In the following, $z$ denotes a complex number, and $x$ and $\varepsilon$ denote real numbers. The imaginary unit is denoted by $i$.

I. Answer the following questions about the function $f _ { n } ( z ) = 1 / \left( z ^ { n } - 1 \right)$. Here, $n$ is an integer greater than or equal to 2.

\begin{enumerate}
  \item For the case that $n = 3$, find all singularities of $f _ { n } ( z )$.
  \item Calculate the residue value at a singularity $p _ { 0 }$ of $f _ { n } ( z )$ and give a simple expression of the residue in terms of $n$ and $p _ { 0 }$.
  \item For a contour $C$ given by the closed curve $| z | = 2$ and oriented in the counter-clockwise direction, evaluate the contour integral $\oint _ { C } f _ { n } ( z ) d z$.
\end{enumerate}

II. Obtain the following limit value:

$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { - \infty } ^ { 1 - \varepsilon } \frac { 1 } { x ^ { 3 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { 1 } { x ^ { 3 } - 1 } d x \right]$$

III. Obtain the following limit value:

$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { 0 } ^ { 1 - \varepsilon } \frac { \cos x } { x ^ { 4 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { \cos x } { x ^ { 4 } - 1 } d x \right]$$

IV. Obtain the following limit value:

$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { 0 } ^ { 1 - \varepsilon } \frac { \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) } { x ^ { 4 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) } { x ^ { 4 } - 1 } d x \right]$$