% Homework for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2010/11/23

\documentclass[a4paper,12pt]{article}

\usepackage{pm1}

\setcounter{homework}{8}

\begin{document}

\begin{homework}
[Answers should be submitted, as both a tex file and a pdf file, to both me and the teaching assistants. You may use this file as a template.]

\question{parity}
Show that
\begin{enumerate}
\item
\begin{equation}
\fn{\delta}{-x} = \fn{\delta}{x}
\end{equation}
\item
\begin{equation}
\fn{\delta'}{-x} = - \fn{\delta'}{x}
\end{equation}
\item
\begin{equation}
x \fn{\delta}{x} = 0
\end{equation}
\item
\begin{equation}
x \fn{\delta'}{x} = - \fn{\delta}{x}
\end{equation}
\end{enumerate}

\question{derivative}
Show that
\begin{enumerate}
\item
\begin{equation}
\fn{\delta}{x} = \fn{\theta'}{x}
\end{equation}
where
\begin{equation}
\fn{\theta}{x} = \left\{
\begin{array}{ccc}
0 & \textrm{for} & x < 0
\\
1 & \textrm{for} & x > 0
\end{array}
\right.
\end{equation}
\item
\begin{equation}
\int_{-\infty}^\infty \d{y} \fn{\delta'}{x-y} \fn{f}{y} = \fn{f'}{x}
\end{equation}
\end{enumerate}

\question{diagrams}
Use PGF to draw diagrams illustrating $\fn{\delta}{x}$ and $\fn{\delta'}{x}$.

\question{fourier}
Show that
\begin{enumerate}
\item
\begin{equation}
\fn{\delta}{x} = \lim_{\epsilon \rightarrow 0^+} \frac{1}{2\pi} \int_{-\infty}^\infty \d{k} e^{ikx - \epsilon k^2}
\end{equation}
\item
\begin{equation}
\fn{\delta}{x} = \lim_{\Lambda \rightarrow \infty} \frac{1}{2\pi} \int_{-\Lambda}^\Lambda \d{k} e^{ikx}
\end{equation}
\end{enumerate}

\end{homework}

\end{document}

