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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 7}
\date{due March 31, 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 $f(t)=\sin t$ on the interval $[0,\tfrac\pi2)$, and
periodically extended outside of this interval, be represented by the
complex Fourier series
\[
  \sum_{n=-\infty}^\infty c_n \, \frac{e^{4nti}}{\sqrt{\tfrac\pi2}}
  \,.
\]
Compute the Fourier coefficients $c_n$.

\item Find the Fourier transform of $f(x)=e^{-|x|}$.

\item Show that $\mathcal F (f(x+a)) = e^{ia\xi} \, \mathcal F(f)$. 

\item 
\begin{itemize}
\item[(E)] Show that $\mathcal F (f') = i\xi \, \mathcal F (f)$.

Hint: Integration by parts.  You may assume that all boundary terms
are zero when integrating by parts.

\item[(A)] By taking the Fourier transform of the equation
\[
  \frac{d^2 u}{d x^2} - u = f \,,
\]
show that the solution $u(x)$ can be written as
\[
  u(x) = \frac{-1}{\sqrt{2\pi}}
  \int_{-\infty}^\infty
  \frac{e^{i\xi x} \, \tilde f(\xi)}{1+\xi^2} \, d\xi \,,
\]
where $\tilde f(\xi)$ is the Fourier transform of $f(x)$.
\end{itemize}

\item Compute the integral
\begin{itemize} 
\item[(E)] $\displaystyle \int_{-2}^2 \delta(2x) \, \cos x \, dx$,

\item[(A)] $\displaystyle \int_{-2\pi}^{2\pi}
  \delta(x^2-\pi^2) \, \cos x \, dx$.
\end{itemize}

\item

\begin{itemize}
\item[(E)] Let the \emph{Heavyside} or \emph{unit step function}
be defined by
\[
  H(x) = 
  \begin{cases}
    0 & \text{for } x<0\\
    \tfrac12 & \text{for } x=0 \\
    1 & \text{for } x >0 \,.
  \end{cases}
\]
Show that $H'(x)=\delta(x)$.

Hint: Integration by parts. 

\item[(A)] One can define (so-called distributional) derivatives of
the $\delta$-function via
\[
  \int_{-\infty}^\infty f(x) \, \delta^{(n)}(x) \, dx
  = (-1)^n \, f^{(n)}(0) 
\]
for any $n$ times differentiable function $f(x)$.

So we may be tempted to write a Taylor series for the
$\delta$-function,
\[
  \delta(x+a) = \sum_{n=0}^\infty
  \frac{\delta^{(n)}(x)}{n!} \, a^n \,.
\]
It appears that the left side is zero except at $x=-a$, while the
right side is zero except at $x=0$.  Resolve this paradox.

\end{itemize}
\end{enumerate}

\end{document}
