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\begin{document}
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\def\blatt{1}
\def\datum{24/02/26}
\def\abgabe{ 10 A.M., Friday, 13/03/2026.}
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\begin{minipage}[m]{110mm}
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{\large\bf MATH3401/3901: Complex Analysis/Advanced Complex Analysis}
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First Semester 2026\\
\datum
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\centerline{{\large\bf  Assignment Number  \blatt}}
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\begin{auf}{\it (3 points)} 
Graph the following regions in the complex plane:\\
{\bf a)} $\{ z\colon 2\Re z \le (\Im z)^2\}$;\\
{\bf b)} $\{ z\colon \pi/4<\Arg z\le 3\pi/4\}$;\\
{\bf c)} $\{ z\colon |z-2i|\ge\pi\}$.
\end{auf} 

\begin{auf}{\it (3 points)} 
Find \textit{all} complex solutions of the following equations:

{\bf a)} $\displaystyle \overline z = z$;\\
{\bf b)} $\displaystyle \overline z +z= 0$;\\
{\bf c)} $\displaystyle \overline z = 4/z$.
\end{auf}

\begin{auf}{\it (3 points)} 
Express the following in the form $x+iy$:\\
{\bf a)} $\displaystyle{\frac{i}{1-i} + \frac{1-i}{i}}$;\qquad
{\bf b)} all the 3rd roots of $64i$;\qquad
{\bf c)} $\displaystyle{\left(\frac{i+1}{\sqrt{2}}\right)^{1337}}$.
\end{auf} 



\begin{auf} {\it (5 points)} 
Recall the following (you do not have to prove it): For $n\in\ZZ_+$, $\sqrt{n}\in\QQ$ if and only if $n$ is a perfect square, i.e., $n=m^2$ for some $m\in\ZZ$.\\
Keeping this in mind, the set ``$\QQ\aadjoin{\sqrt{2}}$" is defined by: 
$\QQ(\sqrt{2})=\{p+q\sqrt{2}\,:\, p,q\in\QQ\}$.\\
{\bf a)} Show that $\QQ(\sqrt{2})$ is a {\it subfield} of $\RR$ (i.e., it is a subset that is a field).\\
Hint 1: remember that $\QQ$ and $\RR$ are fields, and note that $\QQ(\sqrt{2})\subset\RR$: hence you will get many, indeed, most of the properties almost free.\\
Hint 2: make sure you check closure.\\
{\bf b)} Is $\sqrt{3}\in\QQ(\sqrt{2})$? Explain your answer.

\end{auf} 

\begin{auf} {\it (3 points)} 
A field $\FF$ is called {\it ordered} if there exists a distinguished subset $\PP\subset\FF$, closed under addition and multiplication, such that the {\it trichotomy law} holds, i.e., \\
For every $x\in\FF$, precisely one of the following holds:\\
(i)$x\in\PP$;\\
(ii)$x=0$; or\\
(iii)$-x\in\PP$.\\
(Note that we can then define an ordering via $x>y\Leftrightarrow x-y\in\PP$).\\
Prove that $\C$ is not ordered. (Hint: first show that there must hold $-1\notin\PP$). 
\end{auf} 



\begin{auf} {\it (3 points)}
Write one or at most two paragraphs on the {\it Schwarz-Christoffel transformation}. You should also include one relevant picture or diagram, appropriately referenced.
Neither your paragraphs nor your picture should be from Wikipedia. Provide references as appropriate: see the UQ student guide at \href{https://uq.pressbooks.pub/digital-essentials-write-cite-submit/chapter/module-overview}{https://uq.pressbooks.pub/digital-essentials-write-cite-submit/chapter/module-overview/}. Note that it is strongly recommended {\bf not}  to use CoPilot or similar generative AI for this question, or indeed for any question on the assignment. If you do so, you {\bf must} reference your usage fully.\\

(Extra style points if the answer is in verse. Disclaimer: style points are not exchangeable for actual assignment points.)


\end{auf}
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Due: \abgabe
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{\footnotesize Current assignments will be available at\,\,\,{\tt http://www.maths.uq.edu.au/courses/MATH3401/AssignmentsEtc.html }}
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