Fourier Series — Problem Sheet (10 exercises, undergraduate)

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\title{Fourier Series --- Problem Sheet}
\author{}
\date{}

\begin{document}
\maketitle

\exercise
Let $f_1, f_2$ be two odd functions and $g_1, g_2$ two even functions.
\begin{enumerate}[nosep, label=\arabic*.]
  \item Determine the parity of $g_1 \times g_2$, $f_1 \times f_2$ and $f_1 \times g_1$.
  \item Show that $\disp{\int_{-T}^{T} f_1(x)\, \mathrm{d}x = 0}$.
  \item Show that $\disp{\int_{-T}^{T} g_1(x)\, \mathrm{d}x = 2 \int_{0}^{T} g_1(x)\, \mathrm{d}x}$.
\end{enumerate}

\vs

\exercise
Compute $\disp{\int_{0}^{\pi} \sin(t) \cos(2t)\, \mathrm{d}t}$.

\vs

\exercise
Let $f$ be the $2\pi$-periodic function defined by:
\[
\left\{
\begin{array}{l}
f(x) = 1 \text{ on } ]0\,;\,\pi[ \\
f(x) = -1 \text{ on } ]-\pi\,;\,0[ \\
f(0) = f(\pi) = 0
\end{array}
\right.
\]
\begin{enumerate}[nosep, label=\arabic*.]
  \item Sketch the graph of $f$ on $]-2\pi\,;\,2\pi[$.
  \item Show that $\disp{b_n = \frac{4}{n\pi}}$ for every odd $n \in \mathbb{N}^*$ and $b_n = 0$ for every even $n \in \mathbb{N}^*$.
  \item Deduce the Fourier series expansion of $f$.
  \item Show that $\disp{\sum_{n=0}^{+\infty} \frac{(-1)^n}{2n+1} = \frac{\pi}{4}}$.
  \item Using Parseval's identity, compute $\disp{\sum_{n=0}^{+\infty} \frac{1}{(2n+1)^2}}$.
\end{enumerate}

\vs

\exercise
Let $f$ be the $2\pi$-periodic function defined by $f(x) = x^2$ on $[-\pi, \pi]$.
\begin{enumerate}[nosep, label=\arabic*.]
  \item Sketch the graph of $f$ on $[-2\pi, 2\pi]$.
  \item Show that $a_0 = \dfrac{\pi^2}{3}$ and that for every $n \in \mathbb{N}^*$, $\disp{a_n = \frac{4(-1)^n}{n^2}}$.
  \item Deduce the Fourier series expansion of $f$.
  \item Does $f$ satisfy the hypotheses of Dirichlet's theorem?
  \item Deduce the value of $\disp{\sum_{n=1}^{+\infty} \frac{(-1)^n}{n^2}}$.
  \item Show that $\disp{\sum_{n=1}^{+\infty} \frac{1}{n^4} = \frac{\pi^4}{90}}$.
\end{enumerate}

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\exercise
Using Parseval's identity, prove that:
\[ \int_{-\pi}^{\pi} \cos^4 x\, \mathrm{d}x = \frac{3\pi}{4}. \]

\vs

\exercise
Solve the differential equation:
\[ y'' + y = \cos(t). \]

\vs

\exercise
Let $f$ be the $2\pi$-periodic function defined on $]-\pi\,;\,\pi[$ by $f(x) = |x|$.
By expanding $f$ as a Fourier series, recover the values of
\[ \sum_{n=1}^{+\infty} \frac{1}{n^2} \qquad \text{and} \qquad \sum_{n=1}^{+\infty} \frac{1}{n^4}. \]

\vs

\exercise
Let $f$ be a $2\pi$-periodic odd function defined by $f(x) = 1 - \cos(x)$ for $0 \leq x < \pi$, and $f(\pi) = 0$.
\begin{enumerate}[nosep, label=\arabic*.]
  \item Find the Fourier series of $f$ and discuss its convergence.
  \item Deduce $\dfrac{\pi}{4}$ as the sum of a series.
\end{enumerate}

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\exerciseopt{*}
Consider the function $f$ defined on $\mathbb{R}$, even, $\pi$-periodic, with $f(t) = \dfrac{\pi}{2} - t$ on $[0\,;\,\pi/2]$.
\begin{enumerate}[nosep, label=\arabic*.]
  \item Sketch $f$ on $[-\pi\,;\,\pi]$.
  \item Compute the Fourier coefficients of $f$ (distinguish the values of $a_n$ according to the parity of $n$, for $n \neq 0$).
  \item Does $f$ satisfy the Dirichlet conditions?
  \item Let $\disp{S(t) = \frac{\pi^2}{8} - \sum_{p=0}^{+\infty} \frac{1}{(2p+1)^2} \cos\big(2(2p+1)t\big)}$. Express $S(t)$ explicitly on $[0\,;\,\pi/2]$ and on $[\pi/2\,;\,\pi]$.
  \item Compute $\disp{\sum_{p=0}^{+\infty} \frac{1}{(2p+1)^2}}$ using the previous Fourier series.
  \item Then compute $\disp{\sum_{p=0}^{+\infty} \frac{1}{(2p+1)^4}}$.
\end{enumerate}

\vs

\exerciseopt{**}
Solve the differential equation:
\[ y'' + 2y = |\sin(t)|. \]

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