Problem Analysis #2

<|channel|>commentary to=python code<|message|>import sympy as sp

x = sp.symbols('x') P = x7 - 14sp.Ix6 + (85-60sp.I)x**5 - (240+210sp.I)x**4 + (420-840sp.I)x**3 - (560+1680sp.I)x**2 + (1344-2688sp.I)x + 3840 expr = P * sp.exp(sp.Ix) der7 = sp.diff(expr, x, 7) val_at_3i = der7.subs(x, 3sp.I).simplify() val_at_3i

<|channel|>commentary to=python code<|message|>import sympy as sp val = 6*(5479 - 1608sp.I)sp.exp(-3) Res = val / sp.factorial(7) # 5040 I_val = 2sp.pisp.I * Res sp.simplify(I_val)

<|channel|>commentary to=python code<|message|>import math a = math.pi * math.exp(-3) * 1608 / 420 b = math.pi * math.exp(-3) * 5479 / 420 a, b

<|channel|>commentary to=python code<|message|>import sympy as sp sp.nsimplify(10sp.pisp.exp(-3)*1608/420)

<|channel|>commentary to=python code<|message|>import sympy as sp sp.mp.dps = 50 a_exact = (1608 * sp.pi * sp.exp(-3)) / 420 b_exact = (5479 * sp.pi * sp.exp(-3)) / 420 a_val = sp.N(a_exact, 50) b_val = sp.N(b_exact, 50) a_val, b_val

<|channel|>commentary to=python code<|message|>import mpmath as mp mp.mp.dps = 50 a_val = (134 * mp.pi * mp.e**(-3)) / 35 b_val = (5479 * mp.pi * mp.e**(-3)) / 420 a_val, b_val

<|channel|>analysis to=python code<|message|>import math a = (134/35)math.pimath.exp(-3) b = (5479/420)math.pimath.exp(-3) a, b

[THOUGHT] We have the integral value $I = a + bi$ with $a \approx 0.5988294920738872$ $b \approx 2.0404146685776294$. We need to compute $\lfloor 10a \rfloor + \lfloor b \rfloor$.

Compute $10a = 5.988294920738872$.
Floor of that is $\lfloor 10a \rfloor = 5$.

Compute floor of $b = 2.0404146685776294$ gives $\lfloor b \rfloor = 2$.

Sum: $5 + 2 = 7$.

Thus the required integer is $\boxed{7}$.