5.6: Problems

5.1. Find the Fourier Series of each function \(f(x)\) of period \(2 \pi\). For each series, plot the \(N\)th partial sum,

\[S_{N}=\dfrac{a_{0}}{2}+\sum_{n=1}^{N}\left[a_{n} \cos n x+b_{n} \sin n x\right] \nonumber \]

for \(N=5,10,50\) and describe the convergence (is it fast? what is it converging to, etc.) [Some simple Maple code for computing partial sums is shown below.]

a. \(f(x)=x,|x|<\pi\).
b. \(f(x)=\dfrac{x^{2}}{4},|x|<\pi\).
c. \(f(x)=\pi-|x|,|x|<\pi\).
d. \(f(x)=\left\{\begin{array}{c}\dfrac{\pi}{2}, \quad 0<x<\pi \\ -\dfrac{\pi}{2}, \pi<x<2 \pi\end{array}\right.\)
e. \(f(x)=\left\{\begin{array}{l}0,-\pi<x<0 \\ 1,0<x<\pi\end{array}\right.\)

A simple set of commands in Maple are shown below, where you fill in the Fourier coefficients that you have computed by hand and \(f(x)\) so that you can compare your results. Of course, other modifications may be needed.

> restart: 
> f:=x: 
> F:=a0/2+sum(an*cos(n*x)+bn*sin(n*x),n=1..N): 
> N:=10: plot({f,F},x=-Pi..Pi,color=black);

5.2. Consider the function \(f(x)=4 \sin ^{3} 2 x\)

a. Derive an identity relating \(\sin ^{3} \theta\) in terms of \(\sin \theta\) and \(\sin 3 \theta\) and express \(f(x)\) in terms of simple sine functions.
b. Determine the Fourier coefficients of \(f(x)\) in a Fourier series expansion on \([0,2 \pi]\) without computing any integrals!

5.3. Find the Fourier series of \(f(x)=x\) on the given interval with the given period \(T\). Plot the \(N\)th partial sums and describe what you see.

a. \(0<x<2, T=2\).
b. \(-2<x<2, T=4\).

5.4. The result in problem \(5.1 \mathrm{~b}\) above gives a Fourier series representation of \(\dfrac{x^{2}}{4}\). By picking the right value for \(x\) and a little arrangement of the series, show that [See Example 5.8.]

a. \(\dfrac{\pi^{2}}{6}=1+\dfrac{1}{2^{2}}+\dfrac{1}{3^{2}}+\dfrac{1}{4^{2}}+\cdots\)

b. \(\dfrac{\pi^{2}}{8}=1+\dfrac{1}{3^{2}}+\dfrac{1}{5^{2}}+\dfrac{1}{7^{2}}+\cdots\)

5.5. Sketch (by hand) the graphs of each of the following functions over four periods. Then sketch the extensions each of the functions as both an even and odd periodic function. Determine the corresponding Fourier sine and cosine series and verify the convergence to the desired function using Maple.

a. \(f(x)=x^{2}, 0<x<1\)
b. \(f(x)=x(2-x), 0<x<2\).
c. \(f(x)=\left\{\begin{array}{l}0,0<x<1 \\ 1,1<x<2\end{array}\right.\)
d. \(f(x)=\left\{\begin{array}{c}\pi, \quad 0<x<\pi, \\ 2 \pi-x, \pi<x<2 \pi\end{array}\right.\)