\documentclass{article}
\usepackage{exam3}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{epstopdf}
\usepackage{hyperref}
\usepackage{enumerate}

\paper{PHYS 2210}               % <- do not include FC, FT etc

\title{Homework 15}
\time{Start of class on Thurs. May 3}
\semester{Spring}
\year{2012}                     % <- default is the current year
%\campus{City}
%\note{ Be sure to show your work and explain what you are doing. (Correct answers, for which we cannot follow the work, are worth no credit). }
\begin{document}

\begin{questions}

\question Complete your final project.  ({\it Note:} although it is my general policy to drop your lowest single homework score this term, this project is NOT one that you can drop!) 


Only one group member should submit the final report. Be sure to include all group members' names on the front page.  To submit, create a .pdf document and upload it to the D2L ``DISCUSSION" area we have specifically created for this purpose. (It appears as the very last discussion topic - look for it. It is NOT the same as the one where your progress report went two weeks ago)  

{\bf Please also SEPARATELY upload your mathematica notebook (or other working code).} 

So you'll upload two things: report, and code. 

For more details see \url{http://www.colorado.edu/physics/phys2210/phys2210_sp12/project_info.html}  

Danny and I have created a ``sample" project report (available on D2L) so you can see an example of what we have in mind.  No need to copy our format exactly,  it's just there to provide you with helpful ideas. 





\question AFTER completing your project please visit  the online  survey at   

{\url{http://goo.gl/g5cDc}}  

Note: As always, full credit just for inputting your name and ID.  Because we have a bunch of questions about the project, the due date for this question is extended to Friday May 4 at 5 PM. Please don't forget to do it! (It will still be open right up to the final, though after Friday we can't give homework credit for it - but we will still appreciate and make use of your thoughtful feedback) 

 \ 

 \hrule
  \ 

{\bf NOTE: The rest of this set is entirely optional. It is not to be turned in (it's not extra credit), but I will provide solutions to these questions online.  Although the detailed mathematics of Fourier transforms will not be heavily featured on the final exam, some basic conceptual or computational ideas (which these problems will help you work through) might appear. This is the last topic we cover for which you have responsibility on the final} 





\begin{enumerate}
\item  Show that if a function $f(x)$ is even,  then its Fourier transform  $g(\alpha)$ (as defined in Boas Ch 7, Eq 12.2) is also an even function.
% \item  Show that in this case f(x) and g($\alpha$) can be written as Fourier cosine transforms and obtain Boas Eq 12.15 (Assuming  that f(x) is identical to Boas' $f_c(x)$, how is $g_c(\alpha)$ from Boas' Eq 12.15 related to g($\alpha$) of Boas' Eq 12.2? )
 
 %{\it Please note the typo in the top equation of Boas 12.15, the argument of $g_c$ should be $\alpha$, not $x$.} 
 
  \begin{center}%[htc]
%\centering
{\includegraphics[width=1.8in]{Hom15Fig1.pdf}}%\caption{}
\end{center}

For the function shown above: 

\item  Qualitatively, how do you expect  $g(\alpha)$,  the Fourier transform of $f(x)$, to change if you make $a$ bigger or smaller?
\item Find the Fourier transform $g(\alpha)$.  
\item Sketch  $g(\alpha)$ (or plot it in Mathematica if you prefer).  Was your prediction in part (a) correct?  Explain.
\item I claim that the Fourier Transform of a gaussian function with standard deviation $\sigma$, $f(x) = e^{- x^2/(2 \sigma^2)}$, is itself also a Gaussian function, $g(\alpha)$.  How do you expect the standard deviation of $g(\alpha)$ to depend on $\sigma$? 
%\item  (4 pts) Substitute back the function you found for  $g(\alpha)$ into the definition of $f(x)$ and compute $\int_0^\infty \frac{1-\cos\alpha x}{\alpha^2} d\alpha$
    \end{enumerate}


\end{questions}
\end{document}

