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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due March 15
    (Thursday) }} \rhead{\thepage} \cfoot{{\sf MthSc 434 $|$
    Advanced Engineering Mathematics $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}


\item Consider the function $f(x)=L-x$ defined on the interval
  $[-L,L]$ and extended to be $2L$-periodic. 
  \begin{enumerate}
    \item Sketch $f(x)$ on the interval $[-7L,7L]$.
    \item Compute the complex Fourier series of $f(x)$.
    \item Find the real Fourier series of $f(x)$.  [\emph{Hint}: Use
      $a_n=c_n+c_{-n}$, and $b_n=i(c_n-c_{-n})$.]
    \item Sketch the Fourier series of $f(x)$.  [It will be
      the same as the answer to Part~(a) \emph{except} at the points of
      discontinuity.]
  \end{enumerate}


\item Consider the $2\pi$-periodic function defined on $[-\pi,\pi]$ by
  \[
  f(t)=\left\{\begin{array}{llc} 0 && -\pi\leq t<0, \\
  t && 0\leq t\leq\pi, \end{array}\right.
  \]
  \begin{enumerate}
    \item Sketch the graph of $f(t)$ on  $[-7\pi,7\pi]$.
    \item Compute the complex Fourier series of $f(t)$.
    \item Solve the differential equation $x''(t)+\omega_0^2
      x(t)=f(t)$. Look for a particular solution of the form
      \[
      x_p(t)=\sum_{n=-\infty}^\infty c_ne^{inx}\,.
      \]
  \end{enumerate}


\item Consider the function defined by
  \[
  f(x)=x^2\qquad\mbox{for } -\pi<x\leq\pi\,.
  \]
  and extended to be periodic of period $T=2\pi$.
  \begin{enumerate}
  \item Find the complex form Fourier series of $f(x)$. Feel free to use a
    computer to find the indefinite integral $\int x^2e^{-inx}\,dx$.
    \item Find the real form of the Fourier series.
    \item Use Part~(b), along with the real version of Parseval's
      identity to compute
      $\displaystyle\sum_{n=1}^\infty\frac{1}{n^4}$.
  \end{enumerate}


\item Consider the $2\pi$-periodic function $f$ defined on
  $[-\pi,\pi]$ by $f(x)=|x|$.
  \begin{enumerate}
  \item What is $f(\pi)$?
  \item Compute the Fourier series of $f$.
  \item Plug $x=\pi$ into the Fourier series and use this to compute
    $\displaystyle\sum_{n=0}^\infty\frac{1}{(2n+1)^2}$.
\end{enumerate}


\item Consider a complex Fourier series
  \[
  f(x)=\sum_{n=-\infty}^\infty c_n e^{in\pi x/L}\,,
  \]
  Prove the complex version of \emph{Parseval's identity}, which says that
  \[
  \frac{1}{2L}\int_{-L}^L
  (f(x))^2\,dx=\sum_{n=-\infty}^\infty c_n^2\,.
  \] 
  Observe that this is in a sense an infinite-dimensional version of
  the Pythagorean theorem, because
  \[
  ||f||^2:=\<f,f\>:=\frac{1}{2L}\int_{-L}^L
  (f(x))^2\,dx=\sum_{n=-\infty}^\infty c_n^2:=\sum_{n=-\infty}^\infty
  |\<f,e^{in\pi x/L}\>|^2\,.
  \]

  
\end{enumerate}

\end{document}
