Problem Analysis #2

[THOUGHT] We need the contour integral along the upper semicircle $C$ of radius $5/2$ from $z=5/2$ to $z=-5/2$. Parametrize $z=\frac{5}{2}e^{i\theta}$ with $\theta$ increasing from $0$ to $\pi$. Then $dz=\frac{5}{2}ie^{i\theta}\,d\theta$, and $f(z)=z^2=\left(\frac{5}{2}\right)^2e^{2i\theta}$. Compute

Evaluate $\int_0^\pi e^{3i\theta}\,d\theta=\frac{e^{3i\pi}-1}{3i}=\frac{-1-1}{3i}=-\frac{2}{3i}=\frac{2i}{3}$. So the integral is

Thus $p=125$, $q=12$, and $p+q=237$. [/THOUGHT] [PYTHON] from sympy import symbols, integrate, exp, pi

theta = symbols("theta", real=True) R = 5/2 integrand = (R**2) * exp(2thetaI) * (R * exp(theta*I)*I) val = integrate(integrand, (theta, 0, pi)) print(val) [/PYTHON]

[THOUGHT] Using the provided Python output as ground truth, the integral evaluates to $-\frac{125}{12}$, so $p+q=237$. \boxed{237}