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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. We will discuss the methods for computing the integrals in P3 \& P4 on Tuesday 3/9. 
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\item This problem gives another formula for the residue at a pole that is sometimes convenient. Suppose that $f$ has a pole of order $m$ at $z_0 \in \C$. Prove that
\[
\mathrm{Res}_{z = z_0} f(z) = \frac{1}{(m-1)!}\lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}} [(z-z_0)^m f(z)].
\]


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\item For each function $f(z)$, classify all of its isolated singularities as either: removable, essential, or a pole. If an isolated singularity is a pole, make sure so specify the order. Compute the residue at every isolated singularity.\footnote{Note: computing a Laurent series is not always the easiest way to identify the type of singularity. }
\begin{enumerate}[(a)]
	
	\item $f(z) = \frac{1}{\sin z}$;
	
	\item $f(z) = \frac{1}{\sin^2 z}$;
	
	\item $f(z) = \sin\left( \frac{1}{z} \right)$;
	
	\item $f(z) = \frac{z}{\cos z}$;
	
	\item $f(z) = \frac{1}{e^z - 1}$;
	
	\item $f(z) = ze^{\frac{1}{z-2}}$;
	
	\item $f(z) = \frac{e^z-1}{z}$. 
	
\end{enumerate}




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\item Compute the following improper integral using complex analysis:

\[
\int_0^\infty  \frac{x^2}{x^6+1} \, dx.
\]




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\item Let $a>0$. Compute the following improper integral using complex analysis:

\[
\int_0^\infty  \frac{\cos ax}{x^2+1} \, dx.
\]









	
\end{enumerate}


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