Answer the following questions concerning complex functions defined over the $z$-plane ( $z = x + i y$ ), where $i$ denotes the imaginary unit.

I. For the function $f ( z ) = \frac { z } { \left( z ^ { 2 } + 1 \right) ( z - 1 - i a ) }$, where $a$ is a positive real number:

\begin{enumerate}
  \item Find all the poles and respective residues of $f ( z )$.
  \item Using the residue theorem, calculate the definite integral
$$\int _ { - \infty } ^ { \infty } \frac { x } { \left( x ^ { 2 } + 1 \right) ( x - 1 - i a ) } d x$$
\end{enumerate}

II. Consider the function $g ( z ) = \frac { z } { \left( z ^ { 2 } + 1 \right) ( z - 1 ) }$ and the closed counter-clockwise path of integration $C$, which consists of the upper half circle $C _ { 1 }$ with radius $R \left( z = R e ^ { i \theta } , 0 \leq \theta \leq \pi \right)$, the line segment $C _ { 2 }$ on the real axis $( z = x , - R \leq x \leq 1 - r )$, the lower half circle $C _ { 3 }$ with its center at $z = 1 \left( z = 1 - r e ^ { i \theta } , 0 \leq \theta \leq \pi \right)$, and the line segment $C _ { 4 }$ on the real axis $( z = x , 1 + r \leq x \leq R )$. Here, $e$ denotes the base of the natural logarithm, and let $r > 0 , r \neq \sqrt { 2 }$ and $R > 1 + r$.

Answer the following questions.

\begin{enumerate}
  \item Calculate the integral $\int _ { C } g ( z ) d z$.
  \item Using the result from Question II.1, calculate the following value
$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { - \infty } ^ { 1 - \varepsilon } g ( x ) d x + \int _ { 1 + \varepsilon } ^ { \infty } g ( x ) d x \right]$$
\end{enumerate}