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 $n$, ${\lambda }_n=(\frac{n\pi }{L}{)}^2$ is an eigenvalue and $X_n=c_n\text{sin}(\frac{n\pi }{L}x)$ is an eigenfunction for the boundary value problem $X^{″}+\lambda X=0$, $X(0)=0=X(L)$. This means general solution is a linear combinations of $X_n=c_n\text{sin}(\frac{n\pi }{L}x)$. We have the form $\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n\text{sin}(\frac{n\pi }{L}x)$. 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  $$\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\text{cos}(\frac{n\pi x}{L})+b_n\text{sin}(\frac{n\pi x}{L})).$$ On the set of points where this series converges, it defines a function $f(x)$ whose value at each point $x$ is the sum of the series for that value of $x$. The series is said to be the Fourier series of $f(x)$.  

(b) The coefficients $a_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\text{cos}(\frac{n\pi x}{L})dx$, $n=0,1,2,...,$ and $b_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\text{sin}(\frac{n\pi x}{L})dx$, $n=1,2,3,...$. which are known as the Euler-Fourier formulas.

(c) A function is periodic} with period $T>0$ if the domain of $f(x)$ contains $x+T$ whenever it contains $x$, and if $f(x+T)=f(x)$. In particular, $\text{sin}(n\pi x/L)$ and $\text{cos}(n\pi x/L)$ are periodic with period $T=2L/n$. Furthermore, $f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\text{cos}(\frac{n\pi x}{L})+b_n\text{sin}(\frac{n\pi x}{L}))$ has period $2L$. 

Example 1: Sketch the graph for 3 periods. Find the Fourier series. $f(x)=-x$, $-L<x<L$, $f(x+2L)=f(x)$.

Exercise 1: Sketch the graph for 3 periods. Find the Fourier series. $f(x)=x$, $-L<x<L$, $f(x+2L)=f(x)$.

Example 2: Sketch the graph for 3 periods. Find the Fourier series.  $$\begin{array}{rl}f(x)& =\{\begin{array}{ll}0& -L\le x<0\\ 1& 0\le x<L\end{array},\end{array}$$  $-L<x<L$, $f(x+2L)=f(x)$.

Exercise 2: Sketch the graph for 3 periods. Find the Fourier series.  $$\begin{array}{rl}f(x)& =\{\begin{array}{ll}1& -L\le x<0\\ 0& 0\le x<L\end{array},\end{array}$$ $-L<x<L$, $f(x+2L)=f(x)$.

Example 3: Sketch the graph for 3 periods. Find the Fourier series.  $$\begin{array}{rl}f(x)& =\{\begin{array}{ll}0& -\pi \le x<0\\ -x& 0\le x<\pi \end{array},\end{array}$$  $-\pi <x<\pi$, $f(x+2\pi )=f(x)$.

Exercise 3: Sketch the graph for 3 periods. Find the Fourier series.  $$\begin{array}{rl}f(x)& =\{\begin{array}{ll}x& -\pi \le x<0\\ 0& 0\le x<\pi \end{array},\end{array}$$ $-\pi <x<\pi$, $f(x+2\pi )=f(x)$.

Group work:

1. Sketch the graph for 3 periods. Find the Fourier series.  $$\begin{array}{rl}f(x)& =\{\begin{array}{ll}2& -2\le x<0\\ x+2& 0\le x<2\end{array},\end{array}$$  $-2<x<2$, $f(x+4)=f(x)$.

2. Sketch the graph for 3 periods. Find the Fourier series.  $$\begin{array}{rl}f(x)& =\{\begin{array}{ll}x+2& -2\le x<0\\ 2-x& 0\le x<2\end{array},\end{array}$$  $-2<x<2$, $f(x+4)=f(x)$.

3. Sketch the graph for 3 periods. Find the Fourier series.  $$\begin{array}{rl}f(x)& =\{\begin{array}{ll}L& -L\le x<0\\ L+x& 0\le x<L\end{array},\end{array}$$ $-L<x<L$, $f(x+2L)=f(x)$.