Answer the following questions. Here, for any complex value $z$, $\bar{z}$ is the complex conjugate of $z$, $\arg z$ is the argument of $z$, $|z|$ is the absolute value of $z$, and $i$ is the imaginary unit.\subsection*{I.}Sketch the region of $z$ on the complex plane that satisfies$z\bar{z} + \sqrt{2}(z + \bar{z}) + 3i(z - \bar{z}) + 2 \leq 0.$\subsection*{II.}Consider the complex-valued function$f(z) = \frac{z^2 - 2}{(z^2 + 2i)z^2}.$

• [(1)] Find all the poles of $f(z)$ as well as the orders and residues at the poles.

• [(2)] By applying the residue theorem, find the value of the integral $I_1 = \oint_C f(z)\,dz,$ where the integration path $C$ is the circle in the counterclockwise direction given by $|z + 1| = 2$. \subsection*{III.}

• [(1)] Let $g(z)$ be a complex-valued function satisfying $\lim_{|z|\to\infty} g(z) = 0$ for 0 $$\leq$$ \arg z $$\leq$$ $$\pi$$$. Let C_R$be the semicircle, with radius$R, in the upper half of the complex plane with center at the origin. Show that $$ \lim_{R\to\infty} \int_{C_R} e^{iaz} g(z)\,dz = 0, $$ where a$ is a positive real number.

• [(2)] Find the value of the integral $I_2 = \int_0^{\infty} \frac{\sin x}{x}\,dx.$ \subsection*{IV. Final quantity}In I, the region for $z$ is a disk. Let $R$ be its radius and $c$ be its center. In II(2), let $R_{\mathrm{in}}$ be the sum of the residues of $f(z)$ over all poles lying inside the circle $|z + 1| = 2$. In II(2) and III(2), let $I_1$ and $I_2$ be the values of the corresponding integrals.Define$S := (R^2 + |c|^2)^2 \left(\frac{|I_1|}{I_2}\right)^3 \cdot \frac{1}{|R_{\mathrm{in}}|},\qquad\mathcal{K} := 999\,S^2 + 7.$Compute the integer $\mathcal{K}$.\