% Homework for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2010/10/4

\documentclass[a4paper,12pt]{article}

\usepackage{pm1}

\setcounter{homework}{5}

\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{saddle}

\begin{equation} \label{Ixt}
\fn{I}{x} = \int_{-\pi}^{\pi} e^{i x \sin t} dt
\end{equation}
Use the transformation
\begin{equation} \label{zt}
z = e^{it}
\end{equation}
to express $\fn{I}{x}$ as a holomorphic integral over a closed curve.
Then use
\begin{enumerate}
\item
contour integration to determine $\fn{I}{x}$,
\item
the saddle point approximation to obtain the asymptotic form of $\fn{I}{x}$ as $x \to \infty$.
\end{enumerate}


\question{path}

In the path integral formulation of quantum mechanics, the amplitude for a particle moving from $x_\mathrm{i}(t_\mathrm{i})$ to $x_\mathrm{f}(t_\mathrm{f})$ is given by
\begin{equation}
\int \mathinner{d[x(t)]} \exp \left\{ \frac{i}{\hbar} \mathinner{S[x(t)]} \right\}
\end{equation}
where the integral is over all paths $x(t)$ going from $x_\mathrm{i}(t_\mathrm{i})$ to $x_\mathrm{f}(t_\mathrm{f})$.
Derive classical physics.


\question{sum}

Calculate
\begin{equation}
\sum_{n=0}^\infty \frac{n!}{x^n}
\end{equation}
for $x = 10$.

Use PGF to draw a diagram illustrating your answer.

\end{homework}

\end{document}

