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\begin{document}
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\def\blatt{4}
\def\datum{01/05/2026}
\def\abgabe{10:00AM, Friday, 22/05/2026}
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\begin{minipage}[m]{110mm}
\begin{center}
{\large\bf MATH3401: Complex Analysis}
\end{center}
\end{minipage}\hfill\noindent
\begin{minipage}{50mm}\raggedleft
First Semester 2026\\
\datum
\end{minipage}

\bigskip\bigskip
\centerline{{\large\bf  Assignment Number  \blatt}}
\bigskip
\hrule
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\begin{auf}{\it (5 points)}\\ 
%\,\, Let $\Gamma$ be the real axis in $\C$. Find a 
%function $u$ which is harmonic in the upper half-plane, and which 
% satisfies the boundary conditions 
%$u(z)=1$ for $z=(x,0)\in\Gamma$, $x \ge 0$ and
%$u(z)=0$ for $z=(x,0)\in\Gamma$, $x < 0$. 
% \end{auf}
%
%\begin{auf}{\it (5 points)}\\ 
%\,\, Let $\Gamma$ be the unit circle in $\C$. Find a 
%function $u$ which is harmonic in ${\rm int}\, \Gamma$ and which
% satisfies the boundary conditions 
%$u(z)=1$ for $z\in\Gamma$, $\Im z \ge 0$ and
%$u(z)=0$ for $z\in\Gamma$, $\Im z < 0$. 
% \end{auf}


\begin{auf} {\it (2 points)} 
Evaluate
\[
\int_{C}\frac{\sin z}{(z+1)^7}\,dz,
\]
where $C$ is the circle of radius $4$, centre $0$, positively oriented.
\end{auf} 

\begin{auf} {\it (2 points)}\\ 
Verify that the following functions $u$ are harmonic, and in each case
give a conjugate harmonic function $v$ (i.e., $v$ such that 
$u+iv$ is analytic).\\
(a)\,\, $e^x \cos y$,\\
(b)\,\, $x^2-y^2-2y$.
\end{auf}

\begin{auf} {\it (4 points)} \\
(a) Suppose that $U$ solves a Neumann problem for Laplace's equation on a 
domain $\Omega\subset\R^n$, $n\ge 2$. Show that $U+c$ also solves this problem for any $c\in\RR$.\\
(b) Does the same result hold for the corresponding Dirichlet problem?
\end{auf}
\begin{auf}{\it (2 points)}\\ 
Find a power-series expansion of the function 
$\displaystyle{f(z)=\frac 1{3-z}}$ about the point $4i$, and calculate its
radius of convergence.
\end{auf}

\begin{auf}{\it (2 points)}\\ 
Find a Laurent-series expansion of the function 
$\displaystyle{f(z)=z^{-1} \sinh(z^{-1})}$ about the point $0$, and classify the 
singularity at $0$. 
\end{auf}


\begin{auf}{\it (8 points)}
For each of the following functions, find and classify all singularities, including giving the order if the singularity is a pole. Also calculate the residue at each singularity.\\

 (a) $\frac{\displaystyle 1-\cos z}{\displaystyle  z^2}$
 \hfil
(b) $\frac{\displaystyle z^3}{\displaystyle (2+z)^3 }$\hfil
(c) $\frac{\displaystyle 1 }{\displaystyle (z^2+1)(z-1)^2 }$
\hfil
(d) $\frac{\displaystyle e^z}{\displaystyle (1-z)^4 }$
 
\end{auf}



Due: \abgabe
\vspace{.3cm}

{\footnotesize Current assignments will be available at

{\tt http://www.maths.uq.edu.au/courses/MATH3401/AssignmentsEtc.html }}
%\vspace{1cm}

\end{document}
