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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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\item Let $f:\C \to \C$ be an entire function. Suppose that there exists $M>0$ and $N \in \N$ such that
\[
|f(z)| \leq M(1+|z|^N)
\]
for all $z \in \C$. Prove that $f$ is a polynomial of degree at most $N$.



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\item 
\begin{enumerate}[(a)]
	\item Find the Taylor series expansion for $f(w) = \frac{1}{w}$ on the disk $D_1(1) = \{w \in \C : |w-1| <1\}$. 
	
	\item Let $ \Log z$ be the principal branch of the logarithm. Let $z \in D_1(1)$ and let $C$ be a contour lying interior to $D_1(1)$ and joining $w = 1$ to $w = z$. Find the Taylor series expansion for $\Log z$ on the disk $D_1(1)$ by integrating the Taylor series expansion found in part (a) over the contour $C$.
\end{enumerate}


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\item For each function, compute the Laurent series on the given annulus.

\begin{enumerate}[(a)]
	\item $f(z) = z^3 \sin \left(\frac{1}{z^3}\right)$, $0<|z| < \infty$.
	
	\item $f(z) = \frac{1}{2-z}$, $2<|z|<\infty$.
	
	\item $\frac{z}{1+z^2}$, $0<|z-i| < 2$. 
	
	\item $f(z) = \frac{z}{(z-1)(z-3)}$, $0<|z-1|<2$. 
	
	\item $f(z) = \frac{z}{(z-1)(z-3)}$, $2<|z-1|<\infty$. 
\end{enumerate}



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\item Prove that the function 
\[
f(z) = \begin{cases}
	\frac{1-\cos z}{z^2}, & z\neq 0\\
	\frac{1}{2}, & z = 0
\end{cases}
\]
is entire.





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%\item Determine the first 4 terms of following series.
%\begin{enumerate}[(a)]
%	\item The Laurent series of $\frac{1}{e^z-1}$ on the annulus $0<|z|<2\pi$. 
%	
%	\item The Taylor series of $e^z \cos z$ about $0$.
%	
%	\item The Laurent series of $\csc z = \frac{1}{\sin z}$ on the annulus $0<|z|<\pi$. 
%\end{enumerate}

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\item \textbf{(Optional)}\footnote{Worth 4 points of extra credit, added to you score on this assignment.} Use complex analysis to prove the formula:\footnote{Hint: begin by applying the formula for ${2n \choose n}$ from P5 on PSet4.}
\[
\sum_{n=0}^\infty \frac{\displaystyle {2n \choose n}}{7^n} = \sqrt{\frac{7}{3}}
\]
(This problem is fantastic, and surprisingly relevant.)

%7.3.12




	
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