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\begin{document}

\noindent
{\bf  Additional Notes \#2:
Extending Real Functions to Analytic Functions}

\medskip

Let $y=f(x)$ be a real-valued function
of a real variable. A function $w = F(z)$ 
is an {\em extension of $f$} if the domain
of $F$ contains the domain of $f$, and
they agree on their common domain.

\noindent{\bf Question:} 
How many different analytic extensions 
can a function of a real variable have?

The answer to this is a corollary of a result 
we will prove later in this course:

\medskip

\noindent
{\bf Theorem.}
The zeros of a nonzero analytic function on a domain $D$
are isolated in $D$.

\medskip

{\em Proof.}
See Exercise 57.13.

\medskip

The question above is answered by the 1-variable 
version of the following corollary.

\medskip

\noindent
{\bf Duncan Macleod's Corollary.}
{\rm (There can be only one \ldots)}
Let $D$ be a domain that has nonempty intersection 
with the real line. If $f(x_1,x_2,\ldots,x_n)$ is a function 
of $n$ real variables whose domain is $D\cap {\BR}$,
then there is at most one function $F(z_1,z_2,\ldots,z_n)$
such that 
\begin{enumerate}
\item[(1)] $F$ is differentiable on $D$ with respect to each
of its variables, and
\item[(2)] $F$ agrees with $f$ on $D\cap {\BR}$.
\end{enumerate}

\medskip

{\em Proof.} Suppose instead that there are two 
functions $F_1$ and $F_2$ with these properties.
then we argue by induction on $k$ that the equation 
\[F_1(z_1,\ldots,z_k,x_{k+1},\ldots,x_n) = 
F_2(z_1,\ldots,z_k,x_{k+1},\ldots,x_n)\]
holds whenever all $z_i$ are in $D$ and all $x_j$
are in $D\cap {\BR}$. 

When $k=0$, the claim is true because both $F_1$
and $F_2$ agree with $f$ on $D\cap {\BR}$. 
If the claim is true for $k=m$, then 
for any fixed $z_1,\ldots,z_m\in D$
and $x_{m+2},\ldots,x_n\in D\cap {\BR}$ the function
\[P(z) = F_1(z_1,\ldots,z_m,z,x_{m+2},\ldots,x_n)-
F_2(z_1,\ldots,z_m,z,x_{m+2},\ldots,x_n)\]
is analytic on $D$ and zero on $D\cap {\BR}$.
From the Theorem, $P(z)$ is zero throughout $D$.
This implies that the claim holds for $k = m+1$.
By induction it holds for all $m$, and
when $m = n-1$ the claim asserts that 
$F_1 = F_2$. This completes the proof.

\medskip

\noindent
{\bf Applications.} We can deduce lots of identities
using uniqueness of extensions:
\begin{itemize}
\item $e^{z+w} = e^ze^w$ for all $z, w\in {\BC}$.
\item $\cos^2(z) + \sin^2(z) = 1$ for all $z\in {\BC}$.
\item $\cos(z+w) = \cos(z)\cos(w)-\sin(z)\sin(w)$ for all $z, w\in {\BC}$.
\item $\frac{d(\sin(z))}{dz} = \cos(z)$ for all $z\in {\BC}$.
\item $\sin(z+2\pi) = \sin(z)$ for all $z\in {\BC}$.
\item (Reflection Principle) If $F(z)$ is analytic on a domain $D$
that is symmetric about the real line, and $F$ is real-valued on
the real line, then $\overline{F(z)} = F(\overline{z})$
on $D$.
\end{itemize}

\end{document}