% Homework for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2010/10/7

\documentclass[a4paper,12pt]{article}

\usepackage{pm1}

\setcounter{homework}{2}

\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{antiholomorphic}
Let $\fn{f}{z,z^*}$ be a complex function and $\vect{f}$ be the vector field on the complex plane with Cartesian components $(\Re f,\Im f)$.
Show that the conditions
\begin{equation}
f = \fn{f}{z^*}
\end{equation}
and
\begin{equation}
\vect\nabla \wedge \vect{f} = 0 = \vect\nabla \cdot \vect{f}
\end{equation}
are equivalent.

\question{components}
Let $\fn{f}{z,z^*}$ be a complex function.
Show that
\begin{equation} \label{con}
\left( \nabla \Re f \right) \cdot \left( \nabla \Im f \right) = 0
= \left( \nabla \Re f \right)^2 - \left( \nabla \Im f \right)^2
\end{equation}
if and only if $f$ is a holomorphic or antiholomorphic function, with the holomorphicity or antiholomorphicity determined by the sign of
\begin{equation} \label{sign}
\left( \nabla \Re f \right) \wedge \left( \nabla \Im f \right)
\end{equation}
What are the meanings of \eqs{con}{sign}?

\question{coordinates}
Consider the coordinates $w = u + i v$ defined relative to the Cartesian coordinates $z = x + i y$ by the holomorphic function
\begin{equation}
z = \fn{f}{w}
\end{equation}
\begin{enumerate}
\item
What properties do the coordinates $(u,v)$ have?
\item
Express the Laplacian in terms of $w$ and hence $u$ and $v$.
\item
Use PGF to draw the coordinates $(u,v)$ on the $(x,y)$ plane in the case
\begin{equation}
z = \cosh w
\end{equation}
\end{enumerate}

\end{homework}

\end{document}

