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\title{Introduction to Partial Differential Equations}
\author{Homework 1}
\date{due February 14, 2013}
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\begin{enumerate}

\item Evans, p.\ 85 problem 1
  
\item Evans, p.\ 85 problem 2

\item In the proof of the theorem on the solution of Poisson's
equation we have used that
\[
  D \int_{\mathbb R^n} \Phi(y) \, f(x-y) \, dy  
  = \int_{\mathbb R^n} \Phi(y) \, Df(x-y) \, dy \,.
\]
State why and precisely under which assumptions this manipulation is
permitted. 

\item Consider a function of one complex variable $w \colon \mathbb{C}
  \to \mathbb{C}$ on an open connected subset of the complex plane,
  and write $w=w(z)$ with $w=u+\mathrm{i} v$ and $z=x + \mathrm{i} y$.
  The function $w$ is called \emph{(complex) differentiable} or
  \emph{holomorphic} if $u$ and $v$ have continuous first partial
  derivatives with respect to $x$ and $y$ that satisfy the so-called
  \emph{Cauchy--Riemann equations}
\begin{gather*}
  u_x = v_y \\
  u_y = - v_x
\end{gather*}
It is known that holomorphic functions are infinitely differentiable.

Show that the real and imaginary parts of a holomorphic function are
harmonic.

\end{enumerate}


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