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\textbf{\Large{MAT 536 2020 Final Exam}}
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\begin{flushleft} \textbf{Name:}\underline{\hspace{114mm}} \\
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\textbf{I.D.:}\underline{\hspace{118mm}}\\
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\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
1 & 2 & 3 & 4 & 5 & \text{TOTAL}   \\ \hline
\scriptsize{10 pts} & \scriptsize{10 pts} & \scriptsize{10 pts} & 
\scriptsize{10 pts} & \scriptsize{10 pts} & \scriptsize{50 pts} \\ \hline 
& & & & &  \\ \hline
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\begin{enumerate}
\item[\textbf{1.}] (10 points) Let $f(z)$ be an entire function. 
Given $a, b\in\mathbb{C}$, let $C_{R}$ be the circle of radius $R$ 
with center at the origin such that $|a|, |b|<R$ . Evaluate the integral
$$\frac{1}{2\pi i}\int_{C_{R}}\frac{f(z) dz}{(z-a)(z-b)}.$$
\newpage
\item[\textbf{2.}] (10 points) Suppose that the function $f(z)$ is holomorphic 
in a domain $D$ and $C$ is a closed curve in $D$. Prove that
$$\int_{C}\overline{f(z)}f'(z)dz$$
is purely imaginary.
\newpage
\item[\textbf{3.}] (10 points) Determine whether there is a holomorphic 
function $f(z)$ which maps the unit disk onto itself and satisfies 
$f(0)=\frac{1}{2}$ and $f'(0)=\frac{3}{4}$.
\newpage
\item[\textbf{4.}] (10 points) Evaluate the integral
$$\int_0^{\infty}\frac{x \sin x}{x^2+1} dx.$$
\newpage
\item[\textbf{5.}] (10 points) 
Find the number of roots (counted with multiplicity) of the function 
$g(z) = 6 z^3 + e^z + 1$ inside the unit disk (Hint: use Rouch\'e's theorem).
\end{enumerate}


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