Section 6.5 Fourier Series

Objective:

1. The Fourier Convergence Theorem

In previous section, if a Fourier series \[\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\text{cos}(\frac{n\pi x}{L})+b_{n}\text{sin}(\frac{n\pi x}{L})\right).\]  converges and thereby defines a function [latex]f(x)[/latex] then [latex]f(x)[/latex] is periodic with period 2L, with 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]. What we do is that we begin with a periodic function [latex]f(x)[/latex] of period [latex]2L[/latex] that is integrable on [latex][-L,L].[/latex] We compute [latex]a_{n}[/latex] and [latex]b_{n}[/latex] using the formulas above and construct the associated Fourier series. The question is whether this series converges for each [latex]x[/latex], and if so, whether its sum is [latex]f(x)[/latex].

Theorem:  Suppose that [latex]f(x)[/latex] and [latex]f'(x)[/latex] are piecewise continuous on [latex][-L,L)[/latex]. Further, suppose that [latex]f(x)[/latex] is defined outside[latex][-L,L)[/latex] so that it is periodic with period [latex]2L[/latex]. Then [latex]f(x)[/latex] has a Fourier series \[ f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\text{cos}(\frac{n\pi x}{L})+b_{n}\text{sin}(\frac{n\pi x}{L})\right) \] where [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]. The Fourier series converges to [latex]f(x)[/latex] at all points [latex]x[/latex] where [latex]f(x)[/latex] is continuous, and to [latex][f(x^{+})+f(x^{-})]/2[/latex] at all points [latex]x[/latex] where [latex]f[/latex] is discontinuous.

Example 1: Sketch the graph for 3 periods. Find the Fourier series. Graph where the series convergent.  \begin{align*} f(x) & =\begin{cases} 1 & -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 1: Sketch the graph for 3 periods. Find the Fourier series. Graph where the series convergent.  \begin{align*} f(x) & =\begin{cases} -1 & -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].

Example 2: Sketch the graph for 3 periods. Find the Fourier series. Graph where the series convergent.  \begin{align*} f(x) & =\begin{cases} -x & -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].

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

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

Group work:

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