Problem Analysis #1

[THOUGHT] The integral is evaluated using the residue theorem. The pole is at $z_0 = \frac{205}{69} i$ in the upper half-plane. Closing the contour there gives $I = 2\pi i$ times the residue of $\frac{P(z) e^{i z}}{(z - z_0)^8}$ at $z_0$, which is $\frac{1}{7!} \frac{d^7}{dz^7} [P(z) e^{i z}] \big|_{z=z_0}$.

Use SymPy to define P(z), compute Q(z) = P(z) * exp(Iz), differentiate 7 times, evaluate at z_0, divide by 7!, multiply by 2pi*I, then extract real and imaginary parts numerically to compute floor(10a) + floor(b). [/THOUGHT] [PYTHON] from sympy import *

z = symbols('z') I_sym = I pole = Rational(205,69) * I_sym

P = (z7 -14*I_sym * z6 + (85 -60*I_sym)z**5 + (-240 -210I_sym)z**4 + (420 -840I_sym)z**3 + (-560 -1680I_sym)z**2 + (1344 -2688I_sym)*z + 3840)

Q = P * exp(I_sym * z) d7Q = diff(Q, z, 7) res = d7Q.subs(z, pole) / factorial(7) I_int = 2 * pi * I_sym * res

a = re(I_int).evalf(30) b = im(I_int).evalf(30)

print('a:', a) print('b:', b) print('floor(10a):', floor(10a)) print('floor(b):', floor(b)) print('sum:', floor(10a) + floor(b)) [/PYTHON]

Explanation

The reasoning follows the residue theorem for contour integration. The integral ( I = $\int_{-\infty}^{\infty}$ \frac{P(x)

[THOUGHT] The Python output gives floor(10a) = 7 and floor(b) = 2, so their sum is 9. [/THOUGHT] $\boxed{9}$

Explanation

The integral evaluation relies on the residue theorem, as the integrand has a pole of order 8 at $z_0 = \frac{205}{69} i$ in the upper half-plane. For $\int_{-\infty}^{\infty} f(x) \, dx$ with $e^{i x}$, closing the contour in the upper half-plane yields $I = 2\pi i \times \text{Res}(f, z_0)$.

The residue for a pole of order 8 is $\text{Res} = \frac{1}{7!} \frac{d^7}{dz^7} \left[ P(z) e^{i z} \right] \bigg|_{z=z_0}$.

SymPy computes this exactly:

• Defines $P(z)$ as the given polynomial.
• Forms $Q(z) = P(z) e^{i z}$.
• Differentiates $Q(z)$ seven times.
• Evaluates at $z_0$, divides by $7!$, and multiplies by $2\pi i$.

Numerical evaluation gives $I \approx 0.7208 + 2.2479i$, so $a \approx 0.7208$, $b \approx 2.2479$. Thus, $\lfloor 10a \rfloor = 7$, $\lfloor b \rfloor = 2$, and their sum is 9.