Section 6.5 Fourier Series

Objective:

1. The Fourier Convergence Theorem

In previous section, if a Fourier series $$\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\text{cos}(\frac{n\pi x}{L})+b_n\text{sin}(\frac{n\pi x}{L})).$$  converges and thereby defines a function $f(x)$ then $f(x)$ is periodic with period 2L, with 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 $b_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\text{sin}(\frac{n\pi x}{L})dx$, $n=1,2,3,...$. What we do is that we begin with a periodic function $f(x)$ of period $2L$ that is integrable on $[-L,L].$ We compute $a_n$ and $b_n$ using the formulas above and construct the associated Fourier series. The question is whether this series converges for each $x$, and if so, whether its sum is $f(x)$.

Theorem:  Suppose that $f(x)$ and $f^{\prime }(x)$ are piecewise continuous on $[-L,L)$. Further, suppose that $f(x)$ is defined outside$[-L,L)$ so that it is periodic with period $2L$. Then $f(x)$ has a Fourier series $$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}(a_n\text{cos}(\frac{n\pi x}{L})+b_n\text{sin}(\frac{n\pi x}{L}))$$ where $a_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\text{cos}(\frac{n\pi x}{L})dx$, $n=0,1,2,...,$ and $b_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\text{sin}(\frac{n\pi x}{L})dx$,$n=1,2,3,...$. The Fourier series converges to $f(x)$ at all points $x$ where $f(x)$ is continuous, and to $[f(x^{+})+f(x^{-})]/2$ at all points $x$ where $f$ is discontinuous.

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

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

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

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

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

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

Group work:

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

2. Sketch the graph for 3 periods. Find the Fourier series. Graph where the series convergent.  $$\begin{array}{rl}f(x)& =1-x^2,\end{array}$$  $-1<x<1$, $f(x+2)=f(x)$.