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\begin{document}

\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Homework 10}
\date{Not for credit}
\maketitle

\emph{Note:} This homework comprises miscellaneous subject areas from
the last two weeks of classes.  This homework will not count toward
bonus credit, but it's very much worth working on.  I will publish
sample solutions on Monday, May 18.

\begin{enumerate}

\item Suppose $f, g \colon [0,2\pi] \to \C$ are Riemann-integrable.
Prove the following properties of the Fourier transform.
\begin{enumerate}
\item If $f$ is real-valued, then $f_{-k} = \overline{f_k}$.
\item If $f$ is continuously differentiable, then $(f')_k = \i k \,
f_k$.
\item Let $T_af$ denote the right-translation by $a\in \R$ of $f$,
i.e., $(T_af)(x) = f(x-a)$, with the understanding that $f$ is
periodically extended outside of its fundamental domain $[0,2\pi]$.
Show that
\[
  (T_af)_k = \e^{-\i ka} \, f_k \,.
\]
\item Let
\[
  (f * g)(x) = \int_0^{2\pi} f(y) \, g(x-y) \, \d y
\]
denote the \emph{convolution} of the functions $f$ and $g$, again with
the understanding that the functions are periodically extended outside
of their fundamental domain.  Show that
\[
  (f * g)_k = 2 \pi \, f_k \, g_k \,.
\]
\end{enumerate}

\item Compute the Fourier transform of the ``saw-tooth function''
$f(x) = x$ on $[-\pi,\pi)$, periodically extended outside its
fundamental domain.


\item Which of the following functions are complex-differentiable?
\begin{enumerate}
\item $f(z) = z^2$,
\item $f(z) = \lvert z \rvert^2$,
\item $f(z) = \cos(z)$.
\end{enumerate}

\item Show that if $f(z) = u(x,y) + \i \, v(x,y)$ with $z = x + \i y$
is complex-differentiable (holomorphic) on some domain $D \subset \C$,
then $u$ and $v$ are harmonic functions on $\R^2$, i.e.,
$\Delta u = \Delta v = 0$, where
$\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial
y^2}$ is the Laplace operator on the corresponding domain of $\R^2$.

\emph{Remark:} That adds another item to the list of fundamental
equivalences in complex variable calculus: ``harmonic'' $\iff$
``complex-differentiable (holomorphic)'' $\iff$ ``convergent Taylor
series (analytic)'' $\iff$ ``path independence of complex line integral
(identification with conservative vector field)''.

\item Use the residue theorem to compute
\[
  \int_{\lvert z \rvert = 1} z^2 \, \sin \frac1z \, \d z \,.
\]

\item Let $D$ be a domain in $\R^n$.  A function $f \in C^2(D)$ is
called \emph{harmonic} if $\Delta f=0$, where $\Delta$ is the
\emph{Laplace operator} defined via
\[
  \Delta f = \nabla \cdot \nabla f = \partial_1^2 f + \dots +
  \partial_n^2 f \,.
\]
Show that a harmonic function has the mean value property
\[
  f(x) = \frac1{S(\partial B(x,r))}
  \int_{\partial B(x,r)} f \, \d S
\]
for every $x \in D$ and every $r>0$ sufficiently small such that
$\partial B(x,r) \subset D$, where $B(x,r)$ denotes the ball centered
at $x$ with radius $r$ and
\[
  S(\partial B(x,r)) = \int_{\partial B(x,r)} \d S
\]
is the $(n-1)$-dimensional content of $\partial B(x,r)$.

\emph{Hint:} Proceed in the following steps:
\begin{enumerate}
\item Define
\[
  \phi(r) = \dfrac{\displaystyle \int_{\partial B(x,r)} f \, \d S}%
                  {\displaystyle \int_{\partial B(x,r)} \d S} \,.
\]
Now use a change of variables in numerator and denominator which takes
$B(x,r)$ to $B(0,1)$.

\item Differentiate with respect to $r$, move the differentiation
under the integral, and apply the chain rule.

\item Apply the divergence theorem and use that $f$ is harmonic to
conclude that $\phi'(r) = 0$.

\item Finish the proof by considering the limit $r \to 0$.
  
\end{enumerate}


\end{enumerate}
\end{document}
