\section*{Problem 3}
Answer the following questions. Here, $i , e$, and $\log$ denote the imaginary unit, the base of the natural logarithm, and the natural logarithm, respectively.\\
I. Consider the definite integral $I$ expressed as

$$I = \int _ { 0 } ^ { 2 \pi } \frac { \cos \theta d \theta } { ( 2 + \cos \theta ) ^ { 2 } }$$

\begin{enumerate}
  \item Find a complex function $G ( z )$ of a complex variable $z$ when we rewrite $I$ as an integral of a complex function as
\end{enumerate}

$$\oint _ { | z | = 1 } G ( z ) d z$$

where the integration path is a unit circle in the counter clockwise direction.\\
2. Find all poles and the respective orders and residues.\\
3. Evaluate the integral $I$.\\
II. Let a function of a real variable $\theta$ with real parameters $\alpha$ and $\beta$ be

$$f ( \theta ; \alpha , \beta ) = 1 + e ^ { 2 i \beta } + \alpha e ^ { i ( \theta + \beta ) }$$

Consider the definite integral

$$F ( \alpha , \beta ) = \int _ { 0 } ^ { 2 \pi } d \theta \frac { d } { d \theta } [ \log f ( \theta ; \alpha , \beta ) ]$$

\begin{enumerate}
  \item Find a complex function $G ( z )$ of a complex variable $z$ when we rewrite $F ( \alpha , \beta )$ as an integral of a complex function as
\end{enumerate}

$$\oint _ { | z | = 1 } G ( z ) d z$$

where the integration path is a unit circle in the counter clockwise direction.\\
2. Find all poles and the respective orders and residues.\\
3. Evaluate $F ( \alpha , \beta )$ by classifying cases with respect to $\alpha$ and $\beta$. Ignore the case in which the integration path passes through any poles.