To solve the problem, I will perform the following steps using Python:

1. Analyze the Region (Part I): Verify the center $c$ and radius $R$ of the disk defined by the inequality $z\bar{z} + \sqrt{2}(z + \bar{z}) + 3i(z - \bar{z}) + 2 \leq 0$. I will expand the inequality in terms of $x$ and $y$ (where $z = x + iy$) to identify the circle equation $(x-x_0)^2 + (y-y_0)^2 \leq R^2$.

2. Analyze the Function $f(z)$ (Part II): Define the function $f(z) = \frac{z^2 - 2}{(z^2 + 2i)z^2}$. Find the poles of $f(z)$. Calculate the residues of $f(z)$ at these poles. Identify which poles lie inside the contour $C: |z + 1| = 2$. Calculate $R_{\mathrm{in}}$, the sum of residues inside $C$. Calculate $I_1 = \oint_C f(z)\,dz = 2\pi i \cdot R_{\mathrm{in}}$.

3. Analyze the Integral $I_2$ (Part III): Verify the value of $I_2 = \int_0^{\infty} \frac{\sin x}{x}\,dx$.

4. Compute Final Quantity (Part IV): Calculate $S = (R^2 + |c|^2)^2 \left(\frac{|I_1|}{I_2}\right)^3 \cdot \frac{1}{|R_{\mathrm{in}}|}$. Calculate $\mathcal{K} = 999\,S^2 + 7$.

[PYTHON] import sympy from sympy import symbols, I, sqrt, pi, Abs, re, im, integrate, sin, oo, residue

Part I: Region

x, y = symbols('x y', real=True) z = x + Iy z_bar = x - Iy inequality_expr = zz_bar + sqrt(2)(z + z_bar) + 3I(z - z_bar) + 2

Expand and simplify to find center and radius

expanded = inequality_expr.expand()

The form is x^2 + y^2 + Ax + By + C <= 0

Center is (-A/2, -B/2), Radius is sqrt((A/2)^2 + (B/2)^2 - C)

coeff_x = expanded.coeff(x) coeff_y = expanded.coeff(y) constant_term = expanded.subs({x:0, y:0})

Coefficients of x^2 and y^2 should be 1

A = coeff_x B = coeff_y C = constant_term

center_x = -A/2 center_y = -B/2 radius_sq = center_x2 + center_y2 - C radius = sqrt(radius_sq) c_complex = center_x + I*center_y

print(f"Center c: {c_complex}") print(f"Radius R: {radius}") print(f"|c|^2: {Abs(c_complex)2}") print(f"R^2: {radius2}")

Part II: Function f(z)

z_sym = symbols('z') f = (z_sym2 - 2) / ((z_sym2 + 2*I) * z_sym**2)

Poles

denom = (z_sym2 + 2*I) * z_sym2 poles = sympy.solve(denom, z_sym) print(f"Poles: {poles}")

Residues and checking if inside |z+1| = 2

Center of contour is -1, radius 2.

contour_center = -1 contour_radius = 2

R_in = 0 print("Residues:") for p in poles: res = residue(f, z_sym, p) print(f"Pole {p}: Residue = {res}")

# Check if inside contour
dist = Abs(p - contour_center)
if dist < contour_radius:
    print(f"  -> Inside contour (|z+1| < 2)")
    R_in += res
else:
    print(f"  -> Outside contour")

print(f"Sum of residues inside (R_in): {R_in}") I1 = 2 * pi * I * R_in print(f"I1: {I1}") print(f"|I1|: {Abs(I1)}")

Part III: Integral I2

x_dummy = symbols('x') I2 = integrate(sin(x_dummy)/x_dummy, (x_dummy, 0, oo)) print(f"I2: {I2}")

Part IV: Final Calculation

S = (R^2 + |c|^2)^2 * (|I1|/I2)^3 * (1/|R_in|)

Note: R from Part I is 'radius', c is 'c_complex'

term1 = (radius**2 + Abs(c_complex)**2)**2 term2 = (Abs(I1) / I2)**3 term3 = 1 / Abs(R_in)

S = term1 * term2 * term3 print(f"S: {S}")

K = 999 * S**2 + 7 print(f"K: {K}") [/PYTHON]

[THOUGHT] The problem asks for the integer $\mathcal{K}$ defined by $\mathcal{K} := 999\,S^2 + 7$, where $S$ is a quantity derived from several complex analysis problems.

1. Region Parameters ($R, c$): The inequality is $z\bar{z} + \sqrt{2}(z + \bar{z}) + 3i(z - \bar{z}) + 2 \leq 0$. Substituting $z = x+iy$, this becomes $x^2 + y^2 + 2\sqrt{2}x - 6y + 2 \leq 0$. Completing the square: $(x+\sqrt{2})^2 + (y-3)^2 \leq 3^2$. Thus, the region is a disk with center $c = -\sqrt{2} + 3i$ and radius $R = 3$. We compute $|c|^2 = (-\sqrt{2})^2 + 3^2 = 2 + 9 = 11$ and $R^2 = 9$. The term $(R^2 + |c|^2)^2 = (9 + 11)^2 = 20^2 = 400$.

2. Integral $I_1$ and Residues ($R_{\mathrm{in}}$): The function is $f(z) = \frac{z^2 - 2}{z^2(z^2 + 2i)}$. The poles are at $z=0$ (double) and roots of $z^2 = -2i$ (simple). The roots of $z^2 = -2i$ are $z_1 = -1+i$ and $z_2 = 1-i$. The integration contour is $|z+1| = 2$ (center $-1$, radius $2$).

   • $z=0$: distance to center is $|0 - (-1)| = 1 < 2$ (Inside). Residue is $0$.
   • $z_1 = -1+i$: distance is $|-1+i - (-1)| = 1 < 2$ (Inside). Residue is $-1/2$.
   • $z_2 = 1-i$: distance is $|1-i - (-1)| = |2-i| = \sqrt{5} > 2$ (Outside). Sum of residues inside, $R_{\mathrm{in}} = 0 + (-1/2) = -1/2$. $|R_{\mathrm{in}}| = 1/2$. The integral $I_1 = 2\pi i R_{\mathrm{in}} = 2\pi i (-1/2) = -\pi i$. $|I_1| = \pi$.

3. Integral $I_2$: $I_2 = \int_0^{\infty} \frac{\sin x}{x}\,dx = \frac{\pi}{2}$.

4. Calculate $S$: $S = (R^2 + |c|^2)^2 \left(\frac{|I_1|}{I_2}\right)^3 \frac{1}{|R_{\mathrm{in}}|}$. $S = (400) \left(\frac{\pi}{\pi/2}\right)^3 \frac{1}{1/2}$. $S = 400 \cdot (2)^3 \cdot 2 = 400 \cdot 8 \cdot 2 = 6400$.

5. Calculate $\mathcal{K}$: $\mathcal{K} = 999 S^2 + 7$. $S^2 = 6400^2 = 40,960,000$. $\mathcal{K} = 999 \cdot 40,960,000 + 7$. $\mathcal{K} = (1000 - 1) \cdot 40,960,000 + 7 = 40,960,000,000 - 40,960,000 + 7$. $\mathcal{K} = 40,919,040,000 + 7 = 40,919,040,007$.

[/THOUGHT]

\boxed{40919040007}