Complex Variable Analysis Midterm Exam

1. [4 Points] Define $f(z)$ in the forms $f(z) = u(x, y) + iv(x, y)$ under Cartesian coordinates!

   1. $f(z) = z^3 - z$
   2. $f(z) = \displaystyle\frac{1}{i - z}$
   3. $f(z) = \overline{e^{z^2}}$

2. [9 Points] Find the two roots of complex number $c_k$ of $z_0$ where $k=\{0, 1\}$ with complex number $(-4 - 4\sqrt{3}i)^{1/2}$ and plot into a graph! Define clearly $\sqrt[n]{r_0}, \operatorname{Arg}z_0, z_0$ in polar form!