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 \begin{document}


 \thispagestyle{empty}

 \begin{center}
 {\sc
 California State University, Long Beach \\
 Department of Mathematics\\
 Complex Analysis Comprehensive Examination\\
 February 8, 2025
 }
 \end{center}

 \noindent{\bf Do three problems from each part, for a total of six problems.
 Circle the problems you want to be graded.}



 \begin{center}
 {\bf PART A}
 \end{center}

\noindent {\bf Problem 1}. 
  Compute 
\[
\int_{-\infty}^\infty \frac{x^2}{1+2x^2+x^4} dx.
\]




\noindent {\bf Problem 2}. 
Find the Laurent series for $f(z)=\frac{1}{z(z-1)(z-2)}$ around $z_0=1$.



\noindent {\bf Problem 3}.   
\begin{enumerate}
\item  Compute 
\[
\int_\gamma \frac{z}{(z-1)(z-2)(z-3)}dz
\]
where $\gamma$ is the circle $|z|=4$. 


\item  Compute 
\[
\int_\gamma \frac{z^8}{(z-1)(z-2)...(z-10)}dz
\]
where $\gamma$ is the circle $|z|=11$. 
\end{enumerate}

\noindent {\bf Problem 4}.  
Consider the polynomial $f(z)=z^4+5z+1$. Find out the number of zeros of $f$ in the annulus $\{z\in \mathbb{C}: 1<|z|<2\}$.

\noindent {\bf Problem 5}.
Find the Residue at each pole for the function $f(z)=\frac{1}{z^2 \sin z}$ and evaluate the integral $\int_{\mathbb{C}}f(z) dz$
where $\mathbb{C}$ is the circle $|z|=10$ traversed once in the counterclockwise direction.





 \pagestyle{empty}


\[\]

\[\]


 \begin{center}
 {\bf PART B}
 \end{center}


\noindent {\bf Problem 6}. 
Assume $f$ and $g$ are analytic on $\mathbb C$, and assume there are infinitely many $q$ such that $|q|\leq 1$ and $f(q)=g(q)$. Prove that 
$f=g$ everywhere.








\noindent {\bf Problem 7}. 
Assume $f$ is entire. Prove that if $\underset{z\rightarrow \infty}{ \lim}f(z)=0$, then $f(z)=0$ for any $z$.
  





\noindent {\bf Problem 8}. 
\begin{enumerate}
\item State the fundamental theorem of algebra.
\item Prove the fundamental theorem of algebra using the Liouville's theorem.
\end{enumerate}


\noindent {\bf Problem 9}. 
Assume $f$ is analytic on ${\mathbb C}$, and there is $M>0$ so that 
$|f(z)|\leq M(1+\sqrt{|z|})$ for all $z$. Prove that $f$ is a constant.  

\noindent {\bf Problem 10}. 
\begin{enumerate}
\item  State and prove Morera's Theorem.

\item  Give the statement of a claim or theorem whose proof relies on Morera’s Theorem. (You need not provide proof.)
\end{enumerate}



 
\end{document}