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\noindent
\parbox{2.25in}{\bf Math 5160}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Feb. $19^\mathrm{th}$, 2020}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 A Harmonious Problem] {$\mbox{\quad}$}

\begin{enumerate}[(a)]
\item In the last homework we found the real and imaginary parts of $\sin(z)$ (\#1 Sine of the Times part (b)). We also showed that the derivative of sine was cosine. Verify directly that the real and imaginary parts of $\sin(z)$ are harmonic functions.
\item Expand $\cosh(z)$ into its real and imaginary parts and verify that they are harmonic functions.
\item Show that $u(x,y) = x^3-3xy^2+2x^2-2y^2+3y$ is harmonic and then find a harmonic conjugate for $u(x,y)$.
\item We know that if $v(x,y)$ is a conjugate for $u(x,y)$, then $f(z)=u+iv$ is analytic. Referring to part (c), find a nice formula for $f$ which only involves $z$.
\end{enumerate}

\item[\#2 You Must Conform] You should probably use some software to help you generate nice plots. I recommend my Maple worksheet.

\begin{enumerate}[(a)]
\item Explain where $g(z)=\mathrm{Log}(z)$ is a conformal mapping. Give parametrizations for the circles $x^2+y^2=16$ and $(x-3)^2+(y-2)^2=4$ and a parameterization
for the line through $2i$ and $2$. Then plot these curves and their images under the map $g(z)=\mathrm{Log}(z)$. Adjust parameter domains to make your plots look nice.
\item Explain where $g(z)=z^3+3z^2-9z+1$ is a conformal mapping. Give parametrizations for the circles $x^2+y^2=9$ and $(x+1)^2+y^2=4$ and a parameterization
for the line $y=2x-1$. Then plot these curves and their images under the map $g(z)=z^3+3z^2-9z+1$. Adjust parameter domains to make your plots look nice.
\end{enumerate} 

\item[\#3 M\"obius Just Sounds Cool] Let $f_A(z)=\dfrac{a_{11}z+a_{12}}{a_{21}z+a_{22}}$ where $\displaystyle A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ and $\det(A)\not=0$. 

\begin{enumerate}[(a)]
\item Find a formula for the linear fractional transformation which sends $(1,-1,i)$ to $(\infty,0,1)$. Show a detailed construction of such a formula by constructing maps to $(0,1,\infty)$ and then putting them together to form the desired map (as in the constructive proof from the video). Then create a plot showing some lines, circles mapping to lines, circles.
\item Repeat part (a) for $(i,\infty,0)$ to $(-1,i,\infty)$.
\item Show that $f_A(z)=f_B(z)$ if and only if there exists some $s \in \mathbb{C}$, $s\not=0$, and $B=sA$ (i.e., $A$ and $B$ are off by a nonzero scalar multiple).
         {\it Hint:} Consider $f_{AB^{-1}}(z)$.
\end{enumerate}

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