In the following, $z = x + i y$ and $w = u + i v$ represent complex numbers, where $i$ is the imaginary unit, and $x , y , u$ and $v$ are real numbers.
I. In order to evaluate the integral
$$I = \int _ { - \infty } ^ { \infty } \frac { 1 } { x ^ { 6 } + 1 } \mathrm {~d} x$$
consider the complex function $f ( z ) = \frac { 1 } { z ^ { 6 } + 1 }$.
1. Find all singularities of $f ( z )$. 2. By applying the residue theorem, determine the value of $I$.
II. Two domains, which are banded and semi-infinite on the complex $z$-plane, are defined as:
$$D _ { 1 } = \left\{ x + i y \left\lvert \, 0 \leq x \leq \frac { \pi } { 2 } \right. , y \geq 0 \right\} \text { and } D _ { 2 } = \left\{ x + i y \mid x \geq 0 , - \frac { \pi } { 2 } \leq y \leq 0 \right\}$$
Consider the mapping $w = g ( z )$ from the complex $z$-plane to the complex $w$-plane with an analytic function $g ( z )$. Let $D _ { 1 } ^ { * }$ and $D _ { 2 } ^ { * }$ be the images of $D _ { 1 }$ and $D _ { 2 }$, respectively, through this mapping.
1. When $g ( z ) = \cos z$, sketch the domain $D _ { 1 } ^ { * }$. 2. When $g ( z ) = ( \cosh z ) ^ { 3 }$, sketch the domain $D _ { 2 } ^ { * }$.