\textbf{Problem 3}

In the following, $z$ denotes a complex number and $i$ is the imaginary unit. The real part and the imaginary part of $z$ are denoted by $\operatorname { Re } ( z )$ and $\operatorname { Im } ( z )$, respectively.

\textbf{I.} Answer the following questions.
\begin{enumerate}
  \item Give the solutions of $z ^ { 5 } = 1$ in polar form. Plot the solutions on the complex plane.
  \item The mapping $f$ is defined by $f : z \mapsto f ( z ) = \exp ( i z )$. Plot the image of the region $D = \{ z : \operatorname { Re } ( z ) \geq 0, 1 \geq \operatorname { Im } ( z ) \geq 0 \}$ under $f$ on the complex plane.
  \item Find the residue of the function $z ^ { 2 } \exp \left( \frac { 1 } { z } \right)$ at $z = 0$.
\end{enumerate}

\textbf{II.} Consider the complex function: $f ( z ) = \frac { ( \log z ) ^ { 2 } } { ( z + a ) ^ { 2 } }$, where $a$ is a positive real number. The closed path $C$ shown in Figure 3.1 is defined by $C = C _ { + } + C _ { R } + C _ { - } + C _ { r }$, where $R > a > r > 0$. Here, $\log z$ takes the principal value on the path $C _ { + }$. Answer the following questions.
\begin{enumerate}
  \item Apply the residue theorem to calculate the contour integral $\oint _ { C } f ( z ) \, d z$.
  \item Use the result of Question II.1 to calculate the integral: $\int _ { 0 } ^ { \infty } \frac { \log x } { ( x + a ) ^ { 2 } } \, d x$.
\end{enumerate}