Answer the following questions. Here, for any complex value , is the complex conjugate of , is the argument of , is the absolute value of , and is the imaginary unit.\subsection*{I.}Sketch the region of on the complex plane that satisfies\subsection*{II.}Consider the complex-valued function
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[(1)] Find all the poles of as well as the orders and residues at the poles.
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[(2)] By applying the residue theorem, find the value of the integral where the integration path is the circle in the counterclockwise direction given by . \subsection*{III.}
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[(1)] Let be a complex-valued function satisfying for 0 $$\leq$$ \arg z $$\leq$$ $$\pi$$$. Let C_RR, 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.
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[(2)] Find the value of the integral \subsection*{IV. Final quantity}In I, the region for is a disk. Let be its radius and be its center. In II(2), let be the sum of the residues of over all poles lying inside the circle . In II(2) and III(2), let and be the values of the corresponding integrals.DefineCompute the integer .\
[THOUGHT] To solve the problem, we break it down into parts as follows:
I. Region of z: The inequality ( z\bar{z} + \sqrt{2}(z + \bar{z}) + 3i(z - \bar{z}) + 2 \leq 0 ) is transformed into a circle equation by substituting ( z = x + iy ). This results in the circle ( (x + \sqrt{2})^2 + (y - 3)^2 \leq 9 ), giving a radius ( R = 3 ) and center ( c = -\sqrt{2} + 3i ). The modulus of the center is ( |c| = \sqrt{11} ).
II(1). Poles and Residues of ( f(z) ): The function ( f(z) = \frac{z^2 - 2}{(z^2 + 2i)z^2} ) has poles at ( z = 0 ) (order 2), ( z = 1 - i ), and ( z = -1 + i ) (both order 1). The residues are:
- ( \text{Res}(f, 0) = 0 )
- ( \text{Res}(f, 1 - i) = \frac{1}{2} )
- ( \text{Res}(f, -1 + i) = -\frac{1}{2} )
II(2). Integral ( I_1 ): The contour ( |z + 1| = 2 ) includes poles at ( z = 0 ) and ( z = -1 + i ). The sum of residues inside is ( 0 + (-\frac{1}{2}) = -\frac{1}{2} ). Thus, ( I_1 = 2\pi i \times (-\frac{1}{2}) = -\pi i ), giving ( |I_1| = \pi ).
III(2). Integral ( I_2 ): The integral ( \int_0^\infty \frac{\sin x}{x} , dx ) is known to be ( \frac{\pi}{2} ).
Final Calculation: Using the values:
- ( R = 3 ), ( |c| = \sqrt{11} )
- ( |I_1| = \pi ), ( I_2 = \frac{\pi}{2} )
- ( |R_{\text{in}}| = \frac{1}{2} )
We compute: [ S = (R^2 + |c|^2)^2 \left( \frac{|I_1|}{I_2} \right)^3 \cdot \frac{1}{|R_{\text{in}}|} = (9 + 11)^2 \cdot 2^3 \cdot 2 = 400 \cdot 8 \cdot 2 = 6400 ] [ \mathcal{K} = 999 \cdot S^2 + 7 = 999 \cdot 40960000 + 7 = 40919040007 ]
[PYTHON]
Calculations for the final answer
R = 3 c_modulus = (20.5)2 + 32 R_squared = R2 c_squared = c_modulus sum_R_c = R_squared + c_squared S_squared = sum_R_c2 I1_abs = 3.141592653589793 # pi I2 = 3.141592653589793 / 2 # pi/2 ratio = I1_abs / I2 ratio_cubed = ratio3 R_in_abs = 0.5 S = S_squared * ratio_cubed * (1 / R_in_abs) K = 999 * S**2 + 7 print(int(K)) [/PYTHON]
[THOUGHT] The Python output confirms the calculation of (\mathcal{K} = 40919040007). [/THOUGHT] \boxed{40919040007}