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\begin{document}
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\def\blatt{2}
\def\datum{13/03/26}
\def\abgabe{ 10:00 A.M., Friday, 27/03/2026.}
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\begin{minipage}[m]{110mm}
\begin{center}
{\large\bf MATH3401/3901: Complex Analysis/Advanced Complex Analysis}
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\end{minipage}\hfill\noindent
\begin{minipage}{50mm}\raggedleft
First Semester 2026\\
\datum
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\bigskip\bigskip
\centerline{{\large\bf  Assignment Number  \blatt}}
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\begin{auf} {\it (2 points)} 
Determine the M\"obius transformation (viewed as a mapping on $\overline{\C}$)
 mapping $2$  to $0$, $i$ to $\infty$, and $0$ to $-2i$.
\\ 
\end{auf} 

\begin{auf}{\it (4 points)} 
Let $T$ be a mapping from $\Omega$, a subset of $\C$, to $\C$. A {\it fixed point} 
of $T$ is a point $z$ satisfying $T(z)=z$.\\
{\bf a)} Show:
any M\"obius transformation, apart from the identity, can have
at most 2 fixed points in $\C$. (The identity is the
transformation $z\mapsto z$). \\
{\bf b)} Give examples of M\"obius transformations having (i) 2; (ii) 1 and (iii) no fixed 
points in $\C$. \\
\end{auf} 

\begin{auf}{\it (2 points)} 
For $z\in\C$, show:\\
{\bf a)} $\sin \overline{z}=\overline{\sin{z}}$;\qquad
{\bf b)} $\cosh \overline{z}=\overline{\cosh{z}}$\\
\end{auf} 


\begin{auf}{\it (3 points)} 
Find all solutions $z\in\C$
of the following (express your answers in the form $x+iy$): \\
{\bf a)} $\log z = 4i$;\qquad
{\bf b)} $z^{i}=i$.\\
\end{auf} 

\begin{auf}{\it (5 points)} \\
{\bf a)} Prove that $\displaystyle{\cot^{-1}z =
\frac{-i}{2}\log\left(\frac{z+i}{z-i}\right)}$, and note any restrictions on your domain.\\ 
{\bf b)} Find all solutions $z\in \C$
of $\cot z = 1$ (express them in the form $x+iy$). \\
\end{auf}

\begin{auf}{\it (4 points)} 
Let $\Omega_1$ and $\Omega_2$ be nonempty, closed sets in $\C$. \\
{\bf a)} Show that the set $\Omega_1\cup\Omega_2$ is closed.\\
{\bf b)} If instead $\Omega_2$ is nonempty and open:\\
{\it (i)} could $\Omega_1\cup\Omega_2$
still be closed?\\
{\it (ii)} Need it be closed? \\
Give proofs or examples/counterexamples.
\end{auf} 



Due: \abgabe
\vspace{.3cm}

{\footnotesize Current assignments will be available at

{\tt http://www.maths.uq.edu.au/courses/MATH3401/AssignmentsEtc.html }}
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\end{document}
