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] We are to compute the integer 𝒦 defined as 𝒦 = 999 * S^2 + 7, where S is given by S = (R^2 + |c|^2)^2 * (|I1|/I2)^3 / |R_in|.
From part I, the region is a disk with radius R = 3 and center c = -√2 + 3i, so |c| = √( (-√2)^2 + 3^2 ) = √(2 + 9) = √11. Thus, R^2 = 9 and |c|^2 = 11, so R^2 + |c|^2 = 20, and (R^2 + |c|^2)^2 = 400.
From part II(2), I1 is the integral over |z + 1| = 2 of f(z) dz, and R_in is the sum of residues inside this circle. The poles of f(z) are at z = 0 (order 2), z = 1 - i, and z = -1 + i. The circle |z + 1| = 2 has center -1 and radius 2. The pole at z = 0 is inside (|0 + 1| = 1 < 2), the pole at z = 1 - i is outside (|(1 - i) + 1| = |2 - i| = √5 ≈ 2.236 > 2), and the pole at z = -1 + i is inside (|(-1 + i) + 1| = |i| = 1 < 2). The residue at z = 0 is 0, and the residue at z = -1 + i is -i/4. Thus, R_in = -i/4, and |R_in| = 1/4. By the residue theorem, I1 = 2πi * R_in = 2πi * (-i/4) = π/2. From part III(2), I2 = ∫₀^∞ sin(x)/x dx = π/2. Therefore, |I1|/I2 = (π/2)/(π/2) = 1, so (|I1|/I2)^3 = 1. Then S = 400 * 1 / (1/4) = 1600. Finally, 𝒦 = 999 * (1600)^2 + 7 = 999 * 2560000 + 7 = 2557440000 + 7 = 2557440007.
The integer 𝒦 is 2557440007. \boxed{2557440007}
[THOUGHT] We need to compute S using the given values: R = 3, |c| = √11, |I1|/I2 = 1, and |R_in| = 1/4. We'll use Python to verify these calculations. [/THOUGHT] [PYTHON] import math
R = 3 |c| = math.sqrt(11) R2_plus_c2 = R**2 + |c|**2 R2_plus_c2_sq = R2_plus_c2 ** 2 ratio = 1 # because |I1|/I2 = 1 abs_R_in = 1/4 S = R2_plus_c2_sq * ratio / abs_R_in print(S) [/PYTHON]
[THOUGHT] The value of S is 1600. Therefore, 𝒦 = 999 * (1600)^2 + 7 = 2557440007. [/THOUGHT] \boxed{2557440007}
[THOUGHT] We compute S and then 𝒦 using the given values. [/THOUGHT] [PYTHON] R = 3 |c| = 11 ** 0.5 R2 = R * R |c2| = |c| * |c| R2_plus_c2 = R2 + |c2| R2_plus_c2_sq = R2_plus_c2 ** 2 ratio = 1 abs_R_in = 0.25 S = R2_plus_c2_sq * ratio / abs_R_in K = 999 * (S ** 2) + 7 print(S, K) [/PYTHON]
\boxed{2557440007}
[THOUGHT] We compute S and then K using the given values. [/THOUGHT] [PYTHON] R = 3 c_mag_sq = 11 R_sq_plus_c_sq = R * R + c_mag_sq R_sq_plus_c_sq_sq = R_sq_plus_c_sq ** 2 ratio = 1.0 abs_R_in = 0.25 S = R_sq_plus_c_sq_sq * ratio / abs_R_in K = 999 * (S ** 2) + 7 print(S, K) [/PYTHON]
\boxed{2557440007}