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\usepackage[homework]{quizmachine}

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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 6}
\date{due April 3, 2013, before 12:00}

\quizleft{ESM 2B}
\quizcenter{Quiz 6}
\quizright{April 11, 2013}

\hwpreamble{\noindent
Normal questions and advanced questions (A) are worth $5$ points; easy
questions (E) are worth $4$ points.  Complete either the easy, or the
advanced version, not both.}

\defproblem{
Let $f$ be a periodic function on the interval $[-\pi,\pi]$ with
Fourier representation
\begin{equation*}
  f(x) = \sum_{k = -\infty}^\infty
         c_k \, \E^{\i kx} 
  \qquad \text{where} \qquad
  c_k = \frac1{2\pi} \int_{-\pi}^\pi \E^{-\i k x} \, f(x) \, \d x \,.
\end{equation*}
Show that when $f$ is real, then $c_k^*=c_{-k}$.}

\defproblem{
Let $f$ be a periodic function on the interval $[-\pi,\pi]$ with
Fourier representation
\begin{equation*}
  f(x) = \sum_{k = -\infty}^\infty
         c_k \, \E^{\i kx} 
  \qquad \text{where} \qquad
  c_k = \frac1{2\pi} \int_{-\pi}^\pi \E^{-\i k x} \, f(x) \, \d x \,.
\end{equation*}
Express the Fourier coefficients of
\begin{enumerate}
\item $f(x-x_0)$ where $x_0$ is a constant,
\item $f(-x)$,
\item $f^*(x)$
\end{enumerate}
in terms of the Fourier coefficients $c_k$ of $f$.}

\defproblem{
Let $f$ be a periodic function on the interval $[-\pi,\pi]$ with
Fourier representation
\begin{equation*}
  f(x) = \sum_{k = -\infty}^\infty
         c_k \, \E^{\i kx} 
  \qquad \text{where} \qquad
  c_k = \frac1{2\pi} \int_{-\pi}^\pi \E^{-\i k x} \, f(x) \, \d x \,.
\end{equation*}
Express the Fourier coefficients of
\begin{enumerate}
\item $f'(x)$,
\item $\displaystyle \int_{-\pi}^x f(\xi) \, \d\xi$ assuming that
$c_0=0$
\end{enumerate}
in terms of the Fourier coefficients $c_k$ of $f$.}

\defproblem{
Compute the complex Fourier series of the function $f(x)=\E^x$ on
the interval $[-\pi,\pi]$.}

\defproblem{
\begin{itemize}
\item[(E)] 
Compute the complex Fourier series of the function $f(x)=|x|$ on the interval
$[-\pi,\pi]$. 
\item[(A)] In addition to (E), express $f$ as a Fourier cosine series.
\end{itemize}}

\defproblem{
\begin{itemize}
\item[(E)] Let
\[
\v = 
  \begin{pmatrix}
  1 \\ 2 \\ 0 \\ 2
  \end{pmatrix} \,.
\]
Compute the projection of $\v$ onto the subspace spanned by the
orthonormal vectors
\[
\e_1 = \frac1{\sqrt2}
  \begin{pmatrix}
  1 \\ 0 \\ 0 \\ -1
  \end{pmatrix} \,, \quad
\e_2 = \frac1{\sqrt2}
  \begin{pmatrix}
  0 \\ 1 \\ 1 \\ 0
  \end{pmatrix} \,.
\]

\item[(A)]

An odd function of period $2\pi$ is approximated by a Fourier
sine series having only $N$ terms.  The error in the approximation is
measured by the mean-square deviation
\[
  E_N=
  \int_{-\pi}^\pi
  \biggl[ f(x) - \sum_{n=1}^N b_n \, \sin nx \biggr]^2 \, \d x \,.
\]
By differentiating $E_N$ with respect to the coefficients $b_n$, find
the values of $b_n$ that minimize $E_N$.
\end{itemize}}

\quizzes{100}

\end{document}
