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  \textbf{\color{darkblue}MAT 103A: Complex Analysis (W21)}\quad
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  \fbox{\parbox{5.5in}{
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    \textbf{\large  Your Name: } 
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 \noindent Please review the problem set guidelines in the \href{https://people.ucsc.edu/~jbreland/teaching/W21_MATH103A/build/W21_103A_syllabus.pdf}{syllabus} before turning in your work. 
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\begin{enumerate}[\bfseries P1.]

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%--------------------------Problem 1-----------------------------%%
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\item Let $U$ be a domain. Prove that if $f$ is a real-valued function analytic on $U$, then $f$ is constant on $U$. 


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\item Let $f(z)=u+iv$ be a complex-valued function defined on an open set $U \subseteq \C$. Suppose that the first-order partial derivatives of $\Re f= u$ and $\Im f = v$ exist and are continuous on $U$. Define the differential operators\footnote{These operators are inspired by the following formal (meaningless) calculation. Consider the functions $x(z,\conj{z}) = \frac{z+\conj{z}}{2}$ and $ y(z,\conj{z})=\frac{z-\conj{z}}{2i}$ (note $z = x + iy$) and formally apply the chain rule
	\[
	\begin{array}{c}
	\p{}{z} = \p{}{x}\p{x}{z} + \p{}{y}\p{y}{z} = \p{}{x}\frac{1}{2} + \p{}{y} \frac{1}{2i}, \\ \p{}{\conj{z}} = \p{}{x} \p{x}{\conj{z}} + \p{}{y}\p{y}{\conj{z}}  = \p{}{x} \frac{1}{2}- \p{}{y}\frac{1}{2i}.
	\end{array}
	\]}
\[%NOTE: I have defined the command \p{}{} to produce the differential operator symbol. This is quicker than typing \frac{\partial f}{\partial z}!
\begin{array}{ccc}
	\ds \p{f}{z} \defeq \frac{1}{2} \left( \p{f}{x} -i\p{f}{y}\right) & \text{and}  & \ds \p{f}{\conj{z}} \defeq \frac{1}{2} \left( \p{f}{x} +i\p{f}{y}\right).
\end{array}
\]
Note: we also define $\p{f}{x} \defeq \p{u}{x} +i \p{v}{x}$, and similarly for $\p{}{y}$. 

\begin{enumerate}[(a)]
	\item Show that $f$ is analytic on $U$ if and only if and only if $\p{f}{\conj{z}} = 0$. 
	
	\item If $f$ is analytic on $U$, show that $f' = \p{f}{z}$.
\end{enumerate}

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\item \textbf{(Optional.})\footnote{This is worth 3 points extra credit, added to your score on this assignment.} Let $U$ be an open set and let $f$ be a function that is continuous on $U$ with the property
\[
e^{f(z)}=z, \ z \in U.
\]
\begin{enumerate}[(a)]
	
	\item Show that $f$ is analytic on $U$.\footnote{This shows that a \emph{continuously} defined logarithm on an open set is automatically analytic.}
	
	\item Consider the function $f(z) = \log z \defeq \ln |z| + i \arg z, \ |z|>0, \ -\pi < \arg z \leq \pi$. What, if anything, does part (a) tell you about $f$? Justify your conclusions.
	
\end{enumerate}


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\item Find all solutions to the equation $e^{2z} -2ie^z = 1$. 


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\item \begin{enumerate}[(a)]
	
	\item Find real valued functions $u,v :\R^2 \to \R$ such that $e^z = u(x,y) + iv(x,y)$.
	
	\item Find real valued functions $U,V :\R^2 \to \R$ such that $\cos z = U(x,y) + iV(x,y)$. 
	
	\item Show that $e^{\conj{z}} = \conj{e^z}$ and $\cos \conj{z} = \conj{\cos z}$.
	
\end{enumerate}



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\item Determine the points at which the following functions are analytic:

\begin{enumerate}[(a)]
	\item $e^{\conj{z}}$;
	
	\item $\cos \conj{z}$.
\end{enumerate}



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\item Let $z\in \C$.
\begin{enumerate}[(a)]
	\item Prove that \cancel{$1^z$} $\abs{1^z}$ is single-valued if and only if $\Im z = 0$. 
	
	\item Find a necessary and sufficient condition for $|i^{iz}|$ to be single-valued. 
	
	\item \textbf{(Optional.})\footnote{Worth 1 point extra credit, added to your score on this assignment.} Show by counterexample that the statement is false: $1^z$ is single-valued if and only if $\Im z = 0$.
\end{enumerate}




\end{enumerate}


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