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\section*{Problem Set 3}

Note:  You only need part I of Lecture 8 to do the problems (3)-(5) on subharmonic functions.
\begin{enumerate}[leftmargin=0.7cm]
\item Show that $\frac{R^2-|z|^2}{|Re^{\theta_0}-z|^2}$ is harmonic (in $z$) on $D_R$. [Hint: what function is it the real part of?]
\item (i) Solve the Dirichlet problem on $\overline{D}_1$ for the following functions on $\partial D_1$:
\begin{itemize}[leftmargin=1cm]
\item [(a)] $f(\theta)=\left\{\substack{1, \;\;\;\;\;|\theta|\leq \frac{\pi}{2}\\ -1,\;\;\; \frac{\pi}{2}<|\theta|\leq \pi}\right.$,
\item [(b)] $f(\theta)=1-\tfrac{2}{\pi}|\theta|$,
\item [(c)] $f(\theta)=(|\theta|-\pi)^2-\tfrac{\pi^2}{3}$.
\end{itemize}
First express your answer as a series, then try to identify the holomorphic function of which $u$ is the real part. [Hint: use part I of Lecture 7, on Fourier series.]

\noindent (ii) Using your answer to part (i), evaluate
\begin{itemize}[leftmargin=1cm]
\item [(a)] $1-\tfrac{1}{3}+\tfrac{1}{5}-\tfrac{1}{7}+\cdots$
\item [(b)] $1+\tfrac{1}{9}+\tfrac{1}{25}+\tfrac{1}{49}+\cdots$
\item [(c)] $1+\tfrac{1}{4}+\tfrac{1}{9}+\tfrac{1}{16}+\tfrac{1}{25}+\cdots$
\end{itemize}

\item True or False: If $f$ is subharmonic on a region $\Omega$, then so is $|f|$.
\item Let $\Omega\subseteq \CC$ be open, and $\{f_j\}\subset \underline{\mathcal{H}}(U)$ a sequence of subharmonic functions comverging normally to a function $f\colon U\to \RR$. Is $f$ necessarily subharmonic?
\item Let $F\colon U\to V$ be holomorphic and 1-to-1, and $f\colon V\to \RR$ be subharmonic. (Here $U$ and $V$ are regions.) Prove that $f\circ F$ is subharmonic.
\item Let $\Omega\subseteq \CC$ be a region, and $K\subset \Omega$ a compact subset. Show that there exists a constant $M$ (depending only on $\Omega$ and $K$) such that for every positive harmonic function $u\in \mathcal{H}(\Omega)$ and pair of points $z_1,z_2\in K$, we have $u(z_2)\leq M\cdot u(z_1)$. [Hint: Harnack.]
\end{enumerate}

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