Problems

01

Fourier series of triangle wave

Find the Fourier series of the triangle wave in the figure:

Be sure to account for the period of $4$ rather than $2\pi$. (You should change variables so that the wave covers one cycle on the interval $[-\pi ,\pi ]$ in the new variable, and then change back at the very end to express your final answer.)

02

Full Fourier series with rescaling, even and odd parts

Compute the Fourier series for the function $e^x$ defined on the interval $[-1,1]$.

You should first change to a new independent variable for which that interval corresponds to the interval $[-\pi ,\pi ]$, and change back at the end.

03

Fourier series can teach us about $\pi$

• (a) Compute the Fourier series of the following sawtooth wave with $L=1$:
  
• (b) Evaluate your Fourier series at $(1/2,1/2)$ to establish Leibniz’s famous formula: $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\cdots =\frac{\pi }{4}.$$
• (c) Find a similar series expression for $\frac{{\pi }^2}{8}$ using your solution to Problem 14-01.

04

Fourier series can solve initial value problems

For this problem, you should solve the given ODE by plugging in a generic Fourier series and solving for the coefficients. Fourier coefficients are uniquely determined because Fourier terms are orthogonal.

• (a) $y^{\prime \prime }+{\omega }^{2}y=\sin (nt),y(0)=0=y^{\prime }(0)$
• (b) $y^{\prime \prime }+{\omega }^{2}y=f(t),y(0)=0=y^{\prime }(0)$, where $f(t)$ is the triangle wave in Problem 14-01. (Substitute the Fourier series you found for $f(t)$.)