[THOUGHT] I need to evaluate the integral $I = \int_{-\infty}^{\infty} \frac{P(x) e^{ix}}{(x - 3i)^8} dx$

where $P(x)$ is a degree 7 polynomial.

Since the integrand has a pole at $x = 3i$ (which is in the upper half-plane, since $\text{Im}(3i) = 3 > 0$), and we have $e^{ix}$ in the numerator, I'll use contour integration.

For $e^{ix}$ with $x$ real, when we close the contour in the upper half-plane (since $e^{iz}$ decays as $\text{Im}(z) \to +\infty$), the integral over the semicircular arc vanishes as the radius goes to infinity (by Jordan's lemma, provided the rest of the integrand decays appropriately).

The pole at $z = 3i$ is of order 8, so by the residue theorem: $I = 2\pi i \cdot \text{Res}_{z=3i} \left[ \frac{P(z) e^{iz}}{(z - 3i)^8} \right]$

For a pole of order 8, the residue is: $\text{Res}_{z=3i} = \frac{1}{7!} \lim_{z \to 3i} \frac{d^7}{dz^7} \left[ P(z) e^{iz} \right]$

Let me compute this using Python. [/THOUGHT] [PYTHON] import sympy as sp from sympy import I, exp, factorial, pi, diff, simplify, re, im, floor

z = sp.Symbol('z')

Define P(z)

P = (z7 - 14Iz6 + (85 - 60*I)z**5 - (240 + 210I)z**4 + (420 - 840I)z**3 - (560 + 1680I)z**2 + (1344 - 2688I)*z + 3840)

The function whose 7th derivative we need

f = P * exp(I*z)

Compute the 7th derivative

f_7th = diff(f, z, 7)

Evaluate at z = 3i

f_7th_at_3i = f_7th.subs(z, 3*I)

Simplify

f_7th_at_3i = simplify(f_7th_at_3i)

print("7th derivative at z=3i:", f_7th_at_3i)

Residue

residue = f_7th_at_3i / factorial(7) residue = simplify(residue) print("Residue:", residue)

Integral I = 2pii * residue

I_val = 2 * pi * I * residue I_val = simplify(I_val) print("I =", I_val)

Get real and imaginary parts

a = re(I_val) b = im(I_val) print("a =", a) print("b =", b)

Compute floor(10a) + floor(b)

result = floor(10*a) + floor(b) print("floor(10a) + floor(b) =", result) [/PYTHON]

[THOUGHT] The calculation shows that:

• $a = \frac{134\pi e^{-3}}{35}$
• $b = \frac{5479\pi e^{-3}}{420}$

And the final result is: $\lfloor 10a \rfloor + \lfloor b \rfloor = 7$ [/THOUGHT]

\boxed{7}