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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 6}
\date{due March 24, 2004, before 12:00}
\maketitle

\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.

\begin{enumerate}
\item Let $f$ be a function on the interval $[0,2\pi]$ with Fourier
representation
\begin{equation*}
  f(x) = \frac1{\sqrt{2\pi}} \sum_{k = -\infty}^\infty
         c_k \, e^{ikx} \,.
  \tag{*}
\end{equation*}
Show that if $f$ is real, then $c_k^*=c_{-k}$.

\item Assume $f$ is as in (*).  Find the Fourier coefficients for
\begin{enumerate}
\item $f(x-x_0)$ where $x_0$ is a constant,
\item $f(-x)$,
\item $f^*(x)$,
\item $\displaystyle \int_s^t f(\xi) \, d\xi$, assuming that $c_0=0$,
\item $f'(x)$.
\end{enumerate}

\item Compute the Fourier cosine series of $f(x)=|x|$ on the interval
$[-\pi,\pi]$.

\item 
\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 square deviation
\[
  E_N=
  \int_{-\pi}^\pi
  \left[ f(x) - \sum_{n=1}^N b_n \, \frac{\sin nx}{\sqrt{\pi}}
  \right]^2
  dx \,.
\]
By differentiating $E_N$ with respect to the coefficients $b_n$, find
the values of $b_n$ that minimize $E_N$.
\end{itemize}

\item Give the value to which the Fourier series of the
function $f(x) = (x+\pi)^2$, defined on the interval $[-\pi,\pi]$,
converges at each of the following points:  $x=-\pi, 0, \pi, 2\pi$.

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

\end{document}
