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\begin{document}
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\def\blatt{3}
\def\datum{27/03/2026}
\def\abgabe{10:00AM, Friday, 01/05/2026}
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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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\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 (6 points)} 
Show the following limits:\\
\vspace{.6cm}
{\bf a)} $\displaystyle{\lim_{z\to\infty}\frac{4z^5}{z^5-142z}=4}$;\qquad
{\bf b)} $\displaystyle{\lim_{z\to\infty}\frac{z^4}{z^2+142z}=\infty}$;\\
{\bf c)}  $\displaystyle{\lim_{z\to\infty}\frac{(az+b)^3}{(cz+d)^3}=\frac {a^3}{c^3}}  $\qquad if $a,b,c,d\in\C, c\neq 0$.
\end{auf} 

\begin{auf} {\it (3 points)} 
Show that the following functions are defined on all of $\C$, but are nowhere analytic 
(here $z= x+iy$):\\
{\bf a)} $z\mapsto 2xy+i(x^2+y^2)$\qquad
{\bf b)} $z\mapsto e^ye^{ix}$.
\end{auf} 


\begin{auf} {\it (4 points)} 
Determine where the function $z\mapsto x^3+i(1-y)^3$ is:\\
  {\bf a)} analytic;\qquad\qquad
 {\bf b)} differentiable \quad
(here $z= x+iy$).
\end{auf}

\begin{auf} {\it (3 points)} For $z=x+iy$, define $f(z)=\sqrt{|xy|}$.\\
{\bf a)} Show that $f$ satisfies the Cauchy-Riemann equations at the origin. (Note: you will need to use the definition of the partial derivative to calculate $u_x(0,0)$ and $u_y(0,0)$).\\
{\bf b)} Show that $f$ is not differentiable at the origin. (Hint: approach on a suitable line).\\
{\bf c)} Explain why this doesn't contradict any of the results from class.\\
\end{auf}

\begin{auf} {\it (4 points)} \\
(a) Define precisely what it means for a curve in $\C$ to be {\it rectifiable}.\\
(b) Give an example of a non-rectifiable curve in $\C$. You don't have to prove it is non-rectifiable.\\
{\it Note: you will need to provide at least one reference, properly cited. This is not allowed to be wikipedia.}

\end{auf} 

\begin{auf} {\it (2 bonus points)} 
Consider sequences defined as follows:
\bean
&x_0=0, \quad  x_1 = x_2 =1,\quad  x_3=x_4=1-\frac{1}{2}, \quad x_5=x_6=1-\frac{1}{2}+\frac{1}{4}, \, \cdots \quad \text{and}\\
&y_0=y_1=0, \quad y_2 = y_3 =1,\quad y_4=y_5=1-\frac{1}{2},\quad y_6=y_7=1-\frac{1}{2}+\frac{1}{4},\, \cdots .
\eean
Set $z_n = x_n+iy_n$, and define $I_n$ to be the closed line segment (i.e., including end-points) from $z_n$ to $z_{n+1}$, for $n\in\NNO$. Finally, set:
 \[
\Omega=\displaystyle{\bigcup_{n=0}^\infty I_n},\quad 
\text{and } \Lambda=\Omega\cup\{\textstyle{\frac 23+\frac 23 i\}}.
\]


{\bf a)} Show that $\Lambda$ is connected. (A sketch is not a proof, but could be helpful in fixing ideas.)\\
{\bf b)} Show that $\Lambda$ is {\it not} piecewise affinely path connected, in the sense defined in class (Lecture 12).\\
{\bf c)} Do these results contradict anything from class? Explain.

\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}
