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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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 \bigskip
 
 \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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\item Use formal division and multiplication of power series to determine the first 4 nonzero terms of following series. It is not necessary to show the steps of the division/multiplication, but you should briefly explain which power series are being dividing/multiplied, and why it is justified. 
\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 Let $C$ be the positively oriented unit circle. Compute the following integrals. Justify your computations. 
\begin{enumerate}[(a)]
	\item $\int_C \frac{1}{e^z-1} \, dz$;
	
	\item $\int_C e^z \cos z \, dz$;
	
	\item $\int_C \csc z \, dz$. 
\end{enumerate}

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\item For each function $f(z)$, compute $\ds \Res_{z=0} f(z)$. Justify your computations. 

\begin{enumerate}[(a)]
	
	\item $f(z) = z \cos \left( \frac{1}{z} \right )$;
	
	\item $f(z) = \frac{\csc z}{z^4}$.
\end{enumerate}

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\item For each function $f(z)$, compute $\int_C f(z) \, dz$ where $C$ is the positively oriented circle $|z| = \frac{31}{10}$.\footnote{Hint: several of these functions appeared in PSet 6.} Justify your computations. 

\begin{enumerate}[(a)]
	
	\item $f(z) = z \cos \left( \frac{1}{z} \right )$;
	
	\item $f(z) = \frac{\csc z}{z^4}$;
	
	\item $f(z) = \frac{1}{2-z}$; 
	
	\item $f(z) = \frac{z}{(z-1)(z-3)}$;
	
	\item $f(z) = \frac{z}{1+z^2}$. 
\end{enumerate}
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\item Let $C$ be the unit circle with positive orientation. Let $n \geq 0$ be a nonnegative integer. 

\begin{enumerate}[(a)]
	\item The function $f(z) = z^n e^{\frac{1}{z}}$ has an isolated singularity at $z_0 = 0$. Find the Laurent series expansion for $f$ on the annulus $0<|z|<\infty$. 
	
	\item Use the theory of residues to compute the integral 
	\[
	\int_C z^n e^\frac{1}{z} \,dz.
	\]

	\item Prove that\footnote{Hint: write $e^{z + \frac{1}{z}} = e^{\frac{1}{z}} \sum_{n=0}^\infty \frac{z^n}{n!}$, then apply the theorem on integrating power series.} 
	\[
	\int_C e^{z + \frac{1}{z}}\, dz = 2\pi i \sum_{n=0}^\infty \frac{1}{n!(n+1)!}.
	\]
\end{enumerate}


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\item Consider the function 
\[
f(z) = \frac{2z^3+1}{(z^2-1)(z^2+1)}.
\]
Show that $f(z)$ has no antiderivative on the domain $D = \{z \in \C : |z| > 1\}$.\footnote{Hint: consider a circle $C$ that lies in $D$ and contains all the singularities of $f$ in it's interior. Compute $\int_C f(z) \, dz$ by computing the residue at $\infty$. Use the theorem that says $\Res_{z=\infty}f(z) = \Res_{z=0}\frac{1}{z^2}f\left( \frac{1}{z} \right)$. }










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\item \textbf{(Optional.)}\footnote{Worth 4 points of extra credit, added to you score on this assignment.} 
\begin{enumerate}[(a)]
	
	\item Suppose that $f$ is analytic everyone on $\mathbb{C}$, except at a finite number of (isolated) singular points $z_1,\ldots,z_n$. Show that
	\[
	\sum_{k=1}^n \Res_{z = z_k} f(z) + \Res_{z = \infty} f(z) = 0.
	\]
	
	\item Let 
	\[
	p(z) = a_0 + a_1z + \cdots +a_nz^n, \quad a_n \neq 0
	\] be a polynomial of degree $n \geq 2$. Let $C$ be a positively oriented simple closed contour whose interior contains all the zeros of $p(z)$. Show that\footnote{Hint: show that $\Res_{z = \infty} \frac{1}{p(z)} = 0$.}
	\[
	\int_C \frac{1}{p(z)}\, dz =0.
	\]
	
	\item Using (a) and (b), compute the integral
	\[
	\int_C \frac{1}{(z^{2021}+1)(z-3)} \, dz
	\]
	where $C$ is the positively oriented circle $|z| = 2$. 
\end{enumerate}




	
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