[/latex], [latex]f(x+2\pi)=f(x)[/latex].

Exercise 3: Sketch the graph for 3 periods. Find the Fourier series.  \begin{align*} f(x) & =\begin{cases} x & -\pi\leq x \lt 0\\ 0 & 0\leq x \lt \pi \end{cases}, \end{align*} [latex]-\pi \lt x \lt \pi[/latex], [latex]f(x+2\pi)=f(x)[/latex].

Group work:

1. Sketch the graph for 3 periods. Find the Fourier series.  \begin{align*} f(x) & =\begin{cases} 2 & -2\leq x \lt 0\\ x+2 & 0\leq x \lt 2 \end{cases}, \end{align*}  [latex]-2 \lt x \lt 2[/latex], [latex]f(x+4)=f(x)[/latex].

2. Sketch the graph for 3 periods. Find the Fourier series.  \begin{align*} f(x) & =\begin{cases} x+2 & -2\leq x \lt 0\\ 2-x & 0\leq x \lt 2 \end{cases}, \end{align*}  [latex]-2 \lt x \lt 2[/latex], [latex]f(x+4)=f(x)[/latex].

3. Sketch the graph for 3 periods. Find the Fourier series.  \begin{align*} f(x) & =\begin{cases} L & -L\leq x \lt 0\\ L+x & 0\leq x \lt L \end{cases}, \end{align*} [latex]-L \lt x \lt L[/latex], [latex]f(x+2L)=f(x)[/latex].