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\begin{document}

\noindent
{\bf  Additional Notes \#1:
The Existence of Harmonic Conjugates}

\bigskip

Let $u(x,y)$ be a function that is harmonic in a domain 
$D$. We learned a procedure
to find a harmonic conjugate $v(x,y)$ of $u(x,y)$
in $D$ {\em if a harmonic conjugate exists}. We also saw an
example showing that the harmonic conjugate in $D$ need not exist.
(Our example: $u(x,y) = \ln(\sqrt{x^2+y^2}) = \ln(|z|)$ 
is harmonic
in the punctured plane $D = {\BC}-\{0\}$, but has no
harmonic conjugate in $D$. We showed that no harmonic 
conjugate exists by noting that there is no continuous
way to assign values to $v(x,y)$ on the unit circle.)

Our example can be modified to produce other domains which
have harmonic functions with no conjugate. Suppose that
$D$ is a domain that contains a simple closed curve 
$C$, but which does not contain all the interior
points of $C$. If $z_0$ is in the interior of
$C$ but is not in $D$, then $u(x,y) = \ln(|z-z_0|)
= \ln(\sqrt{(x-x_0)^2+(y-y_0)^2})$ is harmonic in $D$.
By a modification of the arguments we have seen, this function
has no harmonic conjugate in $D$ (since there is
no continuous way to assign values to $v(x,y)$ on $C$).
This leads us naturally to the following definition:

\medskip

\noindent
{\bf Definition.}
A domain $D$ is {\em simply connected} if whenever $C$
is a simple closed curve in $D$, then the interior
of $C$ is also in $D$.
\medskip

Our observations above say that if $D$ is a domain that
is not simply connected,
then there exists a function $u(x,y)$ that is harmonic in $D$
but has no harmonic conjugate in $D$. Conversely,
we have the following theorem:

\medskip

\noindent
{\bf Theorem.}
If $u(x,y)$ is harmonic 
in a simply connected domain $D$, 
then $u(x,y)$ has a harmonic conjugate
in $D$.

\medskip

All proofs of this are beyond the level of our course.
However, the theorem statement when $D = {\BR}^2$ or when
$D$ is the unit square can be found in many calculus 
books (see Theorem 15.3.3 of Edwards and Penney, for example).
The full theorem can be derived from
this special case by using the Riemann Mapping Theorem
(also beyond the level of this course).

\bigskip
\bigskip

\noindent
{\bf Things to remember:}
\begin{itemize}
\item What {\em simply connected} means.
\item That harmonic functions on simply connected domains have
harmonic conjugates.
\item That harmonic functions on domains that are not simply connected 
may or may not have harmonic conjugates.
\item Know procedure to construct $v(x,y)$ from $u(x,y)$.
\item Know that the purpose for learning to 
construct harmonic conjugates is to enable us 
to construct analytic functions from harmonic functions.
\end{itemize}
\end{document}
