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\begin{document}
\lhead{PHYS 2210   Fall 2011   Homework Set 12}
\chead{}
\rhead{ Due at 4 PM on \textbf{Dec 1st, 2011}}

%\maketitle

Show and explain all of your work!   Correct answers for which we cannot follow your work are worth no credit.  


\begin{enumerate}


\item For this problem, consider the periodic function $f(x)$: (one period of the function is specified here)
\begin{eqnarray*}
f(x) & = & -x, \mbox{    for } -\frac{\pi}{2}<x<0 \\
& = & x,    \mbox{    for   }  0<x< \frac{\pi}{2} \\
\end{eqnarray*}

\begin{enumerate}
\item (2 pt)  Find the Fourier Series expansion for $f(x)$.  Be sure to include a sketch of the function.
\item (1 pt)  To check that your solution is correct, use a computer, plot the sum of the \textbf{first 3 non-zero terms} in this series from $-\pi$ to $\pi$.  It should resemble the sketch that you drew in part a.
\end{enumerate}


\item For this problem, consider the periodic function $g(x)$:  (one period of the function is specified here)
\begin{eqnarray*}
g(x) & = & 1, \mbox{    for } -1<x<1 \\
& = & 0,    \mbox{    for   }  1<x<5  \\
\end{eqnarray*}
\begin{enumerate}
\item (2 pt)  Find the Fourier Series expansion for $g(x)$.  Be sure to include a sketch of the function.
\item (1 pt)  To check that your solution is correct, use a computer, plot the sum of the first \textbf{first 4 non-zero terms}  in this series from -10 to 10.  It should resemble the sketch that you drew in part a.
\end{enumerate}

\item (0.5 pt)  Show that the functions $x^{2}$ and $\sin{x}$ are orthogonal on the interval (-1,1).  (Hint:  You should not need to work out the integral.  What do you know about even and odd functions?)

\newpage 

\item  Consider the following function:
\begin{eqnarray*}
f(x) & = & \sin{x}, \mbox{    for } -\frac{\pi}{2}<x<\frac{\pi}{2} \\
& = & 0,    \mbox{    for   }   |x|>\frac{\pi}{2}  \\
\end{eqnarray*}
\begin{enumerate}
\item (1 pt) Demonstrate that the  Fourier  transform of $f(x)$ is $g(\alpha)= \dfrac{-i \alpha \cos{(\frac{\alpha \pi}{2})}}{\pi(1-\alpha^{2})}$.  Hint: It might help to express $e^{-i \alpha x}$ in terms of sines and cosines.  Recalling the properties of integrals of odd and even functions can also save you some work.
\item (0.5 pt) Using your answer for $g(\alpha)$, express $f(x)$ as a Fourier integral (i.e. substitute your result for $g(\alpha)$ into the Fourier integral for $f(x)$).  Leave the answer as an integral.  Do not evaluate the integral.  
\end{enumerate}



\item (2 pt) Evaluate the following integrals:
\begin{enumerate}
\item $\int^{\pi}_{0}\mbox{sin}(x)\delta(x-\dfrac{\pi}{2})dx $
\item  $\int^{\infty}_{-\infty} e^{x} \delta(3x)dx $
\item $\int^{2 \pi}_{0}\mbox{cos}(x)\delta(x+\pi)dx $
\item $\int^{10}_{-10}x^{2}\delta(1-x)dx $
\end{enumerate}


\end{enumerate}
\end{document}  