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  \textbf{\color{darkblue}MAT 103A: Complex Analysis (W21)}\quad
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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. Make use of theorems like the Triangle Inequality for Contour Integrals, the Fundamental Theorem of Contour Integrals, and the Cauchy-Goursat theorem, where applicable. 
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%--------------------------Problem 1-----------------------------%%
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\item Let 
\[
\ds f(z) = \frac{z^2+2}{(z^2+3)(z^2 + 2z + 1)}
\]
 and let $C_R$ denote the semicircle of radius $R$ parameterized by $z(t) = Re^{it}$ with $t \in [0,\pi]$. Show that
\[
\lim_{R\to \infty} \int_{C_R}f(z) \, dz = 0. 
\]

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\item Let $C$ be a positively oriented simply closed contour and let $R$ be the region consisting of $C$ and its interior. 

\begin{enumerate}[(a)]
	\item Show that the area $A$ of the region $R$ is given by the formula\footnote{Hint: Green's theorem}
	\[
	A = \frac{1}{2i} \int_C \conj{z} \, dz.
	\]
	
	\item Compute the area $A$ of the region enclosed by the \emph{cardioid} $C$ with parameterization $z(t) = \frac{1}{2} + e^{it} + \frac{1}{2}e^{2it}$ where $t \in [0,2\pi]$. 
\end{enumerate}


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\item Let $C$ be a closed contour and let $z_0 \in \C$ be a point not lying on $C$. The \emph{winding number} of $C$ about $z_0$ is defined by the integral
\[
n(C,z_0) = \frac{1}{2\pi i}\int_C \frac{1}{z-z_0} \, dz.
\]
\begin{enumerate}[(a)]
	\item Compute $n(C_1,z_0)$ where $C_1$ is parameterized by $z(t) = z_0 + Re^{it}, \ t\in[0,2k\pi] , k\in \Z, \ R>0$. 
	
	\item Compute $n(C_2,z_0)$, where $C_2$ is any circle and $z_0$ is any point not lying on or interior to $C_2$. 
	
	\item Let $C_3$ be any closed contour and $z_0$ any point not lying on $C_3$, parameterized by $z:[a,b] \to \C$. For any such contour, we can always find real-valued (piece-wise) differentiable functions $r,\theta :[a,b] \to \R$ with $r(t)>0$ such that $z(t) = z_0 + r(t) e^{i\theta(t)}$. Compute $n(C_3,z_0)$. 
	
	\item Give a geometric interpretation of the winding number. You must justify your interpretation. 
	
\end{enumerate}



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\item Let $f(z) = \frac{1}{z^2-1}$. Determine all possible values of the integral 
\[
\int_{C^{(k)}} f(z) \, dz
\]
where $C^{(k)}$ is any circle, traversed $k$-times, and not passing through $1$ and $-1$. You must justify your claim.



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\item Let $a,b \in \C$ and let $C_R$ be the circle of radius $R$ centered at the origin, traversed once in the positive orientation. If $|a|<R<|b|$, show that
\[
\int_{C_R} \frac{1}{(z-a)(z-b)}\, dz = \frac{2\pi i}{a-b}.
\]

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\item Let $f(z) = \frac{1}{z^2+1}$. Determine whether $f$ has an antiderivative on the given domain $D$. You must prove your claims. 
\begin{enumerate}[(a)]
	\item $D = \C \setminus \{i,-i\}$;
	
	\item $D = \{ z \in \C : \Re z > 0 \}$
\end{enumerate}





	
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