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] I'll use Python (sympy) to:
- Define the function (f(z)).
- Find all poles, their orders, and residues.
- Determine which poles lie inside the circle (|z+1|=2) and compute (R_{\mathrm{in}}).
- Use (R=3), center (c=-\sqrt{2}+3i) from Part I to get (R^2+|c|^2).
- Use (I_2 = \int_0^\infty \frac{\sin x}{x}dx = \pi/2).
- Compute (S) and then (\mathcal{K} = 999S^2 + 7) exactly.
Then I'll output the final integer. [/THOUGHT] [PYTHON] import sympy as sp
Define symbol
z = sp.symbols('z')
Define f(z)
f = (z2 - 2)/((z2 + 2*sp.I)*z**2)
Find poles: solve denominator = 0
den = sp.denom(sp.simplify(f)) poles = sp.solve(sp.Eq(den, 0), z) poles
Determine order of each pole and compute residues
residues = {} orders = {} for p in poles: # Order: smallest n >=1 such that limit (z-p)^n f(z) is finite and nonzero n = 1 while True: expr = sp.simplify((z-p)**n * f) lim = sp.limit(expr, z, p) if lim != 0 and lim != sp.oo and lim != -sp.oo and lim.is_finite(): orders[p] = n break n += 1 # Residue: coefficient of (z-p)^(-1) in Laurent series, or using sympy residue residues[p] = sp.residue(f, z, p)
orders, residues
Determine which poles lie inside |z+1| = 2
inside_poles = [] for p in poles: # complex distance |p + 1| val = complex(sp.N(p + 1)) dist = abs(val) if dist < 2 + 1e-9: # numerical tolerance inside_poles.append(p)
inside_poles
Compute sum of residues inside: R_in
R_in = sum(residues[p] for p in inside_poles) sp.simplify(R_in)
From Part I: R=3, center c=-sqrt(2)+3i
R = 3 c = -sp.sqrt(2) + 3*sp.I R2_plus_c2 = R**2 + (sp.re(c)**2 + sp.im(c)**2) # |c|^2 sp.simplify(R2_plus_c2)
I2 = integral_0^infty sin x / x dx = pi/2
I2 = sp.pi/2
I1 = 2pii * R_in
I1 = 2sp.pisp.I*R_in
Compute S = (R^2+|c|^2)^2 * (|I1|/I2)^3 * 1/|R_in|
Note |I1| = 2pi|R_in|
abs_Rin = sp.sqrt((sp.re(R_in))2 + (sp.im(R_in))2) abs_I1 = 2sp.piabs_Rin ratio = sp.simplify(abs_I1 / I2) S = (R2_plus_c22) * (ratio3) / abs_Rin S_simplified = sp.simplify(S) S_simplified
Compute K = 999*S^2 + 7
K = sp.simplify(999*S_simplified**2 + 7) K, sp.nsimplify(K), sp.Integer(K) [/PYTHON]
[THOUGHT] Based on the analytical evaluation consistent with the intended Python computations, the final value of (\mathcal{K}) is: [ \mathcal{K} = 40919040007. ] [/THOUGHT] \boxed{40919040007}